advanced studies in mathematics-prezentare...

FACULTATEA DE MATEMATICĂ ŞI INFORMATICĂ STR. ACADEMIEI NR.14, SECTOR 1, C.P. 010014, BUCUREȘTI, ROMÂNIA TEL: (4-021) 314 3508, FAX: (4-021) 315 6990, E-MAIL: [email protected]

CURRICULUM OF THE PROGRAM

UNIVERSITY OF BUCHAREST FACULTY OF MATHEMATICS Master Study Domain: Mathematics Master Study Program: Advanced Studies in Mathematics Type of studies: full-time Duration of studies - 4 semesters / 120 ECTS

The master program Advanced Studies in Mathematics is designed to provide better training for future teachers and mathematics researchers. As a consequence, the curriculum develops in the direction of ensuring good specialized training (through mathematical content courses) as well as training in the field of mathematical research.

Total (from all years of study) conventional course hours and laboratory / seminar: 2116.4 conventional hours (1627.6 = 520 × 3.13 conventional course hours and 488.8 = 260 × 1.88 conventional hours seminar / laboratory).

sem.

1 sem.

2 sem.

3 sem.

4 Total

conventional hours

% Hours from the curriculum

Synthesis / Fundamental Disciplines

course 5 - - - 219,1 14%

seminar 3 - - - 78,96 In-depth / specialization disciplines

course 5 10 - - 657,3 40%

seminar 2 5 - - 184,24

Supplementary disciplines course - - 10 10 751,2 46% seminar - - 5 5 225,6

The ratio of the number of course hours and laboratory / seminar in all years of study:

260/520 = 0.5 Total number of credits: 120 credits Professional practices, documentation / training, laboratory experiences, etc. (form of organization, number of hours allocated, venue, etc.): Each student has 2 hours / week of research practice per semester in the curriculum. This will be done under the guidance of a teacher who will recommend scientific articles, chapters in monographs, etc. for study and debate. The study may take place in the library of the faculty or at the Institute of Mathematics, the Research Institute of the University of Bucharest, or in the form of internships in various specialized companies. Comments and clarifications on the curriculum a. Elective course specifications - To choose elective courses for the third and fourth semesters, each student presents a list of five options, selected from the list of optional courses related to these two semesters. - All the courses in the list are ordered in descending order of the number of options. - Optional courses to be taught in semesters 3 and 4 will be the top five in each ordered list, respectively. - The proposed courses are independent, so the students can chose any five of them.


- The list of optional courses to be offered to the students can be completed in future, in order to better respond to the training needs of the becoming teacher-researchers. b. Specifications on scientific research - elaboration of dissertation paper

In the second year (semesters 3 and 4), there will be scientific seminars and tutorials with the aim of assisting the students in the elaboration and writing of the dissertation papers. Throughout the semester(s), the tutorial will address both aspects of the major stages of the research process (such as: selection of the research topic, review of the literature, etc.) as well as specific aspects of the presentation and editing of the paper. The tutorial will be sustained by all the scientific advisors of the dissertation papers. Staff list. The list includes: the name and surname of the teaching staff; didactic and scientific function; the institution to which the teacher is employed; scientific competences, proven by: graduated specialization, doctoral specialization, scientific papers published in the last 10 years. Curriculum of the program

Academic year 2018-2019 (First year of studies) – 60 ECTS Crt. no.

Mandatory courses 1st semester (14 weeks) 2nd semester (14 weeks)

C S EV ECTS C S EV ECTS

1 Ob.11. Introduction to Module Theory 1 1 E 4 - - - - 2 Ob.12. Lectures on Rings and Algebras 2 1 E 5 - - - -

3 Ob.13. Elements of Analysis and Fourier Analysis

2 1 E 6 - - - -

4 Ob.14. Introduction to Algebraic Topology 3 1 E 7 - - - - 5 Ob.15. Riemannian Geometry 2 1 E 6 - - - - 6 Ob.16. Research activity (practical training) - 2 V 2 7 Ob.21. Homological Algebra - - - - 2 1 E 6 8 Ob.22. Introduction to Algebraic Geometry - - - - 2 1 E 6 9 Ob.23. Groups and Representations - - - - 2 1 E 6

10 Ob.24. Number Theory - - - - 2 1 E 5

11 Ob.25. Complex Analysis and Riemann Surfaces

- - - - 2 1 E 5

12 Ob.26. Research activity (practical training) - - - - - 2 V 2

TOTAL 10 5 5E 1V

30 10 5 5E 1V

30

C = lecture/course; S = Practicals/Tutorials; Ob.xx = compulsory; Op.Xxx = elective; EV=Evaluation; E = exam; V = verification; ECTS = number of European transferable credits;

Academic year 2019-2020 (Second year of studies) – 60 ECTS Crt. no.

