Curriculum Vitae Agostino Prastaro

Edition April 2014, 1-60

C U R R I C U L U M V I T A E

CURRICULUM VITAE

AGOSTINO PRASTARO

Fig. 1. Agostino Prastaro

UNIVERSITY OF ROMA LA SAPIENZA - ROMA - ITALY

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2 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

University address:Department of Methods and Mathematical Models for Applied Sciences(MEMOMAT)University of Roma La Sapienza, Via A.Scarpa, 16-00161 - Roma, Italy.Phone: +39-06-49766723; Fax: +39-06-4957647E-mails: Prastaro@dmmm.uniroma1.it; Agostino.Prastaro@uniroma1.it.Home page: http://www.dmmm.uniroma1.it/ agostino.prastaro/HOMEPAGEPRAS.htm

Fig. 2. University of Roma La Sapienza Pictures.

Home address:

Fig. 3. Porta Maggiore.

Via L’Aquila, 29-00176 - Roma - Italy. (Phone & Fax: +39-06-7023432 )

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 3

Acknowledgments

Fig. 4. Prastaro’s mathematicians poster 2013-2014.

“...A World Of Mathematicians...For A World With Mathematics... ”

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1. GENERAL INFORMATIONS

•Birth-place and birth-date: Civitavecchia (Roma - Italy), 01-11-1942.•Diploma di maturita scientifica: Trieste - Italy, July 1960.•Doctor degree in Theoretical Physics: University of Torino, Torino - Italy, July

1967.•Research grant in Theoretical Physics: University of Torino, Torino - Italy,

1967–1968.•Researcher in Mathematical Models and Processes of Materials: Montedison

Research Center, Ferrara - Italy, 1968–1975.• School on Tensors and Group Theory Applied to Crystallography - Brooklyn

Crystallographic Laboratory, Cambridge, UK, 1970.•Professor (incaricato) in Mathematical Physics, University of Lecce and Uni-

versity of Calabria, Cosenza - Italy, 1975–1976.•Professor (incaricato) in Mathematical Physics, University of Calabria, Cosenza

- Italy, 1976-1980.•Professor (associate) in Mathematical Physics, University of Calabria, Cosenza

- Italy 1980-1990.•Professor (associate) in Mathematical Physics, University of Roma La Sapienza,

Roma - Italy, 1990–2013.•Courses taught: Rational Mechanics, Institutions of Mathematical Physics,

Mathematical Physics, Mathematical Analysis, Continuum Mechanics, DifferentialGeometry.

•Member in: Italian Mathematical Union (UMI), European Mathematical So-ciety (EMS), National Group of Mathematical Physics (GNFM/INDAM), Ameri-can Mathematical Society (AMS), International Federation of Nonlinear Analysts(IFNA), Mathematical Association of America (MAA), International MathematicalUnion (IMU).

•Reviewer for: Mathematical Reviews, Zentralblatt Mathematics and someother mathematical journals.

•Head of the research-team MIUR-Faculty of Engineering-University of RomaLa Sapienza: “Geometry of PDEs and Applications”.

•Member in National Project in Algebraic Topology and Differential Geometry(1986–2004).

•Member of PhD Committee: “Models and Mathematical Methods for Technol-ogy and Sociology”, University of Roma La Sapienza (1999–2003).

• Fields of Research: Geometry of PDE’s (Differential Geometry, Algebraic Ge-ometry and Algebraic Topology); (Co)bordism in PDE’s and quantum PDE’s; Ge-ometry of PDE’s in Continuum Mechanics; Geometry of PDE’s in Quantum FieldTheory and Quantum Supergravity; Geometry of PDE’s in Mathematical Physics.

•Organizer of many international conferences. The last ones from 2000: WCNA-2000 Organized Session Functional Equations Geometric Analysis, Catania, Italy(2000); Joint Meeting AMS-UMI Special Session Advances in Differential Geome-try of PDE’s and Applications, Pisa, Italy (2002); ICM-2006 Satellite ConferenceAdvances in PDE’s Geometry, Madrid, Spain (2006); WCNA 2008 - OrganizedSession: Advances Geometric Analysis, Orlando, FL - USA (2008).

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• (Some) Invited Speaker or Partecipant and Visiting Professor: Departmentof Mathematics, Univ. P. Sabatier, Toulouse, France (1986); I.H.E.S., Bures-sur-Yvette, France (1988); Department of Mathematics, University of California,Berkely, USA, (1990); University of Torun, Poland (1996); University of Budapest,Hungary (1996); Technical University of Athens, Greece (1996); Third World Con-gress of Nonlinear Analysts (WCNA2000), Catania, Italy (2000); University of De-brecen, Hungary (2000); Department of Mathematics, University of Opava, CzechRepublic (2000); University of Bilbao, Spain (2000); Department of Mathematics,University of Atlanta, USA (2003); Department of Mathematics, Florida Instituteof Technology, Melbourne, USA (2005); ICM-2006 and Facultad de Matematicas,Universidad Complutense de Madrid, Madrid - Spain (2006); Department of Math-ematics, Morehouse College, Atlanta, USA (2007); WCNA 2008 - Orlando, FL -USA (2008).

• Award Sapienza Ricerca 2010. (Awarded publications [58, 59, 60].)•Member in IFNA Board of Global Advisors World Congress of IFNA, 2012- Athens, Greece, June 25-July 1, 2012.•Membership in Roma Sapienza Foundation (from 2014 ).•Publications: 96 scientific publications including 3 monographs as author,

3 as editor and coauthor and 2 patents. (See pages 7, 21 and 27.)[Metric informations from Googlescholar: ”author:Prastaro author:A.”: (i) Works-

number: 116; (ii) Citations-number: 912; (iii) H-index: 17.]1

1Please visit Wikipedia for criticism about H-index.

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Fig. 5. School of Engineering, University of Roma La Sapienza.

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 7

2. SUMMARY OF PRINCIPAL RESULTS

• [1]. This work is inserted in the problematic of the Regge’s trajectories instrong interactions. The paper shows a relation between Regge’s trajectories of s-channel with the ones in the t-channel that allows us to obtain general behavioursof diffusion amplitudes.

• [2, 3, 4, 5, 95, 96]. These are mathematical models for elongational flows,extrusion and flash-spinning of polymers. These models allow us to obtain optimumindustrial conditions of manufacturing.

• [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25]. These worksare devoted to build new mathematics that allow us to describe mechanics in anintrinsic and completely covariant way. This is obtained by introducing new spaces,derivative spaces, that are the natural universal spaces for differential calculus andPDEs. This point of view generalizes previous one introduced by Ehresmann andallows us to treat all the differential objects in algebraic way, i.e., as generalizedtensor-like objects. In this context a generalized form of the Noether theoremfor any PDE, even if of non-Lagrangian type, is obtained. Then this mathematicalmachinery is applied also to describe the physics of continuum media, gauge theoriesand supergravity.

• [22, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 50]. These works extend thegeometric formulation of PDEs, developed in the previous papers, in order to con-sider also singular solutions and to obtain a geometric formulation of quantizationof PDEs. This geometric approach gives an intrinsic formulation of the canonicalquantization of PDEs. All this theory is extended also to super PDEs, i.e., PDEsdefined in the category of supermanifolds. In particular, a geometric formulationof the canonical quantization of PDEs in supergravity is obtained.

• [37, 38, 39, 42, 45, 46, 51, 55, 56, 63, 70, 89]. These works present a generaltheory of integral (co)bordism in PDEs and develop general methods to calculatethe corresponding (co)bordism groups. Such results allow us to characterize alsoglobal solutions by means of algebraic topologic methods, and to recognize tunneleffects in such solutions. In particular, in [55] some improvements are given empha-sizing the role played by singular and weak-solutions and their relations with thebordism groups for smooth solutions. Tricomi equation, heat equation, Ricci-flowequation and d’Alembert equation on finite dimensional smooth manifolds are con-sidered there as interesting examples where to apply the general theory. VariationalPDE’s, and their global solutions characterizations, by means of integral bordismgroups, are given in [56].

The methods proposed are constructive, as they give us suitable tools to buildsolutions. In particular in [39, 42, 44, 45, 46, 63] the Navier-Stokes equation it isalso carefully considered and it is proved for such an equation existence of global(smooth) solutions. In this way it is solved an old well known problem in thetheory of PDEs, remained open for many decades. (See also [65] where some furtherimprovements are given.)2

2First results on the geometric theory of the Euler equation and the Navier-Stokes equationwere obtained in [15, 16, 17, 19].

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In [51] some applications of previous results on integral (co)bordism groups andcanonical quantization of PDEs are considered. In particular fully covariant canon-ical quantizations of elementary particles interacting with the gravitational field,also for massive neutrinos, are obtained. (See also [89].)

• [32, 33, 34, 40, 41]. Interesting applications of the geometric theory of PDE’s,and integral bordism group theory for PDE’s, as formulated by A.Prastaro, areconsidered to characterize local and global solutions of the d’Alembert equationand its generalizations on Rn. For such equations integral bordism groups areexplicitly calculated and global solutions built. Sophisticated solutions with tunneleffects are recognized.

• [37, 43, 44, 49, 54, 57, 58, 59, 60, 76, 78]. Here the new concepts of quan-tum manifolds and quantum PDEs are introduced, and for such structures it isformulated a new noncommutative mathematics, (differential geometry, algebraicgeometry, geometric theory of quantum PDEs and determination of their integral(co)bordism groups), that extends previous one for commutative PDEs. Applica-tions to many interesting quantum PDEs are given. In particular, in [44] the conceptof category of quantum quaternionic manifolds is introduced and for PDEs builtin such a category theorems of existence of local and global solutions are proved.Extensions to the quantum Navier-Stokes equation of previous results, obtained in[39, 42, 44, 45, 46, 63] for the Navier-Stokes equation, are given in [49]. Moreover,in [54, 57, 58, 59, 60, 69] the theory of quantum PDEs is extended to quantumsuper PDEs and applied to quantum Yang-Mills PDEs and quantum supergravityPDEs. These papers extend previous results of the integral (co)bordism groups the-ory for quantum PDE’s, to super quantum PDE’s. Furthermore, quantized (super)PDE’s are identified with quantum (super) PDE’s in a canonical way. A proof thatquantum (super) PDE’s are a more general approach in the description of quantumphysics, that goes beyond the point of view of the quantization of (super) classicaltheories (PDE’s), is given. Similarly to the commutative case, integral (co)bordismgroups for quantum (super) PDE’s, are related to weak, singular and smooth solu-tions, showing algebraic relations between such groups. Such a theory is applied tomany interesting PDE’s, and in particular, to quantum super Yang-Mills equations.Characterizations of important and very sophisticated solutions, as quantum tunneleffects and quantum black holes evaporation processes, are obtained by means ofintegral bordism groups.

The new algebraic topologic methods introduced allowed also to solve some im-portant problems in Mathematics. In particular, in [54, 60] the open problem toprove existence of global smooth solutions for quantum (super)Yang-Mills equationswith mass-gap is solved.

• [47, 48]. Here some interesting applications of the geometric theory of quantumPDE’s are considered. In fact, generalizations in the category of quantum manifoldsof previous results on d’Alembert equation [32, 33, 34], are given.

• [89, 91]. These monographs present, for the first time, a systematic formulationof the geometric theory of noncommutative PDE’s which is suitable enough tobe used for a mathematical description of quantum dynamics and quantum fieldtheory. A geometric theory of supersymmetric quantum PDE’s is also considered,in order to describe quantum supergravity. Covariant and canonical quantizationsof (super) PDE’s are shown to be founded on the geometric theory of PDE’s andto produce quantum (super) PDE’s by means of functors from the category of

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 9

commutative (super) PDE’s to the category of quantum (super) PDE’s. Globalproperties of solutions to (super) (commutative) PDE’s are obtained by means oftheir integral bordism groups. In particular the (quantum) Navier-Stokes problemand the (quantum)Yang-Mills problem are considered showing that their solutionscan be obtained in the framework of the integral bordism groups for such equations.

• [61, 62]. In these two papers the theory on the integral bordism groups forPDE’s, formulated by Prastaro, is applied to some interesting PDE’s. In particu-lar, in the first part a new general theory is developed that recognizes natural websstructures on PDE’s. These structures are important to solve (generalized) Cauchyproblems. Some applications interesting PDE’s of the Mathematical Physics arealso given (wave equation, Korteweg de Vries equation). In the second part applica-tions of the general theory of webs on PDE’s is applied to some problems concerningthe Riemannian geometry. In particular are considered generalized Yamabe prob-lems and the proof of the Poincare conjecture for 3-dimensional compact Riemann-ian manifolds, via the Ricci-flow equation. The new algebraic topologic methodsintroduced by A.Prastaro in the theory of PDE’s, allowed to give a new proof thatthe Poincare conjecture is true.

• [64, 65, 66, 67, 68]. In these papers an unified geometric theory of stabilityfor PDE’s and solutions of PDE’s, is formulated in the framework of Prastaro’sgeometric theory of partial differential equations, i.e., by using integral bordismgroups. Relations with the Ljapunov’s stability theory, and the classic Ulam prob-lem for approximate homomorphisms are stressed. (See also [52, 53].) Examplesfor some important PDE’s are given. In particular the theory is applied to newanisotropic MHD-PDE’s encoding dynamics for incompressible plasma fluids withnuclear energy production. Such PDE’s are related, on the ground of their integra-bility properties, to crystallographic groups, (extended crystal PDE’s).

• [69, 70, 71]. In these papers one further extends the theory of quantum (su-per) PDE’s, previously developed in [37, 43, 44, 49, 54, 57, 58, 59, 60, 76, 78] tosystematically adapt the algebraic topologic surgery techniques to quantum super-manifolds and to solutions of quantum super manifolds. In particular in [70], byusing also the Prastaro’s geometric theory of quantum super PDE’s, one formulatesand proves the quantum Poincare conjecture. This generalizes to the category ofquantum super PDE’s, the well known Poincare conjecture for 3-dimensional closedRiemannian manifolds. (See also [62] for a proof of this conjecture that is differ-ent by one by G. Perelman.) Furthermore in [71] one generalizes to the categoryof quantum supermanifolds QS , the variational calculus for variational problems,constrained by PDE’s, as formulated in [56]. In this way one formulates also a newquantum gravity theory that extends to the category QS the Einstein’s GeneralRelativity. There, and in [69] solutions of quantum gravity equations, interpretingnuclides and quantum black-holes, are considered in details. In particular quantumblack-holes are solutions that describe very high energy level production of parti-cles, where the effects of strong-quantum-gravity become dominant. A new conceptof quantum propagator is introduced that allows us to recognize Green kernels as“linear approximations” of such non-linear quantum propagators. More preciselyin [60] we have seen how to identify solutions of quantum super Yang-Mills equa-tions that come from the Dirac quantization of super classic counterpart of suchequations, i.e. (classic Dirac solutions). Here we go beyond classic Dirac solutions

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and recognize solutions related to a new nonlinear geometric definition of quantumsuper Yang-Mills equation propagators.

• [72, 73]. In these papers one gives a geometric formulation of singular PDE’stheory on the ground of the PDE’s geometric theory by A.Prastaro, i.e. by consid-ering integral bordism groups and surgery techniques. Many interesting applica-tions are given by explicitly solving (singular) boundary value problems of physicalrelevance. A particular attention is reserved to singular ODE’s, where a generalcharacterization of their singular points is obtained. Many examples of (singular)boundary value problems for ODE’s are given too. Relations with the stabilitytheory for PDE’s and PDE’s solutions are given by utilizing the recent geometricstability theory given by A.Prastaro in [64, 65, 66, 67, 68].

• [74]. In this paper we show that between PDE’s and crystallographic groupsthere is an unforeseen relation. In fact we prove that integral bordism groups ofPDE’s can be considered extensions of crystallographic subgroups. In this respectwe can consider PDE’s as extended crystals. Then an algebraic-topological obstruc-tion (crystal obstruction), characterizing existence of global smooth solutions forsmooth boundary value problems, is obtained. These results, are also extended tosingular PDE’s, introducing (extended crystal singular PDE’s). An application tosingular MHD-PDE’s, is given extending some our previous results on such equa-tions, and showing existence of (finite times stable smooth) global solutions crossingcritical zone nuclear energy production

• [75, 76]. Aim of the second paper (partially announced in [75]) is to special-ize our study to quantum supergravity Yang-Mills PDE’s (quantum SG-Yang-MillsPDE’s). This type of equations have been previously introduced by us in somerecent works and appears very useful to encode quantum dynamics unifying, justat quantum level, gravity with the other fundamental forces of Nature, i.e., elec-tromagnetic, weak and strong forces. These equations extend, at quantum level,some superclassical ones, well known in literature about supergravity. In fact su-pergravity, as has been usually considered, is a classical field theory, that, in somesense comes from a generalization of Charles Ehresmann and Elie Cartan’s differ-ential geometry. Then classical supergravity requires to be quantized. But in thisway one discards nonlinear phenomena. In fact this quantization is obtained bymeans of so-called quantum propagators, that are just associated to linearizationsof classical PDE’s. Our formulation, instead, works directly on noncommutativemanifolds (quantum supermanifolds), and the quantization is not more necessary.In fact, whether it is performed in this noncommutative framework, it can bee seenas a linear approximation of a more general nonlinear integration. In some previ-ous papers this important aspect has been carefully proved. Here we characterizequantum super PDE’s like extended crystals, in the sense that their integral bor-dism groups can be considered as extensions of crystallographic subgroups. Thisapproach generalizes our previous one for commutative PDE’s, and allows us toidentify an algebraic topologic obstruction to the existence of global smooth so-lutions for PDE’s in the category QS . Furthermore, for such solutions we studytheir stability properties from a geometric point of view. Section 3. Here we con-sider “quantum gravity” in the category QS , and encoded by suitable quantum

Yang-Mills equations (quantum SG-Yang-Mills PDE’s), say (YM). In this way weare able to characterize quantum (super)gravity like a secondary object, associ-

ated to some geometric fundamental objects (fields), solutions of (YM). Then the

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 11

mass properties of such solutions are directly pointed-out, without the necessity todirectly assume symmetry breaking Higgs-mechanisms. However, we recognize a

constraint in (YM), that gives a pure quantum geometrodynamic mechanism able

to justify mass acquisition (or loss) to a quantum solution of (YM). Furthermore,

nuclear particles and nuclides can be seen as suitable p-chain solution of (YM),and their energetic, thermodynamic and stability properties characterized.

• [77, 78, 79, 80]. The famous Poincare’s conjecture is about n-dimensionalmanifolds, with n = 3, (R. S. Hamilton, G. Perelman, A. Prastaro [74, 62]), butthere are also generalizations of this conjecture for higher dimension manifolds. Forn = 4 the Poincare conjecture has been proved by M. Freedman and for n ≥ 5, byS. Smale. More recently has been given a generalization for quantum superman-ifolds by A. Prastaro, that has also proved it in [70]). Nowadays, one can statethat a generalized Poincare conjecture can be proved, or disproved, depending onthe particular category C in which it is formulated. This problem aroses in theframework of the geometric topology, but in order to be solved it was necessary togo outside that framework and recast the problem in a theory of PDE’s. But themore recent results by A. Prastaro [70, 74, 62]), have proved that it was necessaryto return inside the algebraic topology framework, applied to the PDE’s geometrictheory. Really, it was soon evident that remaining in a pure algebraic topologicapproach it was not enough to solve this conjecture. In fact, a fundamental ideato solve this problem is to ask whether it is possible find a smooth manifold, V ,that without singular points bords a 3-dimensional compact, closed, smooth, sim-ply connected manifold N with S3, when N is homotopy equivalent to S3. Thebordism theory is able to state that a smooth manifold V such that ∂V = N

∪S3,

there exists, since the nonoriented and oriented 3-dimensional bordism groups Ω3

and +Ω3 respectively, are both trivial: Ω3 = +Ω3 = 0. However, by simply lookingto the above bordism groups it is impossible to state if V has singular points (i.e.,has holes) or it is a cylinder. By the way, more informations can be obtained bythe h-cobordism theory. More precisely the h-cobordism theorem in a category C ofmanifolds, states that if the compact manifold V has ∂V = N0 ⊔ N1, such thatthe inclusion maps Ni → V , i = 0, 1, are homotopy equivalences, (i.e., V is a h-cobordism), and π1(Ni) = 0, then V ∼=C N0 × [0, 1]. This theorem holds for n ≥ 5in the category of smooth manifolds (S. Smale) and for n = 4 in the category oftopological manifolds (M. Freedman). But it does not work for n = 3 !

A very important angular stone, in the long history about the solution of thePoincare conjecture, has been the introduction, by R. S. Hamilton, of a new ap-proach recasting the problem in to solving a PDE, the Ricci flow equation, andasking for nonsingular solutions there, that starting from a Riemannian manifold(N, γ) arrive to the 3-dimensional sphere S3, respectively identified with initial andfinal Cauchy manifolds in the Ricci flow equation. In that occasion the Mathemat-ical Analysis, or more precisely the Functional Analysis, entered in the Poincareconjecture problem. This approach has had many improvements until the papersby G. Perelman. More recently, A. Prastaro, by using his algebraic topologic the-ory of PDE’s was able to give a pure geometric proof of the Poincare conjecture.Let us emphasize that the usual geometric methods for PDE’s (Spencer, Cartan),were able to formulate for nonlinear PDE’s, local existence theorems only, untilthe introduction, by A. Prastaro, of the algebraic topologic methods in the PDE’sgeometric theory. These give suitable tools to calculate integral bordism groups

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in PDE’s, and to characterize global solutions. Then, on the ground of integralbordism groups, a new geometric theory of stability for PDE’s and solutions ofPDE’s has been built. These general methodologies allowed to A. Prastaro to solvefundamental mathematical problems too, other than the Poincare conjecture andsome of its generalizations, like characterization of global smooth solutions for theNavier-Stokes equation and global smooth solutions with mass-gap for the quantumYang-Mills superequation. (See [38, 45, 46, 54, 60, 63, 70, 71, 74, 91, 81, 62].)

The main purpose of this paper is to emphasize some problems related to exoticheat PDE’s recently focused.3

The following theorem is a direct issue from results contained in [74].

Theorem 2.1. Any 3-dimensional compact, closed simply connected smooth man-ifold, homotopy equivalent to S3, is diffeomorphic to S3.4

The main purpose of [78] is to show how, by using the PDE’s algebraic topology,introduced by A. Prastaro, one can prove the Poincare conjecture in any dimensionfor the category of smooth manifolds, but also to identify exotic spheres. In theframework of the PDE’s algebraic topology, the identification of exotic spheresis possible thanks to an interaction between integral bordism groups of PDE’s,conservation laws, surgery and geometric topology of manifolds. With this respectwe shall enter in some details on these subjects, in order to well understand andexplain the meaning of such interactions. So the paper splits in three sections otherthan this Introduction. 2. Integral bordism groups in Ricci flow PDE’s. 3. Morsetheory in Ricci flow PDE’s. 4. h-Cobordism in Ricci flow PDE’s. The main resultis contained just in this last section and it is Theorem 2.2 that by utilizing thepreviously considered results states (and proves) the following.5

Theorem 2.2. The generalized Poincare conjecture, for any dimension n ≥ 1 istrue, i.e., any n-dimensional homotopy sphereM is homeomorphic to Sn: M ≈ Sn.

For 1 ≤ n ≤ 6, n = 4, one has also that M is diffeomorphic to Sn: M ∼= Sn.But for n > 6, it does not necessitate that M is diffeomorphic to Sn. This happenswhen the Ricci flow equation, under the homotopy equivalence full admissibilityhypothesis, (see below for definition), becomes a 0-crystal.

Moreover, under the sphere full admissibility hypothesis, the Ricci flow equationbecomes a 0-crystal in any dimension n ≥ 1.

3The Ricci flow equation can be considered a generalization of the classical Fourier’s heatequation ut − κuxx = 0. In this paper we call exotic heat equations, PDE’s that, like the Ricci

flow equation, are of the type F j ≡ ujt −fj(ui

k) = 0, 1 ≤ i, j ≤ m, with the length |k| of the multi-

index k ∈ 1, · · · , n, given by 0 ≤ |k| ≤ s ≤ r, where F j : Jr(W ) → R are analytic functions oforder r ≥ 0 on a fiber bundle π : W ≡ R× E(M) → R×M , where E(M) is a vector bundle overM , with M an analytic manifold of dimension n. Let us emphasize that the structure of exotic

heat equation is the more suitable to use in order to prove (generalized) Poincare conjectures.In fact, the idea to use PDE’s to solve the Poincare’s conjecture, was the initial motivation tointroduce and study the well-known Yamabe equation. But that road did not turn out a luckychoice to prove the conjecture, even if the Yambe equation is a very important equation to study

conformal problems in Riemannian geometry. (See, e.g., [62].)4This last result agrees with the Hauptvermuntung conjecture that was proved for (n = 2, 3)-

dimensional manifolds and disproved for (n ≥ 4)-dimensional manifolds. (See A. Casson, J.Milnor, E. Moise, T. Rado, D. Sullivan, W. Tuschmann.)

5Results of this paper agree with previous ones by J. Cerf, M. Freedman, M. A. Kervaire and

J. W. Milnor, E. Moise and S. Smale, and with the recent proofs of the Poincare conjecture by R.S. Hamilton, G. Perelman, and A. Prastaro.

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In [79] a theorem characterizing global solutions of exotic 8-d’Alembert equationis given. This theorem allows us to state that two diffeomorphic exotic 7-sphere,identified with two Cauchy manifolds in (d′A)8 over R8, bound singular solutionsonly, but they cannot bound smooth solutions. (Compare with the situation inthe Ricci flow equation on compact, simply connected 7-dimensional Rimennianmanifolds [78].)

In [80] are generalized results of the previous three papers to any PDE and relat-ing them to general algebraic topological properties of PDE’s, like integral bordismgroups, conservation laws, spectral sequences, algebraic topological spectra. In par-ticular it is emphasized their relations with exotic Cauchy manifolds, that motivatethe name “exotic” given to these equations. Theorem 2.2 is generalized to any PDE.The main results in this paper is just the generalization to dimension n = 4, of suchtheorem, obtained by means the proof of the smooth Poincare conjecture. This isthe generalization of the Poincare conjecture in the category of smooth manifoldsin dimension n = 4. This was a very important open problem, remained unsolvedalso after the solution of the famous Poincare conjecture.

• [81]. This paper aims to further develop the A. Prastaro’s geometric theoryof quantum PDE’s, by considering three different (even if related) subjects in thistheory. The first is a way to characterize quantum PDE’s by means of suitableHeyting algebras. Nowadays these algebraic structures are considered important inorder to characterize quantum logics and quantum topoi. Really we prove that toany quantum PDE can be associated a Heyting algebra, naturally arising from thealgebraic topologic structure of the PDE’s and that encodes its integral bordismgroup.

Another aspect that we shall consider is the extension of the category Q of quan-tum manifolds, or QS of quantum supermanifolds, to the ones Qhyper of quantumhypercomplex manifolds. These generalizations are obtained by extending a quan-tum algebra A, in the sense of A. Prastaro, by means of Cayley-Dickson algebras.In this way one obtains a new category of noncommutative manifolds, that are use-ful in some geometric and physical applications. In fact there are some fashionedresearch lines, concerning classical superstrings and classical super-2-branes, whereone handles with algebras belonging just to some term in the Cayley-Dickson con-struction. Thus it is interesting to emphasize that the Prastaro’s geometric theoryof quantum PDE’s can be directly applied also to PDE’s for such quantum hyper-complex manifolds. This allows us to encode quantum micro-worlds, by a generaltheory that goes beyond the classical simple description of classical extended ob-jects, and solves also the problem of their quantization.

Let us emphasize that in some previous works we have formulated a geomet-ric theory of quantum manifolds that are noncommutative manifolds, where thefundamental algebra is a suitable associative noncommutative topological algebra,there called quantum algebra. Extensions to quantum supermanifolds and quantum-quaternionic manifolds are also considered too. Furthermore, we have built a geo-metric theory of quantum PDE’s in these categories of noncommutative manifolds,that allows us to obtain theorems of existence of local and global solutions, andconstructive methods to build such solutions too.

On the other hand it is well known that the sequences R ⊂ C ⊂ H ⊂ O ⊂ S,where S is the sedenionic algebra, fit in the so-called Cayley-Dikson construction,R ⊂ C ⊂ H ⊂ O ⊂ S ⊂ · · · ⊂ Ar ⊂ · · · , where Ar is a Cayley algebra of

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dimension 2r: Ar∼= R2r , r ≥ 0, A0 = R. Thus we can also consider quantum

hypercomplex algebras A⊗

Ar, 1 ≤ r ≤ ∞, where A is a quantum algebra in thesense of Prastaro, obtaining the natural inclusions A

⊗R Ar → A

⊗R Ar+1. It

is important to note that the Cayley algebras Ar are not associative for r ≥ 3,(even if they are exponential associative6) hence also the corresponding quantumhypercomplex algebras are non-associative for r ≥ 3, despite the associativity of thequantum algebra A. This fact introduces some particularity in the theory of PDE’son such algebras. The purpose of this paper is just to study which new behaviourshave PDE’s on the category Qhyper of quantum hypercomplex manifolds.

Let us emphasize that a first justification to use the category of quantum (super)manifolds to formulate PDE’s that encode quantum physical phenomena, arisesfrom the fact that quantized PDE’s can be identified just with quantum (super)PDE’s, i.e., PDE’s for such noncommutative manifolds. However, to formulateequations just in the category Q (or QS) allows us to go beyond the point of viewof quantization of classical systems, and capture more general nonlinear phenomenain quantum worlds, that should be impossible to characterize by some quantizationprocess. (See Refs. [58, 59, 60, 70, 71, 75, 76].) In fact the concept of quantumalgebra (or quantum superalgebra) is the first important brick to put in order tobuild a theory on quantum physical phenomena. In other words it is necessary toextend the fundamental algebra of numbers, R, to a noncommutative algebra A,just called quantum algebra. The general request on such a type of algebra can beobtained on the ground of the mathematical logic. (See [89, 91].) In fact, we haveshown that the meaning of quantization of a classical theory, encoded by a PDE Ek,in the category of commutative manifolds, is a representation of the logic L(Ek) ofthe classic theory, into a quantum logic Lq. More precisely, L(Ek) is the Booleanalgebra of subsets of the set Ω(Ek)c of solutions of Ek: L(Ek) ≡ P(Ω(Ek)c). (Theinfinite dimensional manifold Ω(Ek)c is called also the classic limit of the quantumsitus of Ek.) Furthermore, Lq is an algebra A of (self-adjoint) operators on alocally convex (or Hilbert) space H: Lq ≡ A ⊂ L(H). Then to quantize a PDEEk, means to define a map L(Ek) → Lq, or an homomorphism of Boolean algebrasq : P(Ω(Ek)c) → Pr(H), where Pr(H) is a Boolean algebra of projections on H.This construction allowed us also to prove that a quantization of a classical theory,can be identified by a functor relating the category of differential equations forcommutative (super)manifolds, with the category of quantum (super) PDE’s.

We can also extend a quantum algebra A, when the particular mathematical (orphysical) problem requires it useful. Then the extended algebra does not necessitateto be associative. This is, for example, the case when the extension is made bymeans of some algebra in the Cayley-Dickson construction, obtaining a quantum-Cayley-Dickson construction:

Q0 // Q1

// Q2 // · · ·

// Qr // · · ·

where Qr ≡ A⊗

R Ar, hence Q0 = A, Q1 = A⊗

R C and Q2 = A⊗

R H, etc.The main of this paper is just to show that the Prastaro’s algebraic topologictheory of quantum PDE’s, formulated starting from 80s, directly applies to thesenon-associative quantum algebras arising in the above quantum-Cayley-Dicksonconstruction.

6i.e., zn+m = znzm, ∀z ∈ Ar and n,m ∈ N.

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 15

Finally, the last purpose of this paper is to extend the concept of exotic PDE’s,recently introduced by A. Prastaro, for PDE’s in the category of commutativemanifolds, also for the ones in the category of quantum PDE’s. In particular, weprove also smooth versions of quantum generalized Poincare conjectures.

In the following we list the main results of this paper, assembled for sections. 2. Itis shown that to any quantum hypercomplex PDE Ek ⊂ Jk

n(W ), can be associateda topological spectrum (integral spectrum) and a Heyting algebra (integral Heytingalgebra) encoding some algebraic topologic properties of such a PDE. This allows usto give a new constructive point of view to the actual approach to consider quantumlogic in field theory by means of topoi. 3. Here the Prastaro’s formal geometrictheory of PDE’s is extended from the category of quantum (super)manifolds tothe ones for quantum hypercomplex manifolds. Global solutions of PDE’s in thecategory Qhyper, are characterized by means of suitable bordism groups. 4. Analgebraic topologic characterization of singular PDE’s in the category Qhyper isgiven. 5. The concept of exotic PDE’s, perviously introduced by A. Prastarofor PDE’s in the category of commutative manifolds, is extended to the categoryQhyper . Global solutions for exotic PDE’s in the category Qhyper, that allow toclassify smooth solutions starting from quantum homotopy spheres are classified.In particular, an integral h-cobordism theorem in quantum Ricci flow PDE’s isproved.

• [82, 83, 84]. In order to encode strong reactions of the high energy physics,by means of quantum nonlinear propagators in the Prastaro’s geometric theoryof quantum super PDE’s, some related geometric structures are further developedand characterized. In particular super-bundles of geometric objects in the categoryQS of quantum supermanifolds are considered and quantum Lie derivative of sec-tions of super bundle of geometric objects are calculated. Quantum supermanifoldswith classic limit are classified with respect to the holonomy groups of these lastcommutative manifolds. A theorem characterizing quantum super manifolds withstructured classic limit as super bundles of geometric objects is obtained. A the-orem on the characterization of chi-flow on suitable quantum manifolds is proved.This solves a previous conjecture too. Quantum instantons and quantum solitonsare defined are useful generalizations of the previous ones, well-known in the lit-erature. Quantum conservation laws for quantum super PDEs are characterized.Quantum conservation laws are proved work for evaporating quantum black holestoo. Characterization of observed quantum nonlinear propagators, in the observedquantum super Yang-Mills PDE, by means of conservation laws and observed en-ergy is obtained. Some previous results by A. Prastaro about generalized Poincareconjecture and quantum exotic spheres, are generalized to the category Qhyper,S ofhypercomplex quantum supermanifolds. (This is the first part of a work divided intwo parts. For part II see [83].)

In the second part decomposition theorems of integral bordisms in quantumsuper PDEs are obtained. In particular such theorems allow us to obtain repre-sentations of quantum nonlinear propagators in quantum super PDE’s, by meansof elementary ones (quantum handle decompositions of quantum nonlinear prop-agators). These are useful to encode nuclear and subnuclear reactions in quan-tum physics. Prastaro’s geometric theory of quantum PDE’s allows us to obtainconstructive and dynamically justified answers to some important open problemsin high energy physics. In fact a Regge-type relation between reduced quantum

16 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

mass and quantum phenomenological spin is obtained. A dynamical quantum Gell-Mann-Nishijima formula is given. An existence theorem of observed local andglobal solutions with electric-charge-gap, is obtained for quantum super Yang-Mills

PDE’s, (YM)[i], by identifying a suitable constraint, (YM)[i]w ⊂ (YM)[i], quan-tum electromagnetic-Higgs PDE, bounded by a quantum super partial differential

relation (Goldstone)[i]w ⊂ (YM)[i], quantum electromagnetic Goldstone-boundary.An electric neutral, connected, simply connected observed quantum particle, iden-

tified with a Cauchy data of (YM)[i], it is proved do not belong to (YM)[i]w.

Existence of Q-exotic quantum nonlinear propagators of (YM)[i], i.e., quantumnonlinear propagators that do not respect the quantum electric-charge conservationis obtained. By using integral bordism groups of quantum super PDE’s, a quantumcrossing symmetry theorem is proved. As a by-product existence of massive pho-tons and massive neutrinos are obtained. A dynamical proof that quarks can bebroken-down is given too. A quantum time, related to the observation of any quan-tum nonlinear propagator, is calculated. Then an apparent quantum time estimatefor any reaction is recognized. A criterion to identify solutions of the quantum superYang-Mills PDE encoding (de)confined quantum systems is given. Supersymmet-ric particles and supersymmetric reactions are classified on the ground of integral

bordism groups of the quantum super Yang-Mills PDE (YM). Finally, existenceof the quantum Majorana neutrino is proved. As a by-product, the existence of anew quasi-particle, that we call quantum Majorana neutralino, is recognized madeby means of two quantum Majorana neutrinos, a couple (νe, ˜νe), supersymmetricpartner of (νe, νe), and two Higgsinos. (Part I and Part II are unified in arXiv.)

in the third part quantum nonlinear propagators in the observed quantum super

Yang-Mills PDE, (YM)[i], are further characterized. In particular, a criterion thatassures the zero lost quantum electric-charge is obtained. In a previous work [?],we have characterized observed quantum nonlinear propagators V of the observed

quantum super Yang-Mills PDE, (YM)[i], proving that the total quantum electric-charge of incident particles in quantum reactions does not necessitate to be thesame of the total quantum electric-charge of outgoing particles. This allowed usto define Q-exotic quantum nonlinear propagators ones where there is a non-zerolost quantum electric-charge, Q[V ] ∈ A, in the corresponding encoded reactions.

(A is the fundamental quantum superalgebra in (YM)[i]) This important phenom-

ena, that is related to the gauge invariance of (YM)[i], was non-well previouslyunderstood, since the gauge invariance was wrongly interpreted. Really just thegauge invariance is the main origin of such phenomenon, beside the structure of thequantum nonlinear propagator. This fundamental aspect of quantum reactions in

(YM)[i], gives strong theoretical support to the guess about existence of quantumreactions where the “electric-charge” is not conserved. This was quasi a dogmain particle physics. However, there are in the world many heretical experimentalefforts to prove existence of decays like the following e− → γ + ν, i.e. electrondecay into a photon and neutrino. In this direction some first weak experimentalevidences were recently obtained. Some other exotic decays were also investigated,as for example the exotic neutron’s decay: n→ p+ ν + ν. (See References quotedin the paper.) With this respect, one cannot remark the singular role played, in thehistory of the science in these last 120 years, by the electron, a very small and light

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 17

particle. In fact, at the beginning of the last century was just the electron to pro-duce a break-down in the Maxwell and Lorentz physical picture of the world, untilto produce a completely new point of view, i.e. the quantum physics. Now, after120 years the electron appears to continue do not accept the place that physicistshave reserved to it in the world-puzzle.