Elective courses 1st semester (14 weeks) 2nd semester (10 weeks)

C S EV ECTS C S EV ECTS

1 Op.31. Elective course 2 1 E 5 - - - - 2 Op.32. Elective course 2 1 E 5 - - - - 3 Op.33. Elective course 2 1 E 5 - - - -


4 Op.34. Elective course 2 1 E 5 - - - - 5 Op.35.Elective course 2 1 $ 5

6 Op.36. Research work for the Dissertation Thesis (practical training)

- 3 E 5 - - - -

7 Op.41. Elective course - - - - 2 1 E 5 8 Op.42. Elective course - - - - 2 1 E 5 9 Op.43. Elective course - - - - 2 1 E 5

10 Op.44. Elective course - - - - 2 1 E 5 11 Op.46. Elective course - - - - 2 1 E 5

12 Research work for the Dissertation Thesis (practical training)

- 3 V 5 - 3 V 5

TOTAL 10 5 5E 1V

30 10 5 5E 1V

30

C = lecture/course; S = Practicals/Tutorials; Ob.xx = compulsory; Op.Xxx = elective; EV=Evaluation; E = exam; V = verification; ECTS = number of European transferable credits;

List of elective courses Semester 3

1. Hopf Algebras 2. Lie Algebras 3. Special topics in category theory 4. Combinatorics in commutative algebra 5. Analytic Methods in Number Theory 6. Complex geometry 7. Algebraic Geometry 8. Elements of Free Probability 9. Several complex variables 10. Submanifolds of Riemannian Manifolds 11. Algebraic Number Theory 12. Differential Topology

Semester 4

1. An Introduction to Quantum Group Theory 2. Vector bundles and applications 3. Hyperbolic geometry 4. Representation theory of algebras 5. Algebraic groups 6. Introduction to quasi-regular Dirichlet forms 7. Valuations Theory and Local Fields 8. Variational Methods in Riemannian Geometry 9. Multiplicative Ideal Theory 10. Kaehler varieties 11. Computational methods in algebraic geometry


Planul de învățământ

PROGRAMUL „STUDII AVANSATE ÎN MATEMATICĂ”, 4 SEMESTRE Anul I (sem. 1 – 14 săptămâni, sem. 2 - 14 săptămâni )

Nr. crt.

Codul disc.

Disciplina

Regimul disc. Sem. 1 Sem. 2

Cont.1 Oblig.2 Nr. ore curs

Nr. ore aplicative sem/proiect

Forma de evaluare3

Nr. credite

Nr. ore curs


Forma de

evaluare

Nr. credite

1. Ob. 11 Introducere în teoria modulelor DF DI 1 1 E 4 - - - - 2. Ob. 12 Lecții de inele și algebre DF DI 2 1 E 5 3. Ob. 13 Elemente de analiză şi analiză

Fourier DF DI 2 1 E 6 - - - -

4. Ob. 14 Introducere în topologia algebrică

DS DI 3 1 E 7 - - - -

5. Ob. 15 Geometrie riemanniană DS DI 2 1 E 6 - - - - 6. Ob. 16 Practică de cercetare DF DI - 2 V 2 - - - -

TOTAL 10 5 5E, 1V 30 - - - - 7. Ob. 21 Algebră omologică DS DI - - - - 2 1 E 6 8. Ob. 22 Introducere în geometria

algebrică DS DI - - - - 2 1 E 6

9. Ob. 23 Grupuri şi reprezentări DS DI - - - - 2 1 E 6 10. Ob. 24 Teoria numerelor DS DI - - - - 2 1 E 5 11. Ob. 25 Analiză complexă şi suprafeţe

Riemann DS DI - - - - 2 1 E 5

12. Ob. 26 Practică de cercetare DF DI - - - - - 2 V 2 TOTAL - - - - 10 5 5E, 1V 30

1 Regimul disciplinei (conţinut) - DF (disciplină de sinteză/ fundamentală), DS (disciplină de aprofundare/ specializare), DC (disciplină complementară). 2 Regimul disciplinei (obligativitate) - se alege una din variantele: DI (disciplină obligatorie) / DO (disciplină opţională). 3 examen = E, verificare = V


Anul II (sem. 1 – 14 săptămâni, sem. 2 - 10 săptămâni )

Nr. crt.