Aim of the third paper is to further characterize a criterion to recognize un-der which constraints quantum nonlinear propagators preserve quantum electriccharges between incoming and outgoing particles. The main result of this third

part is the existence of a sub-equation˜

(YM)[i]• ⊂ (YM)[i], were live solutionsstrictly respecting the conservation of the quantum electric charge. Instead, so-

lutions bording Cauchy data contained in the sub-equation˜

(YM)[i]• ⊂ (YM)[i],

that are globally outside˜

(YM)[i]•, can violate the conservation of the quantumelectric charge. This effect is interpreted caused by the quantum supergravity. Theaction of the quantum supergravity is able to guarantee existence of such more gen-eral quantum nonlinear propagators in quantum super Yang-Mills PDEs. In factquantum supergravity can deform quantum nonlinear propagators in order that

they can produce such exotic solutions of (YM)[i]. In other words, quantum ex-otic strong reactions exist as a by-product of quantum supergravity that producesnon-flat quantum nonlinear propagators. In the standard model quantum super-gravity is completely forgotten. Without quantum supergravity, exotic quantumpropagators cannot be realized ! The main results are the following. • A criterionto recognize under which constraints quantum nonlinear propagators have zero lostquantum electric-charge. Our main result is the identification of a sub-equation˜

(YM)[i]• ⊂ (YM)[i], that is formally integrable and completely integrable, suchthat all quantum reactions encoded there, are characterized by non-Q-exotic quan-tum nonlinear propagators. • A theorem proving that Q-exotic quantum nonlinear

propagators of (YM)[i] are solutions with exotic-quantum supergravity, i.e., hav-

ing non zero observed quantum curvature components RjαK . • A justification of the

so-called quantum entanglement phenomenon on the algebraic topologic structureof quantum nonlinear propagators. We prove that the EPR paradox is completelysolved in the framework of the Algebraic Topology of quantum PDE’s, as formu-lated by A. Prastaro. In fact, EPR paradox is related to a macroscopic modelof physical world (Einstein’s General Relativity (GR)). In order to reconcile thismodel with quantum mechanics, it is necessary to extend GR to a noncommuta-tive geometry, as made by the Prastaro’s Algebraic Topology of quantum (super)PDE’s. In fact the logic of microworlds is not commutative, hence it is naturalthat macroscopic mathematical models cannot justify quantum dynamics. In otherwords, the incompleteness of quantum mechanics (QM) is the complementary in-completeness of the GR, since they talk at different levels: microscopic the first(QM), macroscopic the second (GR). These different points of view can be recon-ciled by introducing the noncommutative logic of the QM in the geometric point ofview of the GR. But this must be made at the dynamic level ! It is not enough toformulate some noncommutative geometry to encode microworlds ! Therefore thenecessity to formulated a geometric theory of quantum PDE’s as has been realized

by A. Prastaro. • Existence of solutions of (YM)[i] admitting negative (absolute)temperature and their relations with quantum entanglement is given.

18 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

• [85]. The well known Goldbach’s conjecture in number theory, remained un-solved up to now, was one of the most famous example of the Godel ’s incomplete-ness theorem. In this paper we give a direct proof of this conjecture. Some usefulapplications regarding geometry and quantum algebra are also obtained.

Our proof is founded on the experimental observation that fixed an even integer,say 2n, n ≥ 1, and considered the highest prime number p1 ∈ P , that does notexceed 2n, the difference 2n − p1 is often a prime number, or if not, we can pass

to consider the next prime number, say p(1)1 < p1, and find that 2n − p

(1)1 is just

a prime number. (We denote by P the set of prime numbers.) Otherwise, we can

continue this process, and after a finite number of steps, obtain that 2n−p(s)1 = p(s)2 ,

where p(s)2 ∈ P . This process gives us a practical way to find two primes p

(s)1 and

p(s)2 , such that 2n = p

(s)1 + p

(s)2 , hence satisfy the Goldbach’s conjecture. Of course

the question is ”Does this phenomenon is a law and why ?” The main result of thispaper is to prove that our criterion, is mathematically justified.

• [86]. In this paper we consider some problems in Number Theory called Lan-dau’s problems listed by Edmund Landau at the 1912 International Congress ofMathematicians. These problems are the following. 1. Goldbach’s conjecture. 2.Twin prime conjecture. 3. Legendre’s conjecture. 4. Are there infinitely manyprimes p such that p − 1 is a perfect square ? In [85] the proof of the Goldbach’sconjecture has been already given. In this paper we show that by utilizing somealgebraic topologic methods introduced in [85], some Landau’s problems can beproved too. Furthermore, for the above fourth Landau’s problem a Euler-Riemannzeta function estimate is given and settled the problem negatively by evaluating thecardinality of the set of solutions of a suitable Diophantine equation of Ramanujan-Nagell-Lebesgue type.

• [87]. The Riemann hypothesis is the conjecture concerning the zeta Riemannfunction ζ(s), given by B. Riemann (1892). The difficulty to prove this conjecture isrelated to the fact that ζ(s) has been formulated in a some cryptic way as complexcontinuation of hyperharmonic series and characterized by means of a functionalequation that in a sense caches its properties about the identifications of zeros. Ourapproach to solve this conjecture has been to recast the zeta Riemann function ζ(s)to a quantum mapping between quantum-complex 1-spheres, i.e., working in thecategory Q of quantum manifolds as introduced by A. Prastaro. (See on this sub-ject References [70, 81] and related works by the same author quoted therein.) Moreprecisely the fundamental quantum algebra is just A = C, and quantum-complexmanifolds are complex manifolds, where the quantum class of differentiability isthe holomorphic class. In this way one can reinterpret all the theory on complexmanifolds as a theory on quantum-complex manifolds. In particular the Riemannsphere C

∪∞ can be identified with the quantum-complex 1-sphere S1, as con-

sidered in [70, 81]. The paper splits in two more sections. In Section 2 we resumesome fundamental definitions and results about the Riemann zeta function ζ(s).In Section 3 the main result, i.e., the prof that the Riemann hypothesis is true, iscontained in Theorem 3.1. This is made splitting the proof in some steps (lemmas).It is important to emphasize the central role played by Lemma 3.7. This focusesthe attention on the completed Riemann zeta function, ζ(s), that symmetrizes therole between poles, with respect to the critical line of C, and between zeros, withrespect to the x = ℜ(s)-axis. Finally the conclusion can be obtained by extending

ζ(s) to a quantum-complex mapping ζ(s), between quantum-complex 1-spheres.

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 19

Then by utilizing the properties of meromorphic functions between compact Rie-mann spheres, identified with quantum-complex 1-spheres, we arrive to prove thatall (non-trivial) zeros of ζ(s) must necessarily be on the critical line. In fact, the

extension of ζ(s) to ζ(s), reduces zeros of this last meromorphic function to havehave two simple zeros, symmetric with respect to the equator, and two simple poles,symmetric with respect to the critical line. For symmetry properties, this impliesthat also ζ(s) cannot have zeros outside the critical line, hence the same musthappen for ζ(s) for non-trivial zeros.

• [88]. Aim of this paper is to utilize previous Prastaro’s results on quantumsupergravity to prove that the geometric structure of quantum propagator encod-ing Universe at the Planck epoch is the cause of the Universe’s expansion.7 Thisexpansion is not caused by a strange exoteric force, but it is the boundary-effect ofthe quantum nonlinear propagator encoding the Universe. In fact this propagatorhas a boundary with thermodynamic quantum exotic components. The presence ofsuch exotic components produces an increasing of energy contents in the Universe,seen as transversal sections of such a quantum nonlinear propagator. To the in-creasing of energy corresponds an increasing in the expansion of the Universe. Suchexpansion has produced the passage of the Universe from the Higgs-universe, thefirst massive universe at the Planck epoch, to the actual macroscopic one. How-ever, yet in such a macroscopic age the expansion of the Universe can be justifiedby using the same philosophy. This will be illustrated by adopting the Einstein’sGeneral Relativity equations, but taking into account the effect of its quantumorigin (Planck-epoch-legacy). With this respect one can state that the so-calleddark-energy-matter, is nothing else than the increasing in energy produced by thethermodynamic exotic boundary encoding the Universe. Therefore it is a pure ge-ometrodynamic bordism effect that produces an expansion of our Universe also atthe Einstein epoch. Paradoxically this is a consequence of the energy conservationlaw that continues to work whether at the Planck epoch or at the Einstein age.This Prastaro theory gives also a precise mathematical support to some early con-jectures on the continuous creation of matter. (See, e.g., P. A. Dirac (1974), F.Hoyle (1949) and F. Hoyle and J. V. Narlikar (1963).

7Georges Lematre (1927) and Edwin Hubble (1929) first proposed that the Universe is ex-panding. Lemaitre used Einstein’s General Relativity equations and Hubble estimated value ofthe rate of expansion by observed red-shifts. The most precise measurement of the rate of the

Universe’s expansion, has been obtained by NASA’s Spitzer Space Telescope and published inOctober 2012. Very recently (March 17, 2014) some scientists of the Harvard-Smithsonian Centerfor Astrophysics, announced that observations with the telescope Bicep2 (Background Imaging of

Cosmic Extragalactic Polarization), located at the South Pole, allowed to give an experimentalproof of the Big Bang. (See the following link First Direct Evidence of Cosmic Inflation.)

20 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

Fig. 6. Florida Institute of Technology - Melbourne, FL, USA - 2005.

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 21

3. PUBLICATIONS

References

PAPERS

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lations, Nuovo Cimento 57 A(4)(1968), 881–885. DOI: 10.1007/BF02751394.[2] A. Prastaro & P. Parrini, A mathematical model for spinning molten polymer and conditions

of spinning, Tex. Res. J. 15(1975), 118–127. DOI: 10.1177/004051757504500206.[3] A. Prastaro & P. Parrini, Ein mathematisches Model fur das Verspinnen geschmolzener

Polymerer und fur die Spinnbedingungen, Colloid & Polymer Science 255(2)(1977), 624–633. DOI: 10.1007/BF01549886.

[4] A. Prastaro, A mathematical model for spinning viscoelastic molten polymers, Riv. Mat.Univ. Parma 4(2)(1976), 295–313. Zbl 0375.73034.

[5] A. Prastaro, Modello matematico sulla formazione della melt-fracture nei polimeri fusi ,Quad. Ing. Chim. Ital. Suppl. 13(3-4)(1977), 37–44. (Chemical Abstracts: Collective In-dex 2000.)

[6] A. Prastaro, Geometrodynamics of some non-relativistic incompressible fluids, Stochastica

3(2)(1979), 15–31. MR0556645(81b:76014); Zbl 0427.76003.[7] A. Prastaro, Spazi derivativi e fisica del continuo in relativita generale, Atti Accad. Sci.

Torino Suppl. 114(1980/81), 289–292. MR0670263(83h:58013).[8] A. Prastaro, On the general structure of continuum physics.I: Derivative spaces, Boll. Unione

Mat. Ital. (5)17-B(1980), 704–726. MR0590551(81m:73012); Zbl 0438.58004.[9] A. Prastaro, On the general structure of continuum physics.II: Differential operators, Boll.

Unione Mat. Ital. (5)S.-FM(1981), 69–106. MR0641760(83c:73002a); Zbl 0478.58004

[10] A. Prastaro,On the general structure of continuum physics.III: The physical picture, Boll.Unione Mat. Ital. (5)S.-FM(1981), 107–129.MR0641761(83c:73002b); Zbl 0478.58005.

[11] A. Prastaro, On the intrinsic expression of Euler-Lagrange operator , Boll. Unione Mat. Ital.(5)18-A(1981), 411–416. MR0633674(842:58049); Zbl 0471.58012.

[12] A. Prastaro, Dynamic conservation laws and the Korteweg-De Vries equation, Atti convegnosu onde e stabilita nei mezzi continui, Catania 1981, Quaderni CNR-GNFM, Catania (1982),272–274.

[13] A. Prastaro, Spinor super bundles of geometric objects on spinG space-time structures, Boll.

Unione Mat. Ital. (6)1-B(1982), 1015–1028. MR0683489(84c:53036); Zbl 0501.53023.[14] A. Prastaro, Gauge geometrodynamics, Riv. Nuovo Cimento 5(4) (1982), 1–122. DOI:

10.1007/BF02740593. MR0693882(84e:83045); Zbl 0695.58028.[15] A. Prastaro, Geometry and existence theorems for incompressible fluids, Geometro-

dynamics Proceedings 1983, A. Prastaro (ed.), Pitagora Ed., Bologna (1984), 65–90.MR0823718(87g:58034).

[16] A. Prastaro, Geometrodynamics of non-relativistic continuous media.I: Space-time structures, Rend. Sem. Mat. Univ. Politec. Torino40(2)(1982), 89–117.

MR0724201(85e:53095); Zbl 0525.53036.[17] A. Prastaro, Geometrodynamics of non-relativistic continuous media.II: Dynamic and

constitutive structures, Rend. Sem. Mat. Univ. Politec. Torino 43(1)(1985), 91–116.

MR0859851(87m:53091); Zbl 0609.53042.[18] A. Prastaro, A geometric point of view for the quantization of non-linear field theories, Atti

VI Convegno Nazionale di Relativita Generale e Fisica della Gravitazione, Firenze 1984, R.Fabbri and M. Modugno (eds.), Pitagora Ed., Bologna (1986), 289–292.

[19] A. Prastaro, Dynamic conservation laws, Geometrodynamics Proceedings 1985, A. Prastaro(ed.), World Scientific Publishing, Singapore (1985), 283–420. MR0825801(87g:53109);Zbl 0645.58038. [About “Geometrodynamics” see also Wikipedia.]

[20] A. Prastaro & T. Regge, The group structure of supergravity, Ann. Inst. H. Poincare Phys.

Theor. 44(1)(1986), 39–89. MR0834019(87i:83104); Zbl 0588.53066.[21] V. Marino & A. Prastaro, On the geometric generalization of the Noether theorem, Lecture

Notes in Math. 1209, Springer-Verlag, Berlin (1986), 222–234. DOI: 10.1007/BFb0076634.MR0863759(88j:58142); Zbl 0603.53058.

22 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

[22] A. Prastaro, Quantum gravity and group model gauge theory, Journees Relativistes,

Toulouse, France 1986, A. Crumeyrolle (ed.), Univ. Paul Sabatier Toulouse (1986), 213–222.[Recent results on the quantum geometrodynamics. Proceedings General Relativity and Grav-itation, vol.1. July, 4-9, 1983, Padova - Italy. (B. Bertotti, F. de Felice & A. Pascolini eds.)CNR - Roma (1983), 1145.]

[23] V. Marino & A. Prastaro, On the conservation laws of PDE’s, Rep. Math. Phys.26(2)(1987/8), 211–225. DOI: 10.1016/0034-4877(88)90024-9. MR0991720(91b:58097);Zbl 0695.58029.

[24] A. Prastaro, On the quantization of Newton equation, Atti IX Congresso AIMETA, Bari

1988, AIMETA(1988), 13–16.[25] A. Prastaro, Wholly cohomological PDE’s, International Conference on Differential Geometry

and Applications, Dubrovnick (Yu), 1988, Univ. Beograd & Univ. Novi Sad (1989), 305–314.MR1040078(91c:58151); Zbl 0695.58030.

[26] A. Prastaro, Geometry of quantized PDE’s, Differential Geometry and Applications, J.Janyska & D. Krupka (eds.), World Scientific Publishing, Teanek, NJ, (1990), 392–404.MR1062046(91m:58175); Zbl 0796.35006.

[27] A. Prastaro, On the singular solutions of PDE’s, Atti X Congresso Nazionale AIMETA, Pisa

1990, AIMETA(1990), 17–20.[28] A. Prastaro, Cobordism of PDE’s, Boll. Unione Mat. Ital. (7)5-B(1991), 977–1001.

MR1146763(93a:57037); Zbl 0746.57015.

[29] A. Prastaro, Quantum geometry of PDE’s, Rep. Math. Phys. 30(3) (1991), 273–354.DOI:10.1016/0034-4877(91)90063-S. MR1198655(94e:58150); Zbl 0771.58024.

[30] A. Prastaro, Geometry of super PDE’s, Geometry of Partial Differential Equations, A.Prastaro & Th. M. Rassias (eds.), World Scientific Publishing, River Edge, NJ, (1994), 259–

315. MR1340222(96g:58025); Zbl 0879.58080.[31] V. Lychagin & A. Prastaro, Singularities for Cauchy data, characteristics, cocharacteristics

and integral cobordism, Differential Geom. Appl. 4(3)(1994), 283–300. DOI: 10.1016/0926-2245(94)00017-4. MR1299399(96b:58122); Zbl 0808.58039.

[32] A. Prastaro & Th. M. Rassias, On a geometric approach to an equation of J. D’Alembert , Ge-ometry in Partial Differential Equations, A. Prastaro & Th. M. Rassias (eds.), World ScientificPublishing, River Edge, NJ, (1994), 316–322. MR1340223(96g:35131); Zbl 0879.35038.

[33] A. Prastaro, Th. M. Rassias & J. Simsa, Geometry of the J. D’Alembert equation, in Finite

Sums Decompositions in Mathematical Analysis, Th. M. Rassias & J. Simsa (eds.), J. Wiley(1995), 133–159. MR96k:26006; Zbl 0859.26005.

[34] A. Prastaro & Th. M. Rassias, A geometric approach to an equation of J.D’Alembert, Proc. Amer. Math. Soc. 123(5)(1995), 1597–1606. DOI: 10.2307/2161153.

MR1232143(95f:58007); Zbl 0839.58068.[35] A. Prastaro, Geometry of quantized super PDE’s, The Interplay Between Differential Geom-

etry and Differential Equations, V. Lychagin (ed.), Amer. Math. Soc. Transl. 2/167(1995),

165–192. MR1343988(96d:58159); Zbl 0844.58012.[36] A. Prastaro, Quantum geometry of super PDE’s, Rep. Math. Phys. 37(1)(1996), 23–140.

DOI: 10.1016/0034-4877(96)88921-X. MR1394861(97e:58235); Zbl 0887.58064.[37] A. Prastaro, (Co)bordism in PDEs and quantum PDEs, Rep. Math. Phys. 38(3)(1996), 443–

455. DOI: 10.1016/S0034-4877(97)84894-X. MR1437641(97m:58004); Zbl 0885.58094.[38] A. Prastaro, Quantum and integral (co)bordisms in partial differential equations, Acta Appl.

Math. 51(3) (1998), 243–302. DOI: 10.1023/A:1005986024130. MR1437641(99d:58183);Zbl 0924.58103.

[39] A. Prastaro, Quantum and integral bordism groups in the Navier-Stokes equation, New De-velopments in Differential Geometry, Budapest 1996, J. Szenthe (ed.), Kluwer AcademicPublishers, Dordrecht (1998), 343–360. MR1670467(2000h:58065); Zbl 0937.35133.

[40] A. Prastaro & Th. M. Rassias, On the set of solutions of the generalized d’Alembert equa-

tion, C. R. Acad. Sci. Paris 328(I-5)(1999), 389–394. DOI: 10.1016/S0764-4442(99)80177-3.MR1678135(2000a:3508); Zbl 0931.35031.

[41] A. Prastaro & Th. M. Rassias, A geometric approach of the generalized d’Alembert equa-tion, J. Comput. Appl. Math. 113(1-2)(2000), 93–122. DOI: 10.1016/S0377-0427(99)00247-

2. MR1735816(2001c:58079); Zbl 0936.35011.[42] A. Prastaro, (Co)bordism groups in PDEs, Acta Appl. Math. 59(2) (1999), 111–201. DOI:

10.1023/A:1006346916360. MR1741657(2001m:58046); Zbl 0949.35011.

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 23

[43] A. Prastaro, (Co)bordism groups in quantum PDEs, Acta Appl. Math. 64(2)(2000), 111–217.

DOI: 10.1023/A:1010685903329. MR1826643(2002e:58037); Zbl 0978.58016.[44] A. Prastaro, Theorems of existence of local and global solutions of PDEs in the category

of noncommutative quaternionic manifolds, Quaternionic Structures in Mathematics andPhysics, S. Marchiafava, P. Piccinni & M. Pontecorvo (eds.), World Scientific Publishing,

Singapore (2001), 329–337. MR1848873(2002f:58007); Zbl 0978.81038.[45] A. Prastaro, Local and global solutions of the Navier-Stokes equation, Steps in Differential

Geometry, Proceedings of the Colloquium on Differential Geometry, 25–30 July, 2000, Debre-cen, Hungary, L. Kozma, P. T. Nagy & L. Tomassy (eds.), Univ. Debrecen (2001), 263–271.

MR1859305(2002d:53008); Zbl 0983.35105.[46] A. Prastaro, Navier-Stokes equation: Global existence and uniqueness. (A geometric way

to solve the “(NS)-problem”.), published as: Addendum I: Bordism Groups and the (NS)-Problem, in Quantized. Partial Differential Equations, World Scientific Publishing, Singapore,

(2004), 377–434.[47] A. Prastaro & Th. M. Rassias, A geometric approach to a noncommutative generalized

d’Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000), 545–550. DOI: 10.1016/S0764-4442(00)00238-X. MR1760436(2001d:58026); Zbl 0966.35105.

[48] A. Prastaro & Th. M. Rassias, Results on the J. d’Alembert equation, Ann. Acad. Paed.Cracoviensis. Studia Math. 1(2001)117–128. Zbl 1137.58308.

[49] A. Prastaro, Quantum manifolds and integral (co)bordism groups in quantum partial dif-

ferential equations, Nonlinear Anal. Theory Methods Appl. 47/4(2001), 2609–2620. DOI:10.1016/S0362-546X(01)00382-0. MR1972386(2004c:35343); Zbl 1042.35610.

[50] A. Prastaro, Dirac quantization, Encyclopaedia Math. Suppl.III., M. Hazwinkel (ed.), KluwerAcademic Publishers, Dordrecht (2002), 127–129. DOI: 10.1007/978-0-306-48373-8.

[51] A. Prastaro, Integral bordisms and Green kernels in PDEs, Cubo 4(2)(2002), 316–370.MR1928829(2003g:58056).

[52] A. Prastaro & Th. M. Rassias, On the Ulam stability in geometry of PDE’s , FunctionalEquations Inequality and Applications, Th. M. Rassias (ed.), Kluwer Academic Publishers,

Dordrecht (2003), 139–147. MR2042561(2004k:58031); Zbl 1059.39024.[53] A. Prastaro & Th. M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal.

Appl. 8(2)(2003), 259–278. MR1994707(2004g:35179); Zbl 1096.39028.[54] A. Prastaro, Quantum super Yang-Mills equations: Global existence and mass-gap, Dynamic

Syst. Appl. 4(2004), 227–232. (Eds. G. S. Ladde, N. G. Madhin and M. Sambandham), Dy-namic Publishers, Inc., Atlanta, USA. ISBN:1-890888-00-1. MR2117787(2005m:81203);Zbl 1067.81097.

[55] A. Prastaro, Geometry of PDE’s.I: Integral bordism groups in PDE’s, J. Math. Anal. Appl.

319(2)(2006), 547–566. DOI: 10.1016/j.jmaa.2005.06.044. MR2227923(2007d:58031);Zbl 1100.35007.

[56] A. Prastaro, Geometry of PDE’s.II: Variational PDE’s and integral bordism groups,

J. Math. Anal. Appl. 321(2)(2006), 930–948. DOI: 10.1016/j.jmaa.2005.08.037.MR2241487(2007d:58032); Zbl 1160.58301.

[57] A. Prastaro, Conservation laws in quantum super PDE’s, Proceedings of the Conferenceon Differential & Difference Equations and Applications (eds. R. P. Agarwal & K. Perera),

Hindawi Publishing Corporation, New York (2006), 943–952. MR23049427(2008b:58041);Zbl 1131.35381.

[58] A. Prastaro, (Co)bordism groups in quantum super PDE’s.I: Quantum supermanifolds, Non-linear Anal. Real World Appl. 8(2)(2007), 505–538. DOI: 10.1016/j.nonrwa.2005.12.008.

MR2289563(2008j:58052); Zbl 1152.58313.[59] A. Prastaro, (Co)bordism groups in quantum super PDE’s.II: Quantum super PDE’s, Non-

linear Anal. Real World Appl. 8(2)(2007), 480–504. DOI: 10.1016/j.nonrwa.2005.12.007.MR2289562(2008j:58053); Zbl 1152.58312.

[60] A. Prastaro, (Co)bordism groups in quantum super PDE’s.III: Quantum super Yang-Mills equations, Nonlinear Anal. Real World Appl. 8(2)(2007), 447–479. DOI:10.1016/j.nonrwa.2005.12.006. MR2289561(2009b:58082); Zbl 1152.58311.

[61] R. Agarwal & A. Prastaro, Geometry of PDE’s.III(I): Webs on PDE’s and inte-gral bordism groups. The general theory, Adv. Math. Sci. Appl. 17(1)(2007), 239–266.MR237378(2009j:58026); Zbl 1143.53017.

24 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

[62] R. Agarwal & A. Prastaro, Geometry of PDE’s.III(II): Webs on PDE’s and integral bordism

groups. Applications to Riemannian geometry PDE’s, Adv. Math. Sci. Appl. 17(1)(2007),267–285. MR2337379(2009j:58027); Zbl 1140.53005.

[63] A. Prastaro, Geometry of PDE’s.IV: Navier-Stokes equation and integral bordismgroups, J. Math. Anal. Appl. 338(2)(2008), 1140–1151. DOI:10.1016/j.jmaa.2007.06.009.

MR2386488(2009j:58028); Zbl 1135.35064.[64] A. Prastaro, (Un)stability and bordism groups in PDE’s, Banach J. Math. Anal. 1(1)(2007),

139–147. MR2350203(2009e:58036); Zbl 1130.58014.[65] A. Prastaro, Extended crystal PDE’s stability.I: The general theory, Math. Comput.

Modelling. (2008). DOI: 10.1016/j.mcm.2008.07.020. MR2532085(2011b:58041); Zbl1171.35322.

[66] A. Prastaro, Extended crystal PDE’s stability.II: The extended crystal MHD-PDE’s, Math.Comput. Modelling. (2008). DOI: 10.1016/j.mcm.2008.07.021. MR2532086(2011b:58042);

Zbl 1171.35323.[67] A. Prastaro, On the extended crystal PDE’s stability.I: The n-d’Alembert extended crys-

tal PDE’s, Appl. Math. Comput. 204(1)(2008), 63–69. DOI: 10.1016/j.amc.2008.05.141.MR2458340(2010h:58058); Zbl 1161.35054.

[68] A. Prastaro, On the extended crystal PDE’s stability.II: Entropy-regular-solutions inMHD-PDE’s, Appl. Math. Comput. 204(1)(2008), 82–89. DOI: 10.1016/j.amc.2008.05.142.MR2458342(2010h:58059); Zbl 1161.35462.

[69] A. Prastaro, On quantum black-hole solutions of quantum super Yang-Mills equations,Dynamic Syst. Appl. 5(2008), 407–414. (Eds. G. S. Ladde, N. G. Madhin C. Peng& M. Sambandham), Dynamic Publishers, Inc., Atlanta, USA. ISBN: 1-890888-01-6.MR2468173(2010g:83040).

[70] A. Prastaro, Surgery and bordism groups in quantum partial differential equations.I: Thequantum Poincare conjecture, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 502–525. DOI: 10.1016/j.na.2008.11.077. MR2671857(2012b:58057); Zbl 1238.58025.

[71] A. Prastaro, Surgery and bordism groups in quantum partial differential equations.II: Varia-

tional quantum PDE’s, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 526–549. DOI:10.1016/j.na.2008.10.063. MR2671858(2012b:58058); Zbl 1238.58026.

[72] R. P. Agarwal & A. Prastaro, Singular PDE’s geometry and boundary value problems, J. Non-linear Conv. Anal. 9(3)(2008), 417–460. MR2478974(2010b:58030); Zbl 1171.35006.

[73] R. P. Agarwal & A. Prastaro, On singular PDE’s geometry and boundary valueproblems, Appl. Anal. 88(8)(2009), 1115–1131. DOI: 10.1080/00036810902943612.MR2568427(2010k:58033); Zbl 1180.35012.

[74] A. Prastaro, Extended crystal PDE’s. Mathematics Without Boundaries: Surveys in Pure

Mathematics. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dor-drecht London, (to appear). arXiv:0811.3693[math.AT].

[75] A. Prastaro, Quantum extended crystal PDE’s, Nonlinear Studies 18(3)(2011), 447–485.

arXiv:1105.0166[math.AT]. MR2012k:57043; Zbl 1253.35135.[76] A. Prastaro, Quantum extended crystal super PDE’s. Nonlinear Anal. Real World Appl.

13(6)(2012), 2491–2529. DOI: 10.1016/j.nonrwa.2012.02.014. arXiv:0906.1363[math.AT].MR2927202; Zbl 1258.81064.

[77] A. Prastaro, Exotic heat PDE’s, Commun. Math. Anal. 10(1)(2011), 64–81.arXiv:1006.4483[math.GT]. MR2825954; Zbl 06008771.

[78] A. Prastaro, Exotic heat PDE’s.II. Essays in Mathematics and its Applications. In Honor ofStephen Smale’s 80th Birthday. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg

New York Dordrecht London (2012), 369–419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0(Online). DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176[math.AT]. MR2975595.

[79] A. Prastaro, Exotic n-d’Alembert PDE’s and stability. Nonlinear Analysis. Stability, Approx-imation and Inequalities. Series: Springer Optimization and its Applications Vol 68. P. M.

Pardalos, P. G. Georgiev and H. M. Srivastava (Eds.). Springer Optimization and its Ap-plications Volume 68 (2012), 571–586. ISBN 978-1-4614-3498-6. arXiv:1011.0081[math.AT].Zbl 06073130.

[80] A. Prastaro, Exotic PDE’s. Mathematics Without Boundaries: Surveys in InterdisciplinaryResearch. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York DordrechtLondon, (to appear). arXiv:1101.0283[math.AT].

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 25

[81] A. Prastaro, Quantum exotic PDE’s. Nonlinear Anal. Real World Appl. 14(2)(2013), 893–

928. DOI: 10.1016/j.nonrwa.2012.04.001. arXiv:1106.0862[math.AT]. MR2991123. ; Zbl06142846.

[82] A. Prastaro, Strong reactions in quantum super PDE’s. I: Quantum hypercomplex exoticsuper PDE’s.

arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.)[83] A. Prastaro, Strong reactions in quantum super PDE’s. II: Nonlinear quantum propagators.

arXiv:1205.2984[math.AT].[84] A. Prastaro, Strong reactions in quantum super PDE’s. III: Exotic quantum supergravity.

arXiv:1206.4956[math.AT].[85] A. Prastaro, The Landau’s problems.I: The Goldbach’s conjecture proved.

arXiv:1208.2473[math.GM].[86] A. Prastaro, The Landau’s problems.II: Landau’s problems solved.

arXiv:1208.2473[math.GM]. (Part I and Part II are unified in arXiv.)[87] A. Prastaro, The Riemann hypothesis proved. arXiv:1305.6845[math.GM].[88] A. Prastaro, Quantum Geometrodynamic Cosmology.

(Submitted for publication.)

MONOGRAPHS AND TEXTS

[89] A. Prastaro, Geometry of PDEs and Mechanics, World Scientific Publishing, River Edge,

NJ, 1996, 760 pp. ISBN 9810225202. MR1412798(98e:55182); Zbl 0866.35007.[90] A. Prastaro, Elementi di Meccanica Razionale, Edizione 2010, Aracne Editrice, Roma, 2010,

446 pp. ISBN 978-88-548-3601-3.

[91] A. Prastaro, Quantized Partial Differential Equations, World Scientific Publishing,River Edge, NJ, 2004, 500 pp. ISBN 981-238-764-1. MR2086084(2005f:58036); Zbl1067.58022.

BOOKS (EDITOR AND COAUTHOR)

[92] A. Prastaro, Geometrodynamics Proceedings 1983 , Pitagora Ed., Bologna 1984. ISBN 88-

371-0286-0. MR0823711(86m:58007).[93] A. Prastaro, Geometrodynamics Proceedings 1985, World Scientific Publishing, Sin-

gapore 1985. ISBN 9971-978-63-6. BookSG-Contents. MR0825784(86m:58008); Zbl

0637.00006.[94] A. Prastaro & Th. M. Rassias, Geometry in Partial Differential Equations, World Scientific

Publishing, River Edge, NJ, 1994. ISBN 978-981-02-14-07-4. MR1340208(96a:58003); Zbl0867.00017.

PATENTS

[95] A. Prastaro et al., 28/5/1975 - No 23.797 A/75. Process of production of fibrous structureswith high degree of birefringence.

[96] A. Prastaro et al., 11/7/1975 - No 25.334 A/75. Process of production of plexus-filament ofsynthetic polymers by means of flash-spinning of polymers solutions.

26 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

FIG. 7. Poster: ICM 2006 Satellite Conference “Advances PDE’sGeometry”, August 31 - September 2, 2006, Madrid - Spain.

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 27

4. ABSTRACTS OF PUBLICATIONS

ABSTRACTS OF PAPERS

(1) L. Corgnier, A. D’Adda & A. Prastaro, Regge pole vs. resonance dualityand boostrap calculations, Nuovo Cimento 57 A(4)(1968), 881–885.

Abstract. In this note we use the Regge pole vs. resonance duality tomake bootstrap calulations by comparing the behaviour at fixed scatteringangle of Regge like amplitude with that of a sum of resonance contributions.

(2) A. Prastaro & P. Parrini, A mathematical model for spinning molten poly-mer and conditions of spinning, Tex. Res. J. 45(1975), 118–127.

Abstract.The equation of spinning of molten polymers in the stationarynon-isothermal state have been solved by an analytical numerical method soas to obtain temperature T (s) and section A(z) profiles along the spinningaxis z. T (z) and A(z) are thus correlated with the molecular parametersof the molten polymer: viscosity, density, and extrusion temperature, andpolymer mass-flow rate. Furthermore, the correlation has been obtainedbetween fiber quality and steady-state solutions. A critical collection rateand the critical extrusion output have been deduced from such correlations.Above such critical values, breakage of the molten polymer takes place.

(3) A. Prastaro & P. Parrini, Ein mathematisches Model fur das Verspinnengeschmolzener Polymerer und fur die Spinnbedingungen, Colloid & PolymerScience 255(2)(1977), 624–633.

Abstract.This is a translated German version of the previous article.(4) A. Prastaro, A mathematical model for spinning viscoelastic molten poly-

mers, Riv. Mat. Univ. Parma (4)(1976), 295–313.Abstract.A mathematical model for spinning viscoelastic materials is

proposed. This work can be considered the continuation of the papers[2, 3] which treated the case of newtonian materials. The viscoelastic sys-tem, as more differs from the newtonian as the elastic component is present;thus the viscoelastic mathematical model can be not be inferred from theanalysis of the previous paper; on the contrary the viscoelastic model in-cludes, as particular case, the newtonian model. The spinning process wasanalysed by adding the rheological equation for viscoelastic materials tothe set of simultaneous partial differential equations describing a generalmolten spinning process. We gave the steady-state numerical solutions,i.e. the filament croos-section A(z), filament temperature T (z) and fila-ment tension F (z), as function of position z and we related them to theparameters which influence the process of spinning: material parametersand spinning conditions parameters. We related the yarn birefringence ∆nto the same parameters also. Moreover, we proposed to investigate whichbounds impose the spinnability criterion on the viscoelastic paramaters andwhich conditions realize the maximum yarn production with fixed denierand section.

(5) A. Prastaro, Modello matematico sulla formazione della melt-fracture neipolimeri fusi, Quad. Ing. Chim. Ital. Suppl. 13(3-4)(1977), 37–44.

28 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

Abstract.The melt-fracture is one of the most serious problem in theextrusion of thermoplastic materials. This is a flow instability that appearsas an almost regular distorsion of the extruded material. After a panoramaon the known experimental facts, a mathematical model is given in sucha way to emphasize some essential parameters controlling the start of thisphenomenon. In fact has been decovered an adimensional number, Wc,(Weissenberg number), characterizing the start of flow instability in all meltpolymers. This allows us to determine the extrudibility characteristics ofpolymeric materials.

(6) A. Prastaro, Geometrodynamics of some non-relativistic incompressible flu-ids, Stochastica 3(2)(1979), 15–31.

Abstract. In some papers [7, 8, 9, 10] we proposed a geometric formula-tion of continuum mechanics, where a continuum body is seen as a suitabledifferentiable fiber bundle C on the Galilean space-time M , beside a dif-ferential equation of order k, Ek(C), on C and the assignement of a frameψ on M . In the present paper we apply this general theory to some in-compressible fluids. The scope is to demonstrate that also for these moresimple materials our theory is a suitable tool in order to understand betterthe fundamental principles of continuum mechanics.

(7) A. Prastaro, Spazi derivativi e fisica del continuo in relativita generale, AttiAccad. Sci. Torino Suppl. 114(1980/81), 289–292.

Abstract. In order to give an axiomatic description of continuum physics,a derivative space, Dk(V,W ), is introduced which allows us to describethe derivative of order k of a differentiable map f : V → W as section ofthe fiber bundle Dk(V,W ). Derivative operators and functional differentialoperators are seen as useful generalizations of usual differential operators.With this language we recognize a structural order to all physical entitieswhich are characteristic in continuum physics.

(8) A. Prastaro, On the general structure of continuum physics.I: Derivativespaces, Boll. Unione Mat. Ital. (5)17-B(1980), 704–726.

Abstract. In order to give an intrinsic and axiomatic formulation of con-tinuum physics, the derivative space Dk(V,W ) is introduced. This allowsus to describe the k-order derivative of a mapping f : V →W , as a sectionof the fibre bundle Dk(V,W ) → V . This formulation generalizes the con-cept of jet of a mapping. The corresponding differential calculus is carefullydeveloped.

(9) A. Prastaro, On the general structure of continuum physics.II: Differentialoperators, Boll. Unione Mat. Ital. (5)S.-FM(1981), 69–106.

Abstract. In order to give an intrinsic and axiomatic formulation of con-tinuum physics, the differential operators are studied in the language ofderivative spaces [7, 8]. Useful generalizations of differential operators aregiven by introducing derivative operators and functional differential opera-tors. Finally differential equations are considered as suitable subspaces ofderivative spaces.

(10) A. Prastaro, On the general structure of continuum physics.III: The physicalpicture, Boll. Unione Mat. Ital. (5)S.-FM(1981), 107–129.

Abstract.An axiomatic and intrinsic formulation of continuum systems isgiven by using tools of the modern differential geometry. In this framework

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 29

we are able to give a structural ordering to all the physical entities ofcontinuum physics.

(11) A. Prastaro, On the intrinsic expression of Euler-Lagrange operator, Boll.Unione Mat. Ital. (5)18-A(1981), 411–416.