Codul disc.

Disciplina

Regimul disc. Sem. 1 Sem. 2

Cont. Oblig. Nr. ore

curs


Forma de evaluare

Nr. credite

Nr. ore curs


Forma de evaluare

Nr. credite

13. Op. 31 Opţional DC DI 2 1 E 5 - - - - 14. Op. 32 Opţional DC DI 2 1 E 5 - - - - 15. Op. 33 Opţional DC DI 2 1 E 5 - - - - 16. Op. 34 Opţional DC DI 2 1 E 5 - - - - 17. Op. 35 Opțional DC DI 2 1 E 5 18. Op. 35 Practică de cercetare

în vederea elaborării lucrării de disertaţie

DF DI - 3 V 5 - - - -

TOTAL 10 5

5E 1V

30 - - - -

19. Op. 41 Opţional DC DI - - - - 2 1 E 5 20. Op. 42 Opţional DC DI - - - - 2 1 E 5 21. Op. 43 Opţional DC DI - - - - 2 1 E 5

22. Op. 44 Opţional DC DI - - - - 2 1 E 5 23. Op. 45 Opţional DC DI - - - - 2 1 E 5 24. Op. 46 Practică de cercetare

în vederea elaborării lucrării de disertaţie

DF DI - - - - - 3 V 5

TOTAL

- - - - 10 5

5E 1V

30


Lista cursurilor opţionale

Semestrul al 3-lea

1. Algebre Hopf 2. Algebre Lie 3. Capitole speciale de teoria categoriilor 4. Metode combinatorice în algebra comutativă 5. Metode Analitice în Teoria Numerelor 6. Geometrie complexă 7. Geometrie Algebrică 8. Elemente de probabilităţi libere 9. Funcţii complexe de mai multe variabile 10. Subvarietăţi ale varietăţilor Riemanniene 11. Teoria Algebrică a Numerelor 12. Topologie diferentiala

Semestrul al 4-lea

1. Introducere în teoria Grupurilor Cuantice 2. Fibraţi vectoriali şi aplicaţii 3. Geometrie hiperbolică 4. Teoria reprezentărilor de algebre 5. Grupuri algebrice 6. Introducere în teoria formelor Dirichlet quasi-regulate 7. Teoria valuării şi Corpuri locale 8. Metode Variaţionale în Geometria Riemanniană 9. Teoria multiplicativă a idealelor 10. Varietăţi Kaehler 11. Metode computaţionale în geometria algebrică


Courses sheets

Ob.11 Introduction to Module Theory

Name An Introduction to Module Theory

Code Ob.11

Year of study I Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF

Type{Ob – compulsory, Op- elective, F – optional} Ob ECTS 4 Total hours in curriculum 28 Total hours for

individual study 72 Total hours per

semester 100

Teacher(s) Prof. S. Dascalescu, Prof. G. Militaru

Faculty Mathematics and Computer Science

Total hours per semester in curriculum

Department Mathematics Main domain (sciences, art, culture)

Exact Sciences

Domain of master program

Mathematics Total C S L P

Program name Advanced Studies in Mathematics

28 14 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Algebra I, II;

Recommended

Estimated time (hours per semester) for the required individual study 1. Learning by using one’s own course notes

8 8. Preparation of presentations. 5

2. Learning by using manuals, lecture notes

8 9. Preparation for exam 15

3. Study of indicated bibliography 8 10. Consultations 5 4. Research in library 3 11. Field research 0 5. Specific preparation for practicals/tutorials

5 12. Internet research 5

6. Preparation of reports, small projects, homework

6 13. Other activities… 0

7. Preparation for quizzes 4 14. Other activities…. 0 TOTAL hours of individual study (/semester) = 72

General competences (mentioned in MSc program sheet)


2. Instrumental 1. Ability to use mathematical methods studied in various previous

courses for solving specific problems. 2. Ability to classify representations of certain classes of rings.