Abstract.An intrinsic representation of the Euler-Lagrange differentialoperator is given for the global variational calculus on fibered manifolds.This expression is of particular interest to be utilized in the Lagrangianformulation of the field theory, as it gives an explicit derivative dependenceby the field.

(12) A. Prastaro, Dynamic conservation laws and the Korteweg-De Vries equa-tion, Atti convegno su onde e stabilita nei mezzi continui, Catania 1981,Quaderni CNR-GNFM, Catania,(1982), 272–274.

Abstract.Recently we have introduced a geometric method to describeconservation laws more general than Noetherian ones. In particular in[16, 17, 19] we have given a detailed exposition of this method developingthe geometric-differential calculus to build dynamic conserved quantities.In this communication we shall give an account of some of these resultsemphasizing their applications to the Korteweg-de Vries equation.

(13) A. Prastaro, Spinor super bundles of geometric objects on spinG space-timestructures, Boll. Unione Mat. Ital. (6)1-B(1982), 1015–1028.

Abstract. Spinorial fibre bundles are built on space-times manifolds oftype spinG that are fully covariant in the sense of [8]. These fibre bundlesgeneralize that introduced in [8] and are useful in a unified field theory.

(14) A. Prastaro, Gauge geometrodynamics, Riv. Nuovo Cimento 5(4)(1982),1–122.

Abstract. In this paper we have a self-contained unitary geometric devel-opment of the methods and structures on which the gauge theories arebased. We hope that this geometrical framework shall be useful for amore clear understanding of continuum physics. Since the framework issufficiently generalized, it can be applied to all the situations of physicalinterest. Contents: Functors and fibre bundles. Derivative spaces and dif-ferential equations. Derivative spaces and variational calculus. Connectionsand derivative spaces. Geometrodynamics of gauge continuum systems andsymmetries properties. Classification of gauge continuum systems. Spinorsuperbundles of geometric objects and dynamics.

(15) A. Prastaro, Geometry and existence theorems for incompressible fluids, Ge-ometrodynamics Proceedings 1983, A. Prastaro (ed.), Pitagora Ed., Bologna(1984), 65–90.

Abstract.By utilizing a geometric framework to describe the dynamicof a continuous medium [7, 8, 9, 10, 11, 12, 13, 14, 15] we give existencetheorems of local and global solutions for an incompressible fluid. Thediscussion concerns the Euler equation (E) and the non-isothermic Navier-Stokes equation (NS).

(16) A. Prastaro, Geometrodynamics of non-relativistic continuous media.I: Space-time structures, Rend. Sem. Mat. Univ. Politec. Torino 40(2)(1982),89–117.

Abstract. In order to formulate the non-relativistic continuum mechanicsas a unified field theory on Galilei space-time M , the geometrical structure

30 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

of M is considered and the space time resolution of bundles of geometricobjects onM are analysed in detail. In particular, the concept of geometricobject gives rigorous meaning to the concept of observed physical quantity.It clarifies the ambiguity of why “frame dependent” quantities are useful,even essential, in the kinematic of description of continuum mechanicalbodies. Moreover, it clarifies the paradosical nature of “frame indifferentstatements about frame dpendent quantities”. These turn out to be simplystatements about fields of geometric objects which are not tensor fields.

(17) A. Prastaro, Geometrodynamics of non-relativistic continuous media.II:Dynamic and constitutive structures, Rend. Sem. Mat. Univ. Politec.Torino 43(1)(1985), 89–116.

Abstract.An intrinsic formulation of Continuum Mechanics on the affineGalielan space-time M is given, emphasizing the role of the dynamic equa-tion as a geometric structure. In particular, a continuous body is describedas a geometric structure on M . Thus, the study of symnmetry propertiesof this structure allows us to give useful classifications of continuous bodiesand to state generalized forms of Noether’s theorem. These considerationsare applied to incompressible fluids. Existence and uniqueness theoremsfor regular solutions are obtained.

(18) A. Prastaro, A geometric point of view for the quantization of non-linearfield theories, Atti VI Convegno Nazionale di Relativita Generale e Fisicadella Gravitazione, Firenze 1984, Pitagora Ed., Bologna (1986), 289–292.

Abstract.The fundamental geometric structure of any field theory is afiber bundle π : W → M beside a PDE Ek ⊂ JDk(W ). So we shall recog-nize in this geometric structure (W,Ek) suitable properties to interpretatethe meaning of quantization. This communication shortly describe our newpoint of view in this field, showing how it is possible to read the meaningof the quantization in the formal properties of PDEs.

(19) A. Prastaro, Dynamic conservation laws, Geometrodynamics Proceedings1985, A. Prastaro (ed.), World Scientific Publishing, Singapore (1985), 283–420.

Abstract.Part A:PDE and Conservation Laws. Basic results on the for-mal theory of PDE. Pseudogroups and PDE. The Euler-Lagrange operatorfor Lagrangians of any order. Cartan form and Noether conservation lawsfor Lagrangian of any order. Symmetry properties and dynamic conser-vation laws. Part B: Quantized PDE and Conservation Laws. Quantumcharges. Quantum situs. Geometric theory of quantized PDE. Quantumcobordism and Dirac’s approach to quantization. Applications: Klein-Gordon equation; Maxwell equation; Dirac equation; Einstein equation;Yang-Mills equation. Appendix. Topological vector spaces, C∗-algebrasand spectral theory. Local characterization of some geometric structuresrelated to PDE.

(20) A. Prastaro & T. Regge, The group structure of supergravity, Ann. Inst.H. Poincare Phys. Theor. 44(1)(1986), 39–89.

Abstract.An intrinsic description of the ”group manifold approach” tosupergravity is given. Emphasis is placed on some geometric structureswhich allow us to obtain a direct full covariant formulation. In particular,

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 31

the geometric theory of partial differential equations allows us to give a dy-namic description of space-time. Some applications to physically interestingsituations are discussed in detail.

(21) V. Marino & A. Prastaro, On the geometric generalization of the Noethertheorem, Lecture Notes in Math. 1209, Springer-Verlag, Berlin (1986),222–234.

Abstract.Task of this paper is to compare some geometric approaches toobtain conservation laws associated to partial differential equations. Moreprecisely we intend to consider the methods developed by A. Prastaro andA. M. Vinogradov. The differential equations are considered from a geo-metric point of view: namely they are submanifolds of jet-derivative spaceson fiber bundles. To the symmetries of these submanifolds are associatedconservation laws that are not necessarily of Noetherian type.

(22) A. Prastaro, Quantum gravity and group model gauge theory, Journees Rel-ativistes, Toulouse, France 1986, A. Crumeyrolle (ed.), Univ. Paul SabatierToulouse (1986), 213–222.

[Recent results on the quantum geometrodynamics. Proceedings GeneralRelativity and Gravitation, vol.1. July, 4-9, 1983, Padova - Italy. (B.Bertotti, F. de Felice & A. Pascolini eds.) CNR - Roma (1983), 1145.]

Abstract.A new geometric point of view of quantization of PDE’s, foundedon the formal properties of PDE’s, is applied to the quantization of grav-ity coupled with a Yang-Mills gauge field. The quantization of Einsteinequation in the framework of group model gauge theory is discussed.

(23) V. Marino & A. Prastaro, On the conservation laws of PDE’s, Rep. Math.Phys. 26(2)(1987/8), 211–225.

Abstract.The general methods of obtaining conservation laws for (non-linear) partial differential equations (PDE’s) introduced by A. Prastaro in[12, 19] and by A. M. Vinogradov in Soviet Math. Dokl. (5)18(1977), 1200–1204; J. Math. Anal. Appl. (1)100(1984), 1–129, are considered and thegeneral covariance of such methods is studied. In particular, it is shown thatVinogradov’s method fails to be fully covariant in the non-linear case. Therelations between the number of conservation laws for PDEs and the Atiyah-Singer index theorem are studied. A criterion for recognize the whollycohomological character of a PDE is given and the link between spectralsequences and wholly cohomlogical equations is found. Some examples ofinteresting PDEs which arise in physics are also considered.

(24) A. Prastaro, On the quantization of Newton equation, Atti IX CongressoAIMETA, Bari 1988, AIMETA (1988), 13–16.

Abstract. It is proved that the method of formal quantization of PDEsintroduced by the author in some previous papers [18, 25, 28] allows toobtain in intrinsic way the canonical Dirac quantization for particles ofthe lassical mechanics. As examples harmonic oscillator and anharmonicoscillator are considered. These considerations can be generalized to anyPDE defined on fiber bundles.

(25) A. Prastaro, Wholly cohomological PDE’s, International Conference on Dif-ferential Geometry and Applications, Dubrovnick (Yu), 1988, Univ. Beograd& Univ. Novi Sad (1989), 305–314.

32 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

Abstract.PDE’s are geometric objects to which one can associate con-servation laws in relation to their symmetry properties [15, 17, 19, 21, 23].Then, the wholly-cohomological character of a PDE is its possibility torepresent any (n− 1)-dimensional cohomological class of the n-dimensionalbasis manifold by means of a conservation law. In this paper we resumesome recent results in this direction obtained by the author [29] and alsoannounce some new further results for PDEs defined in the category ofsupermanifolds [36].

(26) A. Prastaro, Geometry of quantized PDE’s, Differential Geometry and Ap-plications, J. Janyska & D. Krupka (eds.), World Scientific Publishing,Singapore (1990), 392–404.

Abstract. In this paper we resume some recent results in the directionof the formal quantization of PDE’s obtained by the author [24], and alsoannounce some new further results. The categorial meaning of quantizationof PDE’s is given. Formal quantization results a canonical functor definedon the category of differential equations. Furthermore, a Dirac-quantizationcan be interpreted as a covering in the category of differential equations.A quantum (pre-)spectral measure is a functor that can be factorized bymeans of formal quantization and a (pre-)spectral measure. A relationbetween canonical Dirac-quantization and singular solutions of PDE’s isgiven. It is proved also that knowledge of Backlund correspondeces, aswell conservation laws, can aid the proceeding of canonical quantization ofPDE’s. (See [29].)

(27) A. Prastaro, On the singular solutions of PDE’s, Atti X Congresso NazionaleAIMETA, Pisa 1990, AIMETA (1990), 17–20.

Abstract. In the geometric formal theory of PDE’s we recognize alsothe problem of existence of singular solutions with singularities of Thom-Boardman type, i.e., singularities that can be resolved by means of prolon-gations. Scope of this paper is to give a short account of some fundamentalresults in these directions and apply them to some important classic equa-tions of fluid mechanics: Euler equation (E) and Navier-Stokes equation(NS). Quantum tunneling effects can be described by means of such sin-gular solutions. Furthermore, we show also as singular solutions enter inthe description of canonical quantization of PDE’s. We shall specialize, forsake of coincision, on equations (E) and (NS). (See [29].)

(28) A. Prastaro, Cobordism of PDE’s, Boll. Unione Mat. Ital. (7)5-B(1991),977–1001.

Abstract.Cobordism groups of systems of partial differential equationsof any order are considered and their representations by means of suitablehomology groups are given. This approach generalizes previous one by J.Eliashberg, given for PDE’s of first order, These results are useful also inorder to give an algebraic-topological characterization of the quantum situsor its classic limit, for PDE’s of any order. (See also [19, 70].)

(29) A. Prastaro, Quantum geometry of PDE’s, Rep. Math. Phys. 30(3)(1991),273–354.

Abstract. In this paper we present the formal quantization of PDE’s[18, 19, 22, 24, 26, 27] in categorial language. Formal quantization resultsas a canonical functor defined on the category of differential equations.

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Furthermore, a Dirac-quantization can be interpreted as a covering in thecategory of differential equations. A quantum (pre-)spectral measure is afunctor that can be factorized by means of formal quantization and a (pre-)spectral measure. A relation between canonical Dirac-quantization andsingular solutions of PDE’s is given. It is also proved that the knowledgeof Baklund correspondences, as well as the conservation laws, can aid theprocedure of canonical quantization of PDE’s. Physically interesting ex-amples are considered. In particular, we give the canonical quantization ofan anharmonic oscillator. A general theory of quantum tunneling effects inPDE’s is given. In particular, quantum cobordism has been related withLeray-Serre spectral sequences of PDE’s.

(30) A. Prastaro, Geometry of super PDE’s, Geometry of Partial DifferentialEquations, A. Prastaro & Th. M. Rassias (eds.), World Scientific Publish-ing, River Edge, NJ, (1994), 259–315.

Abstract. Superspaces and supermanifolds are introduced by using theconcept of weak differentiability as usually given for locally convex spaces.This allows us to consider in algebraic way superdual spaces and superderiva-tive spaces. In this way we obtain a good generalization of just known super-structures general enough to develop a formal theory for super PDE’s thatdirectly extends previous ones for standard manifolds of finite dimension.In particular, we give a Goldschmidt-type criterion of formal superintegra-bility for super PDE’s, and show that a geometric theory of singular super-solutions, with singularities of Thom-Boardman type, can be formulated inthe framework of super PDE’s too. Conservation superlaws associated tosuper PDE’s are considered and related with some spectral sequences andwholly cohomological character of these equations.

(31) V. Lychagin & A. Prastaro, Singularities for Cauchy data, characteris-tics, cocharacteristics and integral cobordism, Differential Geom. Appl.4(3)(1994), 283–300.

Abstract.A generalization of the classical concept of characteristic forpartial differential equations (PDE) is given in the framework of the geo-metric formal theory for PDE’s. In particular, it is given a relation betweensingularities of Cauchy data and characteristics in order to obtain integralmanifolds (solutions) generated by means of characteristics. In this di-rection it is shown that to any PDE we can associate a ”dual” equationhaving the same characteristics. These equations can be related by meansof a sort of Backlund transformation. Furthermore, a criterion that relatescharacteristics and integral cobordism (or quantum cobordism [19, 29]) isgiven. Also a relation between quantum cobordism in non-linear PDE’sand Green’s functions is given.

(32) A. Prastaro & Th. M. Rassias, On a geometric approach to an equationof J.D’Alembert, Geometry in Partial Differential Equations, A. Prastaro& Th. M. Rassias (eds.), World Scientific Publishing, Singapore (1994),316–322.

Abstract.Here we announce some firt results on the J. D’Alembert equa-

tion ( ∂2

∂x∂y log f) = 0. More precisely, by using a geometric framework we

prove that the set of smooth functions of two variables f(x, y), solutions of

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the J. D’Alembert equation, is larger than the set of functions of the formf(x, y) = h(x).g(y).

(33) A. Prastaro, Th. M. Rassias & J. Simsa, Geometry of the J. D’Alembertequation, in Finite Sums Decompositions in Mathematical Analysis, Th.M. Rassias & J. Simsa (eds.), J. Wiley (1995), 133–159.

Abstract.This is the last chapter of a book devoted to a very interestingand actual problem in Mathematical Analysis. Here the geometric theory ofPDE’s is considered and applied to the d’Alembert equation in its connec-tion with the problem of representation of functions by (partial) separationof variables.

(34) A. Prastaro & Th. M. Rassias, A geometric approach to an equation ofJ.D’Alembert, Proc. Amer. Math. Soc. 123(5)(1995), 1597–1606.

Abstract.By using a geometric framework of PDE’s we prove that the

set of solutions of the D’Alembert equation (∗) ∂2 lg f∂x∂y = 0 is larger than the

set of smooth functions of two variables f(x, y) of the form (∗∗) f(x, y) =h(x).g(y). This agrees with a previous counterexample by Th. M. Rassiasgiven to a statement by C. M. Stephanos. More precisely, we have thefollowing result: The set of 2-dimensional integral manifolds of PDE (∗)properly contains the ones representable by graphs of 2-jet-derivatives offunctions f(x, y) expressed in the form (∗∗). A generalization of this resultto functions of more than two variables is sketched also by considering the

equation ∂n log f∂x1...∂xn

= 0.

(35) A. Prastaro, Geometry of quantized super PDE’s, The Interplay BetweenDifferential Geometry and Differential Equations, V. Lychagin (ed.), Amer.Math. Soc. Transl. 2/167(1995), 165–192.

Abstract. In this paper we announce some results on the geometrization ofsuper PDE’s, i.e., PDE’s defined in the category of supermanifolds. Theseresults generalize previous ones for PDE’s [29].

(36) A. Prastaro, Quantum geometry of super PDE’s, Rep. Math. Phys. 37(1)(1996),23–140.

Abstract. In order to extend to super PDEs the theory of quqntizationof PDEs as contained in [29] we first develop a geometric theory for superPDEs (see also [30, 35]). Superspaces and supermanifolds are introducedby using the concept of weak differentiability as usually given for locallyconvex spaces. This allows us to consider in algebraic way superdual spacesand superderivative spaces and to develop a formal theory for super PDEsthat directly extends the previous ones for standard manifolds of finitedimension. In particular, we give a criterion of formal superintegrabilityfor super PDEs, and show that a geometric theory of singular supersolu-tions, with singularities of Thom–Boardman type, can be formulated inthe framework of super PDEs too. These results generalize the previousones obtained for ordinary manifolds by H.Goldschmidt and by Moscow’smathematical school. Conservation superlaws associated to super PDEsare considered and related with some spectral sequences and wholly coho-mological character of these equations. Then, the quantization of superPDEs is formulated on the ground of quantum cobordism [25, 39]. This ismade in order to give an intrinsic and fully covariant geometric formulation

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of unified quantum field theory. In particular, a theory of quantum super-gravity is developed. We explain how canonical quantization and quantumtunneling effects arise in super PDEs. Furthermore, we explicitly extendprevious results of Witten and Atiyah in topological quantum field theoryto our geometric framework for super PDEs. Obstructions to existence ofquantum cobords in super PDEs are given by means of supercharacteristicclasses. These results can be considered as a generalization of the recentresults obtained by Gibbons and Hawking.

(37) A. Prastaro, (Co)bordism in PDEs and quantum PDEs, Rep. Math. Phys.38(3)(1996), 443–455.

Abstract. In this paper we announce some recent results on the quantumand integral (co)bordism in PDEs and quantum PDEs. We shall essentiallyprove that the tecnique of (co)bordism, introduced by Pontrjagin and Thomin algebraic topology, can be generalized in the framework of partial dif-ferential equations in order to obtain sufficient criteria that allow to decidewhen a p-dimensional compact closed integral manifold contained in a PDEEk ⊂ Jk

n(W ), is the boundary of a (p + 1)-dimensional integral compactmanifold contained also in Ek (integral bordism) or eventually in the jet-space Jk

n(W ) containing Ek (quantum bordism). Furthermore, we shallprove that such results can be extended to the category of quantum PDEs.Here, by the term “quantum manifold” (and as a consequence of “quantumPDEs”) we mean a new structure that extends globally usual concepts ofquantum spaces, and that is very useful for physical applications.

(38) A. Prastaro, Quantum and integral (co)bordisms in partial differential equa-tions, Acta Appl. Math. 51(3)(1998), 243–302.

Abstract.Characterizations of quantum bordisms and integral bordismsin PDEs by means of subgroups of usual bordism groups are given. Moreprecisely, it is proved that integral bordism groups can be expressed as ex-tensions of quantum bordism groups and these last are extensions of sub-groups of usual bordism groups. Furthermore, a complete cohomologicalcharacterization of integral bordism and quantum bordism is given. Appli-cations to particular important classes of PDEs are considered. Finally, wegive a complete characterization of integral and quantum singular bordismsby means of some suitable characteristic numbers. Some examples of inter-esting PDEs which arise in Physics are also considered where existence ofsolutions with change of sectional topology (tunnel effect) is proved. As an

application, we relate integral bordism to the spectral term E0,n−11 , that

represents the space of conservation laws for PDEs. This gives, also, ageneral method to associate in a natural way a Hopf algebra to any PDE.

(39) A. Prastaro, Quantum and integral bordism groups in the Navier-Stokesequation, New Developments in Differential Geometry, Budapest 1996, J.Szenthe (ed.), Kluwer Academic Publishers, Dordrecht (1998), 343-360.

Abstract. In this paper we announce some results concerning theoremsof existence and classification of solutions of the Navier-Stokes equation(NS). In particular, following our general theory for bordism groups inPDEs, introduced in [37, 38, 70], the quantum and integral bordism groupsof (NS) are explicitly calculated. (See also [42, 45, 46, 63, 65].)

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(40) A. Prastaro & Th.M.Rassias, On the set of solutions of the generalizedd’Alembert equation, C. R. Acad. Sci. Paris 328(I-5)(1999), 389–394.

Abstract.By using a geometric approach we prove that the set of so-lutions of the generalized d’Alembert equation ∂n log f/∂x1 · · · ∂xn = 0,considered in the domain of the (x1, · · · , xn)-space Rn, is larger that theset of the functions that can be represented in the form as f(x1, · · · , xn) =f1(x

2, · · · , xn) · · · fn(x1, · · · , xn−1). Here the recent general method intro-duced by A. Prastaro to calculate integral and quantum (co)bordism groupsin PDE’s [37, 38, 70] is used. This method is very useful in order to prove ex-istence of tunneling effects in PDE’s, i.e., existence of solutions that changetheir sectional topology.

(41) A. Prastaro & Th. M. Rassias, A geometric approach of the generalizedd’Alembert equation, J. Comput. Appl. Math. 113(1-2)(2000), 93–122.

Abstract.The following results are obtained: 1) The set Sol(d′A)n of all

solutions of the equation ∂n log f∂x1...∂x1

= 0, (n-d’Alembert equation), (n ≥ 2),

considered in domains of the (x1, . . . , xn) ∈ Rn, is larger than the setof all functions f that can be represented in the form f(x1, . . . , xn) =f1(x

2, . . . , xn) . . . fn(x1, . . . , xn−1). 2) In the set of solutions Sol(d′A)n

of the n-d’Alembert equation, (d′A)n ⊂ JDn(Rn,R), there are also somemanifolds that have a change of sectional topology (tunneling effect).

(42) A. Prastaro, (Co)bordism groups in PDEs, Acta Appl. Math. 59(2)(1999),111–201.

Abstract.We introduce a geometric theory of PDEs, by obtaining exis-tence theorems of smooth and singular solutions. In this framework, fol-lowing our previous results on (co)bordisms in PDEs [37, 38, 70] we givecharacterizations of quantum and integral (co)bordism groups and relatethem to the formal integrability of PDEs. An explicitly proof that theusual Thom-Pontrjagin construction in (co)bordism theory can be gener-alized also to the case of singular integral (co)bordism in the category ofdifferential equations is given. In fact, we prove the existence of a spectrumthat characterizes the singular integral (co)bordism groups in PDEs. More-over, a general method that associates in a natural way Hopf algebras (fullp-Hopf algebras, 0 ≤ p ≤ n− 1), to any PDE Ek ⊂ Jk

n(W ), just introducedin [38, 70], is further studied. Applications to particular important classesof PDEs are considered. In particular, we carefully consider the Navier-Stokes equation (N) and explicitly calculate their quantum and integralbordism groups. An existence theorem of solutions of (NS) with change ofsectional topology is obtained. Relations between integral bordism groupsand causal integral manifolds, causal tunnel effects, and the full p-Hopfalgebras, 0 ≤ p ≤ 3, for the Navier-Stokes equation are determined.

(43) A. Prastaro, (Co)bordism groups in quantum PDEs, Acta Appl. Math.64(2)(2000), 111–217.

Abstract. In this paper we formulate a theory of noncommutative mani-folds (quantum manifolds) and for such manifolds we develop a geometrictheory of quantum PDEs (QPDEs). In particular, a criterion of formalintegrability is given that extends to QPDEs previous one given by H.Goldschmidt for PDEs [50] and by us for super PDEs [30, 35, 36]. Quan-tum manifolds are seen as locally convex manifolds where the model has

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 37

the structure Am11 × · · · ×Ams

s , with A ≡ A1 × · · · ×As a noncommutativealgebra that satisfies some particular axioms (quantum algebras). A generaltheory of integral (co)bordism for QPDEs is developed, that extends ourprevious for PDEs [37, 38, 42, 70]. Then, non-commutative Hopf algebras,(full quantum p-Hopf algebras, 0 ≤ p ≤ m−1), are canonically associated to

any QPDE Ek ⊂ Jkm(W ), whose elements represent all the possible invari-

ants that can be recognized for such a structure. Many examples of QPDEsare considered where we apply our theory. In particular, we carefully studyQPDEs for supergravity. We show that the corresponding regular solu-tions, observed by means of quantum relativistic frames, give curvature,torsion, gravitino and electromagnetic fields as A-valued distributions onspace-time, where A is a quantum algebra. For such equations canonicalquantizations are obtained and the quantum and integral bordism groupsand the full quantum p-Hopf algebras, 0 ≤ p ≤ 3, are explicitly calculated.Then, the existence of (quantum) tunnel effects for quantum superstringsin supergravity is proved.

(44) A. Prastaro, Theorems of existence of local and global solutions of PDEsin the category of noncommutative quaternionic manifolds, QuaternionicStructures in Mathematics and Physics, Rome 1999, S. Marchiafava, P.Piccinni & M. Pontecorvo (eds.), World Scientific Publishing, Singapore(2001), 329–337.

Abstract. In this paper we apply our recent geometric theory of non-commuttive (quantum) manifolds and noncommutative (quantum) PDEs[37, 38, 42, 70, 43] to the category of quantum quaternionic manifolds.These are manifolds modelled on spaces built starting from quaternionicalgebras. For PDEs considered in such category we determine theoremsof existence of local and global quaternionic solutions. We show also thatsuch a category of quantum quaternionic manifolds properly contains thatof manifolds with (almost) quaternionic structure. So our theorems of ex-istence of quantum quaternionic manifolds for PDEs produce a cascade ofnew solutions with nontrivial topology.

(45) A. Prastaro, Local and global solutions of the Navier-Stokes equation, Stepsin Differential Geometry, Proceedings of the Colloquium on DifferentialGeometry, 25–30 July, 2000, Debrecen, Hungary, L. Kozma, P. T. Nagy &L. Tomassy (eds.), Univ. Debrecen (2001), 263–271.

Abstract.A brief report is given on our recent results [42, 46] provingexistence of (smooth) global solutions of the 3D nonisothermal Navier-Stokes equation (NS), and (non) uniqueness of such solutions for (smooth)boundary value problems.

(46) A. Prastaro, Navier-Stokes equation. Global existence and uniqueness, (Ageometric way to solve the “(NS)-problem”.), published as: Addendum I:Bordism Groups and the (NS)-Problem, in Quantized. Partial DifferentialEquations, World Scientific Publishing, Singapore, (2004), 377–434.

Abstract.Here we report on our recent results on the integral bordismgroups of the 3D nonisothermal Navier-Stokes equation (NS) [39, 42, 45]that proved the existence of global (smooth) solutions. In particular, wego in some further results emphasizing surgery techniques that allow usto better understand this geometric proof of existence of (smooth) global

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solutions for any (smooth) boundary condition. Furthermore, a theoremof nonuniqueness of such solutions for general boundary conditions is givenon the ground of the symmetry properties of (NS) and just by using ourresults on the integral bordism groups of (NS). A comparison with theisothermal case, in the zero viscosity limit condition, (Euler equation), isconsidered. (See also [63].)

(47) A. Prastaro & Th. M. Rassias, A geometric approach to a noncommutativegeneralized d’Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000),545–550.

Abstract. In this paper the authors provide an account of some of theirresults concerning the J. D’Alembert equation especially in a suitable cat-egory of noncommutative manifolds, proving that the geometric theory ofPDE’s introduced by A. Prastaro is an handable framework where problemsin the theory of partial differential equations find their natural solutions.In fact, the J. d’Alembert equation is one such applications.

(48) A. Prastaro & Th. M. Rassias, Results on the J.d’Alembert equation, Ann.Acad. Paed. Cracoviensis. Studia. Math. 1(2001), 117–128.

Abstract.A new m-d’Alembert equation, m ≥ 2, is introduced in thecategory of quantum manifolds [37, 43, 70], that extends the commutativegeneralized d’Alembert equation just considered in [47]. For such a newequation we give theorems of existence of local and global solutions.

(49) A. Prastaro, Quantum manifolds and integral (co)bordism groups in quan-tum partial differential equations, Nonlinear Anal. Theory Methods Appl.47/4(2001), 2609–2620.

Abstract. In this paper it is given a short account of some recent theoremsby A. Prastaro on the existence of local and global solutions of QPDEs, i.e.partial differential equations built in the category of quantum manifolds[43]. This theory is then applied to the quantum Navier-Stokes equation,obtaining a generalization of the previous results by Prastaro on the Navier-Stokes equation [37, 42, 45, 46].

(50) A. Prastaro, Dirac quantization, Encyclopaedia Math. Suppl.III., M. Hazwinkel(ed.), Kluwer Academic Publishers, Dordrecht (2002), 127–129.

Abstract.A panorama on the modern developments of quantizations ofPDEs is given. In particular it is emphasized that on the framework of ageometric theory of PDEs, A. Prastaro has given a formulation of canonicalquantization of partial differential equations, without assuming that theseshould be of variational type and/or linear. Furthermore, the generalizationof this geometric approach to PDEs in the category of quantum manifolds,given more recently by A. Prastaro, has been considered. Relations withother recent works in noncommutative geometry are given.

(51) A. Prastaro, Integral bordisms and Green kernels in PDEs, Cubo Matem-atica Educacional 4(2)(2002), 316–370.

Abstract. Integral bordisms of (nonlinear) PDEs are characterized bymeans of geometric Green kernels and prove that these are invariant for theclassic limit of statistical sets of formally integrable PDEs. Such geomet-ric characterization of Green kernels is related to the geometric approachof canonical quantization of (nonlinear) PDEs, previously introduced by us[29]. Some applications are given where particle fields on curved space-times

CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014 39

having physical or unphysical masses, (i.e., bradions, luxons and massiveneutrinos), are canonically quantized respecting microscopic causality.

(52) A. Prastaro & Th. M. Rassias, On the Ulam stability in geometry of PDE’s,Functional Equations Inequality and Applications, Th. M. Rassias (ed.),Kluwer Academic Publishers, Dordrecht (2003), 139–147.

Abstract.The article is concerned with the problem of unstability of flowscorresponding to solutions of the Navier-Stokes equation in relation withthe stability of a new functional equation that is stable as well as superstablein an extended Ulam sense. In such a framework a natural characterizationof global stable laminar flow is given also.

(53) A. Prastaro & Th. M. Rassias, Ulam stability in geometry of PDE’s, Non-linear Funct. Anal. Appl. 8(2)(2003), 259–278.

Abstract.The unstability of characteristic flows of solutions of PDE’s isrelated to the Ulam stability of functional equations. In particular, weconsider, as master equation, the Navier-Stokes equation. The integral(co)bordism groups, that have recently been introduced by A. Prastaro tosolve the problem of existence of global solutions of the Navier-Stokes equa-tion [37, 38, 39, 42, 70], lead to a new application of the Ulam stability forfunctional equations. This allowed us here to prove that the characteristicflows associated to perturbed solutions of global laminar solutions of theNavier-Stokes equation, can be characterized by means of a stable (as wellsuperstable) functional equation (functional Navier-stokes equation). Insuch a framework a natural criterion to recognize stable laminar solutionsis given also.

(54) A. Prastaro, Quantum super Yang-Mills equations: Global existence andmass-gap, Dynamic Syst. Appl. 4(2004), 227–232. (Eds. G. S. Ladde,N. G. Madhin and M. Sambandham), Dynamic Publishers, Inc., Atlanta,USA. ISBN:1-890888-00-1.

Abstract.Quantum super Yang-Mills equations are considered in theframework of some noncommutative manifolds (quantum supermanifolds)and for such equations existence theorems of local and global solutionsare obtained by using some geometric methods recently introduced byA.Prastaro [37, 43, 44, 70]. In particular, global properties of solutionsare characterized by means of integral bordism groups. A criterion to rec-ognize global solutions with mass gap is given. (See also the book quotedin [74].)

(55) A. Prastaro, Geometry of PDE’s.I: Integral Bordism Groups in PDE’s, J.Math. Anal. Appl. 319(2)(2006), 547–566.

Abstract.We improve some our previous theorems on the calculation ofintegral bordism groups of formally integrable and completely integrablePDE’s, emphasizing the role played by singular solutions and weak solu-tions. Some applications to interesting PDE’s, defined on finite dimensionalmanifolds, are also considered.

(56) A. Prastaro, Geometry of PDE’s.II: Variational PDE’s and integral bordismgroups, J. Math. Anal. Appl. 321(2)(2006), 930–948.

Abstract. In the framework of the geometry of PDE’s, we classify varia-tional equations of any order with respect to their formal properties. Fol-lowing our previous results [55], we relate constrained variational PDEs to

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their integral bordism groups. In this way we are able to characterize globalsolutions of constrained variational PDEs and to relate them to the struc-ture of global solutions for the corresponding constraint equations. Someapplications are also considered.

(57) A. Prastaro, Conservation laws in quantum super PDE’s, Proceedings ofthe Conference on Differential & Difference Equations and Applications,(eds. R. P. Agarwal & K. Perera), Hindawi Publishing Corporation, NewYork (2006), 943–952.

Abstract.Conservation laws are considered for PDE’s built in the cate-gory QS of quantum supermanifolds. These are functions defined on theintegral bordism groups of such equations and belonging to suitable Hopfalgebras (full quantum Hopf algebras). In particular, we specialize our cal-culations on the quantum super Yang-Mills equations and quantum blackholes.

(58) A. Prastaro, (Co)bordism groups in quantum super PDE’s.I: Quantum su-permanifolds, Nonlinear Anal. Real World Appl. 8(2)(2007), 505–538.

Abstract. Following our previous works on noncommutative manifoldsand noncommutative PDE’s [37, 43, 44, 54, 57, 70, 74, 76], we considerin these series of three papers, some further results on quantum super-manifolds and quantum super PDE’s. In particular, in this first part wefocus our attention on quantum supermanifolds. These structures globalizethe notion of quantum superalgebras, obtaining noncommutative manifoldsthat are useful to give a fully covariant description of noncommutativegeometric structures, hence of quantum physics. We study the geome-try of quantum supermanifolds and characterize their (co)homological and(co)bordism properties. Covariant quantizations of super PDE’s are given,obtaining examples of quantum supermanifolds justifying their definitions.

(59) A. Prastaro, (Co)bordism groups in quantum super PDE’s.II: Quantumsuper PDE’s, Nonlinear Anal. Real World Appl. 8(2)(2007), 480–504.

Abstract. Following our previous works on the integral (co)bordism groupsof quantum PDE’s [37, 43, 44, 59, 70, 74], we specialize, now, on quantumsuper partial differential equations, i.e., partial differential equations builtin the category of quantum supermanifolds. These are manifolds modeledon locally convex topological vector spaces built starting from quantumalgebras endowed also with a Z2-gradiation, and a Z2-graded Lie algebrastructure, (quantum superalgebra). Then, we extend to these manifolds,with such richer structure, our previous results, and build a geometric the-ory of quantum super PDEs, that allows us to obtain theorems of existenceof (smooth) local and global solutions in the category of quantum super-manifolds. Some quantum (super) PDE’s, arising from the Dirac quantiza-tion of some classical (super) PDE’s, are considered in some details.

(60) A. Prastaro, (Co)bordism groups in quantum super PDE’s.III: Quantumsuper Yang-Mills equations, Nonlinear Anal. Real World Appl. 8(2)(2007),447–479.

Abstract. In this third part of a series of three papers devoted to the studyof geometry of quantum super PDE’s [58, 59], we apply our theory, devel-oped in the first two parts, to quantum super Yang-Mills equations andquantum supergravity equations. For such equations we determine their

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integral bordism groups, and by using some surgery techniques, we obtaintheorems of existence of global solutions, also with nontrivial topology, forCauchy problems and boundary value problems. Quantum tunnelling ef-fects are described in this context. Furthermore, for quantum supergravityequations we prove existence of solutions of the type quantum black holesevaporation processes just by using an extension to quantum super PDEsof our theory of integral (co)bordism groups. Our proof is constructive,i.e., we give geometric methods to build such solutions. In particular a cri-terion to recognize quantum global (smooth) solutions with mass-gap, forthe quantum super Yang-Mills equation, is given. Finally it is proved thatquantum super PDE’s contain also solutions that come from Dirac quan-tization of their superclassical counterparts. This proves that quantumsuper PDE’s are (nonlinear) generalizations of Dirac quantized superclas-sical PDE’s. Applications of this result to free quantum super Yang-Millsequations are given.

(61) R. Agarwal & A. Prastaro, Geometry of PDE’s.III(I): Webs on PDE’sand integral bordism groups. The general theory, Adv. Math. Sci. Appl.17(1)(2007), 239–266.

Abstract.Web structures are recognized on any partial differential equa-tion (PDE) that bring new insights in the geometric theory of PDE’s. Re-lations with integrability properties of PDE’s, and their integral bordismgroups, are obtained also, emphasizing the role played by singular andweak solutions. Applications to some important PDE’s of the Mathemati-cal Physics are given too.

(62) R. Agarwal & A. Prastaro, Geometry of PDE’s.III(II): Webs on PDE’sand integral bordism groups. Applications to Riemannian geometry PDE’s,Adv. Math. Sci. Appl. 17(1)(2007), 267–285.

Abstract.By using previous results by A.Prastaro on integral bordismgroups of PDE’s, and some issues of the companion paper in [61], we char-acterize in a geometric way local and global solutions of (generalized) Yam-abe equations and Ricci-flow equations. We prove that such results help tofind natural linear and parallel webs on a large category of PDE’s, that areimportant in order to find regular and singular solutions on such PDE’s. Inparticular, by applying algebraic topologic methods on the Ricci-flow equa-tion we definitively prove that the Poincare conjecture on the 3-dimensionalmanifolds is true.

(63) A. Prastaro, Geometry of PDE’s.IV: Navier-Stokes equation and integralbordism groups, J. Math. Anal. Appl. 338(2)(2008), 1140–1151.

Abstract. Following our previous results on this subject [39, 42, 45, 46],integral bordism groups of the Navier-Stokes equation are calculated forsmooth, singular and weak solutions respectively. Then a characterizationof global solutions is made on this ground. Enough conditions to assureexistence of global smooth solutions are given and related to nullity of inte-gral charecteristic numbers of the boundaries. Stability of global solutionsare related to some characteristic numbers of the space-like Cauchy data.Global solutions of variational problems constrained by (NS) are classifiedby means of suitable integral bordism groups too.

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(64) A. Prastaro, (Un)stability and bordism groups in PDE’s, Banach J. Math.Anal. 1(1)(2007), 139–147.

Abstract. In this paper, by using the theory of integral bordism groupsin PDE’s, previously introduced by Prastaro, we give a new interpretationof the concept of (un)stability in the framework of the geometric theory ofPDE’s. A geometric criterium to identify stable PDE’s and stable solutionsof PDE’s is given.