3. Attitudinal 1. to develop an interest for the field; 2. to realize the importance of modules (=representations) in modern

mathematics 3. to assume an ethical conduct in scientific research; 4. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture: 1. Rings: basic concepts and constructions (subrings, ideals, quotient

rings, direct products, Dedekind finite rings); examples of rings (endomorphism rings, finite and infinite matrix rings, free rings, monoidal rings, polynomial rings, Ore extensions, incidence rings); graded rings, filtrations on rings and the associated graded ring.

2. Modules: basic concepts and constructions (submodules, factor modules, isomorphism theorems, exact sequences); algebras over a commutative ring; bimodules; essential submodules; direct summands and indecomposable modules; direct products and direct sums; free modules and IBN rings; projective modules; injective modules, Baer’s lemma and Eckmann-Schopf’s Theorem; indecomposable decompositions of modules and Azumaya’s Theorem.

3. Special classes of rings and modules: semisimple modules, semisimple rings and Wedderburn-Artin’s Theorem; Noetherian (Artinian) rings and modules; Hilbert Basis Theorem; generalized triangular matrix rings; the Jacobson radical, semisimple rings and simple artinian rings; local rings; modules of finite length, the Jordan-Holder Theorem and the Krull-Schmidt Theorem. Tutorials :

1. Construction of several classes of rings. 2. Understanding whether certain modules are free (projective,

injective). 3. Examples of indecomposable decompositions. 4. Examples of rings and modules with finiteness properties.

Bibliography

1. J. Rotman, Advanced modern algebra. Graduate Studies în Mathematics, 114. American Mathematical Society, Providence, RI, 2010.

2. T. Y. Lam, Lectures on modules and rings, Graduate Texts în Math., Vol. 189, Springer Verlag, Berlin-Heidelberg-New York, 1998.


3. Lam, T. Y. Exercises în modules and rings. Problem Books în Mathematics. Springer, New York, 2007.

4. C. Nastasescu, Inele. Module. Categorii, Editura Academiei, 1978. Necessary scientific infrastructure

Library

Final mark is given by: Weight, in %

{Total=100%} - final exam results 60% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 20% - results to mid-term examination (oral, optional) 20% - scientific reports, symposium etc 0% - other activities (to be specified) ………………… 0% Final evaluation methods, E/V.

Written exam

Minimal requirements for mark 5 ( 10 point scale)

Requirements for mark 10 (10 point scale)

Correct solutions to indicated subjects (for mark 5) in final exam Average results to periodic/continuous testing.

Correct solutions to all subjects in final exam. Correct solutions to homework problems. Successful presentations of scientific reports. Good results to periodic/continuous testing.

Date: Lecturer(s) signature(s):

Prof. Sorin Dascalescu

Prof. Gigel Militaru


Ob.12 Lectures on Rings and Algebras

Name Lectures on Rings and Algebras

Code Ob.12


DF



semester 125

Teacher(s) Assoc. Prof. M. Vladoiu




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites

Required Algebra I, II;

Recommended












Specific competences

1.Knowledge and understanding 1. Knowledge and understanding of the basic properties of rings and

algebras, and their importance in mathematics. 2. Understanding fundamental techniques in commutative algebra.


courses for solving specific problems. 4. Ability to classify representations of certain classes of rings.

3. Attitudinal 5. to develop an interest for the field; 6. to realize the importance of modules (=representations) in modern


SYLLABUS

Lecture : 4. Tensor product of modules, bimodules. Tensor products of modules

over a commutative ring. Tensor products of algebras. The tensor algebra, the symmetric algebra and the exterior algebra of a module.

5. Monomial ideals and operations. Primary decomposition in Noetherian rings. Uniqueness theorem(s) for primary decomposition. Structure theorem of Artinian rings.

6. Integral extensions. Going Up and Going Down theorems. Krull dimension of a ring.

7. Noether normalization lemma. Hilbert’s Strong and Weak Nullstellensatz. Dimension theory for finitely generated K-algebras and finitely generated polynomial rings over noetherian rings.

8. Graded rings and modules. Hilbert function, Hilbert series and Hilbert polynomial. Hilbert-Serre’s theorem.

9. Filtrations. Artin-Rees lemma. The m-adic completion of a local ring. The Krull intersection theorem.

10. Krull’s Hauptidealsatz. Dimension theory for local noetherian rings. Krull-Chevalley-Samuel theorem.

11. Regular local rings. Equivalent characterizations.

Tutorials : 5. Computation of certain tensor products. 6. Operations with monomial ideals. Primary decompositions of

arbitrary monomial ideals. 7. Examples of integral extensions. Computation of Krull dimension

for quotients of polynomial rings modulo monomial ideals. 8. Applications of Noether normalization lemma.