(65) A. Prastaro, Extended crystal PDE’s stability.I: The general theory, Math.Comput. Modelling. (2008).

Abstract.This work, divided in two parts, follows some our previousworks devoted to the algebraic topological characterization of PDE’s. Inthis first part, the stability of PDE’s is studied in details in the frameworkof the geometric theory of PDE’s, and bordism groups theory of PDE’s.In particular we identify criteria to recognize PDE’s that are stable (inextended Ulam sense) and in their regular smooth solutions do not occurfinite time unstabilities, (stable extended crystal PDE’s). Applications tosome important PDE’s are considered in some details. (In the second parta stable extended crystal PDE encoding anisotropic incompressible magne-tohydrodynamics is obtained.)

(66) A. Prastaro, Extended crystal PDE’s stability.II: The extended crystal MHD-PDE’s, Math. Comput. Modelling. (2008).

Abstract.This paper is the second part of a work devoted to the alge-braic topological characterization of PDE’s stability and its relation withan important class of PDE’s called extended crystals PDE’s. In fact, theirintegral bordism groups can be considered extensions of subgroups of crys-tallographic groups. This allows us to identify a characteristic class thatmeasures the obstruction to the existence of global solutions. In part I weidentified criteria to recognize PDE’s that are stable (in extended Ulamsense) and in their regular smooth solutions do not occur finite time unsta-bilities, (stable extended crystal PDE’s). Here we study in some details anew PDE encoding anisotropic incompressible magnetohydrodynamics. Astable extended crystal MHD-PDE’s is obtained where in its smooth solu-tions do not occurr unstabilities in finite times. These results are consideredfirst for systems without body energy source and after by introducing alsoa contribution by energy source in order to take into account of nuclear en-ergy production. A condition in order solutions satisfy the second principleof thermodynamics is given.

(67) A. Prastaro, On the extended crystal PDE’s stability.I: The n-d’Alembertextended crystal PDE’s, Appl. Math. Comput. 204(1)(2008), 63–69.

Abstract.Our recent results on extended crystal PDE’s and geometrictheory on PDE’s stability, are applied to the generalized n-d’AlembertPDE’s, (d′A)n, n ≥ 2. We prove that these are extended crystal PDE’sfor any n ≥ 2. For suitable n, (d′A)n becomes an extended 0-crystal PDEand also a 0-crystal PDE. An equation, having all the same smooth solu-tions of (d′A)n, but without unstabilities at “finite time” is obtained foreach n ≥ 2.

(68) A. Prastaro, On the extended crystal PDE’s stability.II: Entropy-regular-solutions in MHD-PDE’s, Appl. Math. Comput. 204(1)(2008), 82–89.

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Abstract. Local and global existence theorems of entropy-regular-solutionsin the geometric framework of MHD-PDE’s, recently introduced by A.Prastaro,are given. Stability properties of such solutions are studied. In particularit is shown how to stabilize the smooth entropy-regular-solutions, in orderto avoid finite-times unstabilities.

(69) A. Prastaro, On quantum black-hole solutions of quantum super Yang-Millsequations, Proceedings Dynamic Systems Appls. 5(2008), 407–414. (Eds.G. S. Ladde, N. G. Madhin C. Peng & M. Sambandham), Dynamic Pub-lishers, Inc., Atlanta, USA. ISBN: 1-890888-01-6.

Abstract.The category Q, (resp. QS), of quantum manifolds, (resp.quantum supermanifolds), introduced by A.Prastaro, gives a natural frame-work where implement a geometric theory of quantum (super) partial dif-ferential equations (PDE’s). The interest for quantum supermanifolds ismotivated by the fact that these structures allow us to describe the uni-fication of all the four fundamental forces (gravity, electromagnetic, weaknuclear, strong nuclear), at the quantum level. In this note we presentsome recent developments in this direction. In particular we will considerquantum black holes as solutions of quantum super Yang-Mills equations.These interpret very high energy level production of particles, where theeffects of strong-quantum-gravity become dominant.

(70) A. Prastaro, Surgery and bordism groups in quantum partial differentialequations.I: The quantum Poincare conjecture, Nonlinear Anal. TheoryMethods Appl. 71(12)(2009), 502–525.

Abstract. In this work, in two parts, we continue to develop the geomet-ric theory of quantum PDE’s, introduced by us starting from 1996. (Thesecond part is quoted in ref.[71].) This theory has the purpose to build arigorous mathematical theory of PDE’s in the category DS of noncommu-tative manifolds (quantum (super)manifolds), necessary to encode physicalphenomena at microscopic level (i.e., quantum level). Aim of the presentpaper is to report on some new issues in this direction, emphasizing aninterplaying between surgery, integral bordism groups and conservationslaws. In particular, a proof of the Poincare conjecture, generalized to thecategory DS , is given by using our geometric theory of PDE’s just in sucha category.

(71) A. Prastaro, Surgery and bordism groups in quantum partial differentialequations.II: Variational quantum PDE’s, Nonlinear Anal. Theory Meth-ods Appl. 71(12)(2009), 526–549.

Abstract.This is the second part of a work devoted to the interplay be-tween surgery, integral bordism groups and conservation laws, in order tocharacterize the geometry of PDE’s in the category QS of quantum (su-per)manifolds. (Part I is quoted in ref.[70].) In this paper we will considervariational problems, in the category QS , constrained by partial differentialequations. We get theorems of existence for local and global solutions. Thecharacterization of global solutions is made by means of integral bordismgroups. Applications to some important examples of the MathematicalPhysics, as quantum super-black-hole solutions of quantum super Yang-Mills equations, are discussed in some details. Quantum supermanifolds

44 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

allow us to unify, at the quantum level, the four fundamental forces, (grav-itational, electromagnetic, weak-nuclear, strong-nuclear), in an unique geo-metric structure. The geometric theory of PDE’s, built in the category QS

of quantum supermanifolds, gives us the right mathematic tool to describequantum phenomena also at very high energy levels, where quantum-gravitybecomes dominant.

(72) R. P. Agarwal & A. Prastaro, Singular PDE’s geometry and boundary valueproblems, J. Nonlinear Conv. Anal. 9(3)(2008), 417–460.

Abstract. Local and global existence theorems for boundary value prob-lems in singular PDE’s are considered. In particular, surgery techniques andintegral bordism groups are utilized, following previous works by A.Prastaroon PDE’s, in order to build global solutions crossing also singular pointsand to study their stability properties.

(73) R. P. Agarwal & A. Prastaro, On singular PDE’s geometry and boundaryvalue problems, Appl. Anal. 88(8)(2009), 1115-1131.

Abstract.A geometric formulation of singular PDE’s is considered. Surgerytechniques and integral bordism groups are utilized, following previousworks by A.Prastaro on PDE’s, in order to build global solutions cross-ing also singular points and to study their stability properties. A detailedproof on the integral characterization of singular ODE’s is given.

(74) A. Prastaro, Extended crystal PDE’s. Mathematics Without Boundaries:Surveys in Pure Mathematics. P. M. Pardalos and Th. M. Rassias (Eds.)Springer-Heidelberg New York Dordrecht London, (to appear).

arXiv:0811.3693[math.AT].Abstract. In this paper we show that between PDE’s and crystallographic

groups there is an unforeseen relation. In fact we prove that integral bor-dism groups of PDE’s can be considered extensions of crystallographic sub-groups. In this respect we can consider PDE’s as extended crystals. Then analgebraic-topological obstruction (crystal obstruction), characterizing exis-tence of global smooth solutions for smooth boundary value problems, isobtained. Applications of this new theory to the Ricci-flow equation andNavier-Stokes equation are given that solve some well-known fundamentalproblems. These results, are also extended to singular PDE’s, introducing(extended crystal singular PDE’s). An application to singular MHD-PDE’s,is given following some our previous results on such equations, and showingexistence of (finite times stable smooth) global solutions crossing criticalnuclear energy production zone.

(75) A. Prastaro, Quantum extended crystal PDE’s, Nonlinear Studies 18(3)(2011),447–485. arXiv:1105.0166[math.AT].

Abstract.Our recent results on extended crystal PDE’s are generalized toPDE’s in the category QS of quantum supermanifolds. Then obstructionsto the existence of global quantum smooth solutions for such equations areobtained, by using algebraic topologic techniques. Applications are consid-ered in details to the quantum super Yang-Mills equations. Furthermore,our geometric theory of stability of PDE’s and their solutions, is also gen-eralized to quantum extended crystal PDE’s. In this way we are able toidentify quantum equations where their global solutions are stable at finite

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times. These results, are also extended to quantum singular (super)PDE’s,introducing (quantum extended crystal singular (super) PDE’s).

(76) A. Prastaro, Quantum extended crystal super PDE’s. Nonlinear Anal. RealWorld Appl. 13(6)(2012), 2491–2529. arXiv:0906.1363[math.AT].

Abstract.We generalize our geometric theory on extended crystal PDE’sand their stability, to the category QS of quantum supermanifolds. Byusing algebraic topologic techniques, obstructions to the existence of globalquantum smooth solutions for such equations are obtained. Applicationsare given to encode quantum dynamics of nuclear nuclides, identified withgraviton-quark-gluon plasmas, and study their stability. We prove thatsuch quantum dynamical systems are encoded by suitable quantum ex-tended crystal Yang-Mills super PDE’s. In this way stable nuclear-chargedplasmas and nuclides are characterized as suitable stable quantum solutionsof such quantum Yang-Mills super PDE’s. An existence theorem of localand global solutions with mass-gap, is given for quantum super Yang-Mills

PDE’s, (YM), by identifying a suitable constraint, (Higgs) ⊂ (YM), Higgsquantum super PDE, bounded by a quantum super partial differential rela-

tion (Goldstone) ⊂ (YM), quantum Goldstone-boundary. A global solution

V ⊂ (YM), crossing the quantum Goldstone-boundary acquires (or loses)mass. Stability properties of such solutions are characterized.

(77) A. Prastaro, Exotic heat PDE’s, Commun. Math. Anal. 10(1)(2011),64–81. arXiv:1006.4483[math.GT].

Abstract.Exotic heat equations that allow to prove the Poincare conjec-ture, some related problems and suitable generalizations too are considered.The methodology used is the PDE’s algebraic topology, introduced by A.Prastaro in the geometry of PDE’s, in order to characterize global solutions.

(78) A. Prastaro, Exotic heat PDE’s.II. Essays in Mathematics and its Appli-cations. In Honor of Stephen Smale’s 80th Birthday. P. M. Pardalos andTh. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London(2012), 369–419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0 (Online).DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176[math.AT].

Abstract.Exotic heat equations that allow to prove the Poincare conjec-ture and its generalizations to any dimension are considered. The method-ology used is the PDE’s algebraic topology, introduced by A. Prastaro inthe geometry of PDE’s, in order to characterize global solutions. In partic-ular it is shown that this theory allows us to identify n-dimensional exoticspheres, i.e., homotopy spheres that are homeomorphic, but not diffeomor-phic to the standard Sn.

(79) A. Prastaro, Exotic n-d’Alembert PDE’s and stability. Nonlinear Analysis.Stability, Approximation and Inequalities. Series: Springer Optimizationand its Applications Vol 68. P. M. Pardalos, P. G. Georgiev and H. M.Srivastava (Eds.). Springer Optimization and its Applications Volume 68(2012), 571–586. ISBN 978-1-4614-3498-6. arXiv:1011.0081[math.AT].

Abstract. In the framework of the PDE’s algebraic topology, previouslyintroduced by A. Prastaro, exotic n-d’Alembert PDE’s are considered. Theseare n-d’Alembert PDE’s, (d′A)n, admitting Cauchy manifolds N ⊂ (d′A)nidentifiable with exotic spheres, or such that ∂N , can be exotic spheres. For

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such equations local and global existence theorems and stability theoremsare obtained.

(80) A. Prastaro, Exotic PDE’s. Mathematics Without Boundaries: Surveysin Interdisciplinary Research. P. M. Pardalos and Th. M. Rassias (Eds.)Springer-Heidelberg New York Dordrecht London, (to appear).

arXiv:1101.0283[math.AT].Abstract. In the framework of the PDE’s algebraic topology, previously

introduced by A. Prastaro, are considered exotic differential equations, i.e.,differential equations admitting Cauchy manifolds N identifiable with ex-otic spheres, or such that their boundaries ∂N are exotic spheres. For suchequations are obtained local and global existence theorems and stabilitytheorems. In particular the smooth (4-dimensional) Poincare conjecture isproved. This allows to complete the previous Theorem 4.59 in [22] also forthe case n = 4.

(81) A. Prastaro, Quantum exotic PDE’s. Nonlinear Anal. Real World Appl.14(2)(2013), 893–928. arXiv:1106.0862[math.AT].

Abstract. Following the previous works on the A. Prastaro’s formula-tion of algebraic topology of quantum (super) PDE’s, it is proved that acanonical Heyting algebra (integral Heyting algebra) can be associated toany quantum PDE. This is directly related to the structure of its globalsolutions. This allows us to recognize a new inside in the concept of quan-tum logic for microworlds. Furthermore, the Prastaro’s geometric theoryof quantum PDE’s is applied to the new category of quantum hypercom-plex manifolds, related to the well-known Cayley-Dickson construction foralgebras. Theorems of existence for local and global solutions are obtainedfor (singular) PDE’s in this new category of noncommutative manifolds.Finally the extension of the concept of exotic PDE’s, recently introducedby A.Prastaro, has been extended to quantum PDE’s. Then a smoothquantum version of the quantum (generalized) Poincare lemma is giventoo. These results extend ones for quantum (generalized) Poincare lemma,previously given by A. Prastaro.

(82) A. Prastaro, Strong reactions in quantum super PDE’s. I: Quantum hyper-complex exotic super PDE’s.

arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.)Abstract. In order to encode strong reactions of the high energy physics,

by means of quantum nonlinear propagators in the Prastaro’s geometrictheory of quantum super PDE’s, some related geometric structures are fur-ther developed and characterized. In particular super-bundles of geometricobjects in the category QS of quantum supermanifolds are considered andquantum Lie derivative of sections of super bundle of geometric objects arecalculated. Quantum supermanifolds with classic limit are classified withrespect to the holonomy groups of these last commutative manifolds. Atheorem characterizing quantum super manifolds with structured classiclimit as super bundles of geometric objects is obtained. A theorem on thecharacterization of chi-flow on suitable quantum manifolds is proved. Thissolves a previous conjecture too. Quantum instantons and quantum soli-tons are defined are useful generalizations of the previous ones, well-knownin the literature. Quantum conservation laws for quantum super PDEs are

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characterized. Quantum conservation laws are proved work for evaporatingquantum black holes too. Characterization of observed quantum nonlinearpropagators, in the observed quantum super Yang-Mills PDE, by means ofconservation laws and observed energy is obtained. Some previous resultsby A. Prastaro about generalized Poincare conjecture and quantum exoticspheres, are generalized to the category Qhyper,S of hypercomplex quantumsupermanifolds. (This is the first part of a work divided in two parts. Forpart II see [83].)

(83) A. Prastaro, Strong reactions in quantum super PDE’s. II: Nonlinear quan-tum propagators.

arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.)Abstract. In this second part, of a work devoted to encode strong re-

actions of the high energy physics, in the algebraic topologic theory ofquantum super PDE’s, (previously formulated by A. Prastaro), decompo-sition theorems of integral bordisms in quantum super PDEs are obtained.(For part I see [82].) In particular such theorems allow us to obtain rep-resentations of quantum nonlinear propagators in quantum super PDE’s,by means of elementary ones (quantum handle decompositions of quantumnonlinear propagators). These are useful to encode nuclear and subnuclearreactions in quantum physics. Prastaro’s geometric theory of quantumPDE’s allows us to obtain constructive and dynamically justified answersto some important open problems in high energy physics. In fact a Regge-type relation between reduced quantum mass and quantum phenomenolog-ical spin is obtained. A dynamical quantum Gell-Mann-Nishijima for-mula is given. An existence theorem of observed local and global solu-tions with electric-charge-gap, is obtained for quantum super Yang-Mills

PDE’s, (YM)[i], by identifying a suitable constraint, (YM)[i]w ⊂ (YM)[i],quantum electromagnetic-Higgs PDE, bounded by a quantum super partial

differential relation (Goldstone)[i]w ⊂ (YM)[i], quantum electromagneticGoldstone-boundary. An electric neutral, connected, simply connected ob-

served quantum particle, identified with a Cauchy data of (YM)[i], it is

proved do not belong to (YM)[i]w. Existence of Q-exotic quantum non-

linear propagators of (YM)[i], i.e., quantum nonlinear propagators that donot respect the quantum electric-charge conservation is obtained. By us-ing integral bordism groups of quantum super PDE’s, a quantum crossingsymmetry theorem is proved. As a by-product existence of massive photonsand massive neutrinos are obtained. A dynamical proof that quarks can bebroken-down is given too. A quantum time, related to the observation of anyquantum nonlinear propagator, is calculated. Then an apparent quantumtime estimate for any reaction is recognized. A criterion to identify solutionsof the quantum super Yang-Mills PDE encoding (de)confined quantum sys-tems is given. Supersymmetric particles and supersymmetric reactions areclassified on the ground of integral bordism groups of the quantum super

Yang-Mills PDE (YM). Finally, existence of the quantum Majorana neu-trino is proved. As a by-product, the existence of a new quasi-particle, thatwe call quantum Majorana neutralino, is recognized made by means of two

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quantum Majorana neutrinos, a couple (νe, ˜νe), supersymmetric partner of(νe, νe), and two Higgsinos.

(84) A. Prastaro, Strong reactions in quantum super PDE’s. III: Exotic quantumsupergravity. arXiv:1206.4956[math.AT].

Abstract. of quantum super PDE’s, quantum nonlinear propagators in

the observed quantum super Yang-Mills PDE, (YM)[i], are further char-acterized. In particular, quantum nonlinear propagators with non-zerolost quantum electric-charge, are interpreted as exotic-quantum supergrav-ity effects. As an application, the recently discovered bound-state calledZc(3900), is obtained as a neutral quasi-particle, generated in a Q-quantumexotic supergravity process. Quantum entanglement is justified by meansof the algebraic topologic structure of quantum nonlinear propagators. Ex-

istence theorem for solutions of (YM)[i] admitting negative local tempera-tures (quantum thermodynamic-exotic solutions) is obtained too and relatedto quantum entanglement.

(85) A. Prastaro, The Landau’s problems. I: The Goldbach’s conjecture proved.arXiv:1208.2473[math.GM].Abstract.We give a direct proof of the Goldbach’s conjecture, (GC), in

number theory, in the Euler’s form. The proof is also constructive, since itgives a criterion to find two prime numbers ≥ 1, such that their sum gives afixed even number ≥ 2. The proof is obtained by recasting the problem inthe framework of the Commutative Algebra and Algebraic Topology. Evenif in this paper we consider 1 as a prime number, our proof of the GC worksalso for the restricted Goldbach conjecture, (RGC), i.e., by excluding 1 fromthe set of prime numbers.

(86) A. Prastaro, The Landau’s problems. II: Landau’s problems solved.arXiv:1208.2473[math.GM].Abstract.Three of the well known four Landau’s problems are solved in

this paper. (In [85] the proof of the Goldbach’s conjecture has been alreadygiven.)

(87) A. Prastaro, The Riemann hypothesis proved.arXiv:1305.6845[math.GM].Abstract.The Riemann hypothesis is proved by extending the zeta Rie-

mann function to a quantum mapping between quantum 1-spheres withquantum algebra A = C, in the sense of A. Prastaro [70, 81]. Algebraictopologic properties of quantum-complex manifolds and suitable bordismgroups of morphisms in the category QC of quantum-complex manifolds areutilized.

(88) A. Prastaro, Quantum Geometrodynamic Cosmology.(Submitted for publication.)

Abstract.By utilizing Prastaro’s quantum supergravity, it is proved thatthe Universe’s expansion at the Planck epoch is justified by the fact thatit is encoded by a quantum nonlinear propagator having thermodynamicquantum exotic components in its boundary. This effect produces also anincreasing of energy in the Universe at the Einstein epoch: Planck-epoch-legacy on the boundary of our Universe. This is the main source of theUniverse’s expansion and solves the problem of the non-apparent energy-matter (dark-energy-matter) in the actual Universe.

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ABSTRACTS OF MONOGRAPHS AND TEXTS

(89) A. Prastaro, Geometry of PDEs and Mechanics, World Scientific Publish-ing, River Edge, NJ, 1996, 760 pp.

Abstract. 1. Algebraic Geometry: Algebraic Complements, Affine spaces,Differential Manifolds, Grassmann Manifolds, Spectral Sequences. 2. Dif-ferential Equations (PDES): Geometry of Differential Equations, OrdinaryDifferential Equations (ODEs), Characteristics of PDEs, Affine PDEs andGreen Functions, Spectral Sequences in PDEs, Tunnel Effects in PDEs,Cobordism Groups in PDEs. 3. Mechanics: Structure of Galilean Space-Time, One-Body Dynamics, Important Formulas, Fundamental Theoremsof Dynamics, Lagrangian Mechanics for Perfect Holonomic Systems, Rigid-Body Dynamics. 4. Continuum Mechanics: Flow, Stress Tensor and Mo-ment of Stress Tensor, Local Dynamic Equations, Thermodynamics of Con-tinuum Media, Rheological Classification of Materials, Rheoptics, Multi-component Continuum Systems, Variational Field Theory. 5. QuantumField Theory: Locally Convex Manifolds and Derivative Spaces, Differen-tial Geometry of Quantum Situs, Mathematical Logic and Quantizationof PDEs, Formal and Dirac Quantizations of PDEs, Canonical Quantiza-tion of PDEs. 6. Geometry of Quantum PDEs: Differential Geometry ofQuantumManifolds, Cohomology of QuantumManifolds, Formal Theory ofQuantum PDEs, Cartan Spectral Sequences of Quantum PDEs, QuantumDistribution Solutions and Singular Solutions of Quantum PDEs, TunnelEffects in Quantum PDEs, Quantum PDEs Non-holonomic Connections,Gauge Quantum PDEs, Supergravity Quantum PDEs.

(90) A. Prastaro, Elementi di Meccanica Razionale, VII edizione, Aracne Ed-itrice, Roma, 2010,446 pp.

Abstract. 1. This monography is addressed to Italian university studentsin Mathematics, Physics and Engineering. It develops with a modern geo-metric language the methods of classical mechanics and geometry of (par-tial) differential equations. The presentation, even if elementary, gives theactual mathematics situation in classical mechanics.

Indice. Algebra: matrici ed applicazioni lineari; prodotto tensorialefra spazi vettoriali; tensori; spazi vettoriali Euclidei; componenti covari-anti e controvarianti; tensori simmetrici ed antisimmetrici; orientazione dispazi vettoriali; gruppi GL(V ), O(V ), SO(V ); equazione agli autovalori;spazio affine; gruppo affine; soluzioni di equazione affine. 2. Equazionidifferenziali: varieta differenziale; spazio tangente; campi tensoriali e loroimmagine reciproca; forma volume ed orientazione di varieta differenziale;integrazione su varieta differenziale; varieta Riemanniana; gruppi di Lie eloro caratterizzazione tramite costanti di struttura; elementi di teoria geo-metrica delle equazioni differenziali. 3. Connessioni differenziali: derivatacovariante; connessione di Levi-Civita; curvatura di una connessione; com-plesso di de Rham e gruppi di comologia; importanti operatori differen-ziali su varieta Riemanniane; componenti fisiche di oggetti geometrici. 4.Spazio-tempo Galileiano: struttura dello spazio-tempo Galileiano; moto;velocita ed accelerazione del moto; osservatore e moto osservato; gruppo

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di Galileo; formule di Frenet; espressione della velocita ed accelerazionetramite prametro di linea; teorema di Coriolis; osservatori inerziali ed os-servatori rigidi; moti rigidi e teorema fondamentale della cimematica deicorpi rigidi; angoli di Euler; precessioni; traiettorie polari (base e ruletta).5. Dinamica di un elemento materiale: vincoli di ordine 0 ≤ n ≤ 2; forze edequazione di Newton; forze conservative; equazione di Lagrange; equazionedi Hamilton e campo vettoriale di Euler; struttura simplettica della mec-canica; simmetrie dinamiche; teorema di Noether generalizzato; sistemi di-namici con un numero finito N di particelle; vincoli con attrito. 6. Teoremifondamentali della dinamica: teorema dell’impulso; teorema della conser-vazione dell’impulso; momento totale e sue proprieta; momento assiale;coppia; lavoro; potenza; energia cinetica; stabilita dell’equilibrio; equazionicardinali della dinamica; teorema di conservazione; principio dei lavori vir-tuali; teorema di Koenig. 7. Meccanica Lagrangiana: Equazione di La-grange e sue forme particolari. Potenziali generalizzati e forza di Lorentz;stabilita ed equazioni di Lagrange linearizzate. Leggi di conservazione ecoordinate ignorabili; equazione di Lagrange e calcolo variazionale; con-figurazioni di stato stazionario e loro stabilita. 8. Meccanica dei sistemirigidi: baricentri di sistemi continui; tensore momento d’inerzia; equazionicardinali per corpo rigido; moto alla Poinsot e sue leggi di conservazione. 9.Esercizi: (ventuno esercizi completamente risolti). 10. Meccanica dei con-tinui: flusso; osservatore proprio di un sistema continuo; oggetti geometriciassociati ad un flusso; tensore degli sforzi e suo momento; equazioni diEuler; equazione dinamica dei sistemi continui; termodinamica covariantedei sistemi continui; classificazione reologica dei sistemi continui; esempi diflussi e deformazioni. 11. Reottica. Bibliografia. 12. Equazioni di Maxwelle relativita ristretta. 13. Esercizi complementari (completamente svolti):Esercizi di Geometria; Esercizi di Meccanica dei sistemi rigidi; Esercizi diMeccanica dei sistemi continui. Indice analitico. Indice dei simboli.

(91) A. Prastaro, Quantized Partial Differential Equations, World Scientific Pub-lishing, River Edge, NJ, 2004, 500 pp.

Abstract.This book contains three chapters and two addenda. Quan-tized PDE’s.I. In this first part we consider quantum (super) manifolds astopological spaces locally identified with open sets of some locally convextopological vector spaces built starting from suitable topological algebrasA, quantum (super)algebras. The noncommutative character of such quan-tum (super)manifolds is given by the underlying noncommutative algebrasA. In fact, here A plays the role of “fundamental algebra of numbers”, likeK = R,C does for usual commutative manifolds. Therefore, quantum (su-per)manifolds are the natural generalizations of manifolds , when one sub-stitutes commutative numbers with noncommutative ones. Commutativemanifolds are contained into quantum (super)manifolds, as quantum (su-per)algebras A are required to containK. This aspect is also reflected by thefact that the class of differentiability Qk

w for pseudogroup structures defin-ing quantum (super)manifolds, contains the usual Ck differentiability formanifolds. In fact, the class of differentiability of such topological manifoldsis defined by requiring weak differentiability and Z-linearity of the deriva-tives, where Z is the center of the underlying quantum (super)algebras.

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We give (co)homological characterizations of quantum (super)algebras andquantum (super)manifolds, by applaying to these noncommutative topo-logical manifolds standard methods of algebraic topology. In particular, wecalculate also (co)bordism groups in quantum (super)manifolds. QuantizedPDE’s.II. Here we give a geometric theory of PDE’s in the category of quan-tum (super)manifolds. This theory is the natural extension of the geometrictheory of PDE’s in the category of commutative (super)manifolds. Empha-sis is put on some new algebraic topological techniques that allow us to cal-culate the integral (co)bordism groups of quantum (super)PDE’s, hence tocharacterize global properties of solutions of quantum (super)PDE’s. Manyapplicatio ns to important equations of quantum field theory are consideredalso. Quantized PDE’s.III. Here we consider a process that allows us to as-sociate to a (super)PDE, defined in the category of (super)commutativemanifolds, a quantum (super)PDE. This process is the covariant quantiza-tion. We describe it in some steps. In fact, we first define quantizationsof PDE’s in the framework of the mathematical logic, by means of evalua-tions of the logic of a PDE Ek, that is the Boolean algebra of subsets of theclassic limit Ω(Ek)c of the quantum situs Ω(Ek) of Ek, into quantum logicsA ⊂ L(H), that are algebras of (self-adjoint) operators on a locally convextopological vector (Hilbert) space H, in such a way to define (pre-)spectralmeasures on Ω(Ek)c: Ω(Ek)c→L(H). We show that these quantizationscan be obtained by means of a geometric process called covariant quan-tization, (or canonical quantization), of PDE’s, that is, roughly speakingthe covariant quantization observed by a physical frame. In fact, in apurely geometric context, we prove that any physical observable deformsthe classical PDE, Ek ⊂ JDk(W ), around its solutions. In this way we canassociate to the Lie filtered (super)algebra of the (super)classical observ-

ables, B, of Ek, a filtered quantum (super)algebra B, defined by means

of distributive kernels, Gq, propagators, canonically associated to Ek. Wecharacterize also the propagators of PDE’s by means oftheir integral bor-dism groups. The final step is the relation between the formal propertiesE∞ · · · → Ek+1 → Ek → · · · of the classical equation Ek, with quantum

ones E∞ · · · → Ek+1 → Ek → · · · . These are obtained in the category

of QPDE’s, where the quantum (super)algebra B, so obtained as covari-ant quantization of Ek, identifies a quantum (super)PDE. Addendum I. Inrefs.[38, 41] are calculated, for the first time, the integral bordism groupsof the 3D nonisothermal Navier-Stokes equation (NS). A direct conse-quence of these results is the proof of existence of global (smooth) solutionsfor (NS). Here we go in some further results emphasizing surgery tech-niques that allow us to better understand this geometric proof of existenceof (smooth) global solutions for any (smooth) boundary condition. Adden-dum II. In the framework of the geometry of PDE’s, we classify variationalequations of any order with respect to their formal properties. A vari-ational sequence is introduced for constrained variational PDE’s that ex-tends previous ones for variational calculus on fiber bundles. Such extendedvariational sequence allows us to locally and globally solve variational prob-lems, constrained by PDE’s of any order, Ek ⊂ Jk

n(W ), by means of some

52 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

cohomological properties of Ek. Moreover, we relate constrained varia-tional PDE’s to the integral (co)bordism groups for PDE’s. In this waywe are able to characterize the structure properties of global solutions ofconstrained variational PDE’s and to relate them to the structure of globalsolutions for the corresponding constraint equations. Contents: QuantizedPDE’s.I. Noncommutative Manifolds: Algebraic topology; Quantum alge-bras; Quantum manifolds; Quantum supermanifolds. Quantized PDE’s.II.Noncommutative PDE’s: Quantum PDE’s; The quantum Navier-Stokesequation; Quantum super PDE’s; The quantum super Yang-Mills equa-tions. Quantized PDE’s.III. Quantizations of commutative PDE’s: Inte-gral (co)bordism groups in PDE’s; Algebraic geometry of PDE’s; Spectralmeasures of PDE’s; Quantizations of PDE’s; Covariant and canonical quan-tizations of PDE’s. Addendum I: Bordism groups and the (NS)-problem.Addendum II: Bordism groups and variational PDE’s. References. Index.

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ABSTRACTS OF BOOKS (EDITOR AND COAUTHOR)

(92) A. Prastaro, Geometrodynamics Proceedings 1983, Pitagora Ed., Bologna1984.

Abstract.The Hamilton-Jacobi Equation for a Hamiltonian Action: S.Benenti. Twisteurs Sans Twisteurs: A. Crumeyrolle. General Covarianceand Minimal Gravitational Coupling in Newtonian Space-Time: H. P. Kun-zle. Canonical Cartan Equations for Higher Order Variational Problems:J. M. Masque. Backlund Problem and Group Theory: J. F. Pommaret.Group Structure of Non-linear Field Theories: J. F. Pommaret. Geometryand Existence Theorems for Incompressible Fluids: A. Prastaro. Relativis-tic Hydrodynamics as a Symplectic Field Theory: W. M. Tulczyjew.

(93) A. Prastaro, Geometrodynamics Proceedings 1985, World Scientific Pub-lishing, Singapore 1985.

Abstract.A Geometrical Interpretation of the 1-cocycles of a Lie Group:S. Benenti, W. M. Tulzyjew. Supermanifolds and Supergravity: Y. Choquet-Bruhat. Self-dual Yang-Mills Fields and the Penrose Trasnform: A. Crumey-rolle. On Smooth and Analytic Functions in Gauge Field Theory: J. Czyz.An Application of Topological Methods to the Study of Periodic Solutions ofHamiltonian Systems: G. F. Dell’Antonio. Introducing Spinors, Isospinors,etc. in Globally Nontrivial Space-times: L. Dabrowski. The Radon Trans-form on Compact Symmetric Spaces: H. Goldschmidt. Unconstrained De-grees of Freedom of Gravitational Field and the Positivity of GravitationalEnergy: J. Kijowski. Polynomial Identities Satisfied by Realizations ofLie Algebras: M. Iosifescu, H. Scutaru. Free Motions in MultidimensionalUniverses: G. Marmo. Harmonically Immersed Lorentz Surfaces: T. Mil-nor. On the Symmetry Properties of Constrained Hamiltonian System:M. Mintchev. On a Property of Higher Order Poincare-Cartan Forms inthe Constructive Approach: J. M. Masque. Covariant Canonical Formal-ism for Gravity Theories: J. E. Nelson, T. Regge. Invariant DifferentialTechniques: A. Nijenhuis. A Utiyama Type Theorem in the C-K-S- GaugeApproach to Gravity: A. Perez-Rendon. Dynamic Conservation Laws: A.Prastaro. The Doulbeault-Kostant Complex and Geometric Quantization:M. Puta. Symplectic Origin of Some Properties of Generally CovariantField Theories: A. Smolski. Symplectic Scattering Theory: S. Sternberg.

(94) A. Prastaro & Th. M. Rassias, Geometry in Partial Differential Equations,World Scientific Publishing, River Edge, NJ, 1994.

Abstract. Some Applications of the Corea Formula to Partial DifferentialEquations: F. Bethuel, J.-M. Chidaglia. Large Solutions for the Equationof Surfaces of Prescribed Mean Curvature: F. Bethuel, O. Rey. OpticalHamiltonian Functions: M. Bialy, L. Polterovich. On the Geometry ofthe Hodge-de Rham Laplace Operator: M. Craioveanu, M. Puta, Th. M.Rassias. Minimal Surfaces in Economic Theory: J. Donato. The MorimotoProblem: B. Doubrov, A. Hushner. Asymptotic Expansions in Spectral Ge-ometry: P. B. Giley. Deformations and Recursion Operators for EvolutionEquations: I. S. Krasil’shchik, P. H. M. Kersten. Geometric Hamiltonian

54 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

Forms for the Kadomtsev-Petviashvili and Zabolotskaya-Khokhlov Equa-tions, B. A. Kupershmidt. Classification of Mixed Type Momnge-AmpereEquations: A. Kushner. Non-Holonomic Filtration: Algebraic and Geo-metric Aspects of Non-Integrability: V. Lychagin, V. Rubtsov. SpencerCohomologies: V. Lychagin, L. Zilbergleit. Hawking’s Relation via FourierIntegral Operators: P. E. Parker. Geometry of Super PDE’s: A. Prastaro.On a Geometric Approach to an Equation of J.d’Alembert: A. Prastaro,Th. M. Rassias. Geometric Prequantization of the Einstein’s VacuumField: M. Puta. On Differential Equations and Cartan’s Projective Con-nections: Y. R. Romanovsky. Smooth Marginal Analysis of Bifurcationof Extremals: Yu I. Sapronov. On the Schrodinger Equation for an N -Electron Atom: C. S. Sharma. Higher Symmetries and Conservation Lawsof Euler-Darboux Equations: V. E. Shemarulin. Strings and Menbranes:K. S. Stelle. Methods for Solving Two-Dimensional Nonstationary MHDEquations at Small Alfven-Mach Numbers: V. S. Titov.

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ABSTRACTS OF PATENTS

(95) A. Prastaro et al., 28/5/1975 - No 23.797 A/75. Process of production offibrous structures with high degree of birefringence.

Abstract.The present invention refers to a process to produce fibrousstructures having an high degreee of monoaxial orientation, by means ofcontrolled extrusion of solutions, emulsions, suspensions of fibrogenous ther-moplastic polymers. In particular, one fixes the optimum conditions of theextrusor design in order to obtain a birefringence higher than 0.1 10−4.

(96) A. Prastaro et al., 11/7/1975 - No 25.334 A/75. Process of production ofsynthetic polymers by means of flash-spinning of polymers solutions.

Abstract.The present invention refers to a process to produce fibrousstructures of synthetic polymers, in the form of plexus-filament of singlelittle fibers, by means of the technique of the “flash-spinning” of polymerssolutions. In particular, one fixes the optimum thermodynamical condi-tions to obtain plexus-filaments that extend ranges previously found byUSA-patents. This has been possible by using a new mathematical modelof flash-spinning, purposely formulated, and put on informatic support. Ofparticular importance has been the discovery of metastable states in theextruded solution, similar to ones used in the bubble-chambers for dedect-ing sub-atomic collisions. These thermodynamical conditions, beside somesuitable dynamical conditions, allow us to get optimum control conditionsin such a process.

56 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

FIG. 8. Prastaro’s mathematicians poster 2005-2006.

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Fig. 9. Prastaro’s mathematicians poster 2007-2008.

58 CURRICULUM VITAE AGOSTINO PRASTARO – APRIL 2014

Fig. 10. Prastaro’s mathematicians poster 2009-2012.

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5. INDEX

General Informations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Summary of Principal results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Papers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Monographs and Texts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Books (Editor and Coauthor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Patents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Abstracts of Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Abstracts of Papers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Abstracts of Monographs and Texts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Abstracts of Books (Editor and Coauthor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Abstracts of Patents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Photos and Posters.Title and A.Prastaro’s picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1University Address. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Home Address. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Acknowledgments (Prastaro’s mathematicians poster 2009-2013). . . . . . . . .3School of Engineering, University of Roma La Sapienza. . . . . . . . . . . . . . . . . . 6Florida Institute of Technology, Melbourne, FL - USA. . . . . . . . . . . . . . . . . .20ICM 2006 SC: Advances in PDE’s Geometry, Madrid - poster. . . . . . . . . . 26Prastaro’s mathematicians poster 2005-2006. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Prastaro’s mathematicians poster 2007-2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Prastaro’s mathematicians poster 2009-2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Prastaro’s mathematicians poster 2013-2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Roma - Spagna Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Fig. 11. Roma - Spagna Square.

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6. CONTENTS

1 - General Informations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 - Summary of Principal Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 - Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .214 - Abstracts of Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275 - Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Curriculum Vitae Agostino Prastaro - Edition April 2014