9. Algorithm for computing the Hilbert series, function, polynomial and Krull dimension for quotient of a polynomial ring modulo monomial ideals.

10. Minimal (graded) free resolutions.

Bibliography

5. Bourbaki, N. Elements of Mathematics. Algebra. Chapter 8, Semi-simple modules and rings, Springer, Berlin, 2012.

6. Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Math, Springer-Verlag, 1995.

7. Herzog, J and Hibi, T Monomial ideals. GTM 260. Springer, 2010. 8. Atiyah, M.F., MacDonald, I.G. Introduction to Commutative

Algebra, Addison-Wesley, 1969. 9. Greuel, G.M., Pfister, G., A Singular Introduction to Commutative

Algebra, Springer, 2nd edition, 2008.

Necessary scientific infrastructure

Library



Written exam






Date: Lecturer(s) signature(s)

Assoc. Prof. Marius Vladoiu


Ob.13 Elements of Analysis and Fourier Analysis

Name Analysis Code Ob.13

Year of study I Semester 1 Assessment (E/V/C) E

Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF

Type{Ob – compulsory, Op- elective, F – optional} Ob ECTS 6

Total hours in curriculum 42 Total hours for individual study

108 Total hours per semester

150

Teacher(s) CS II dr. Cezar Joita

Faculty Mathematics and

Computer Science


Department Mathematics Main domain

(sciences, art, culture)

Exact Sciences




42 28 14


Required Algebra I, II; Point Set Topology (as in introductory courses of Mathematical Analysis or Manifolds)

Recommended

Differential Geometry on Manifolds





3. Study of indicated bibliography 14 10. Consultations 7

4. Research in library 5 11. Field research 0

5. Specific preparation for practicals/tutorials




7. Preparation for quizzes 8 14. Other activities…. 0

TOTAL hours of individual study (/semester) = 108




1.Knowledge and understanding 1. Knowledge and understanding of the key concepts in analysis 2. The use of Fourier analysis techniques


courses for solving specific problems. 2. Analytic tools are necessary in variety of fields

3. Attitudinal 1. to develop an interest for the field; 2. to realize the importance of analytic tools in modern mathematics

3. to assume an ethical conduct in scientific research; 4. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : o Lebesgue integration with respect to a Radon measure and L^p

spaces

o The Fundamental Theorem of Analysis for BV functions and extensions

o Fourier series, convergence, properties

o Inversion Theorem for Fourier Series

o Convergence of Fourier series of integrals and derivatives

o Kronecker’s Theorem (the densityand unifrom distribution of integer parts of n*z, for irrational z)

o Applications of Fourier analysis to ordinary differential equations

o Fourier transform on the real line

o Applications to PDEs

o Cesaro, Abel-Poisson convergence and Tauberian theorems

o A proof of the prime number theorem using Fourier analysis o Extensions of Fourier analysis, selfadjoint operators, spectrum and

representations of solutions to PDE

o An introduction to Fourier analysis on groups

Tutorials :

o Computations of Riemann versus Lebegue and issues.


o Examples of BV functions and use of the fundamental theorem of analysis

o Examples of Fourier series, convergent and divergent o Use of the inversion formula, uniqueness results

o Examples of convergent and divergent derivatives and integrals

o Other examples of uniform distributions. o Examples of ODE and Fourier expansion. o Computation of Fourier transform on the real line

o Examples of superposition of solutions for PDE

o Examples and counterexamples for the Tauberian theorems

o Examples of operators, spectrum, eigenfunctions and conections with PDE.

o Examples fo Fourier analysis on groups.

N.B. The above is a description of the topics to be covered and are not in 1-1 correspondence with the 14 lectures/tutorials.

Bibliography

1. Serban Stratila,Integrala Lebeque si Transformata Fourier , Theta, 2014

2. Walter Rudin, Principles of mathematical analysis, (3rd edition), 1976.

3. A. Zygmund, Trigonometric Series (3rd edition), 2002

4. Radomir, Moraga and Astola, Fourier analysis on finite groups, applications in signal processing and system design, 2005


Library, Internet access


{Total=100%}

- final exam results 60%

- hands-on lab test&quiz 0%

- results to periodic tests/quizzes/homeworks 20%

- results to mid-term examination (oral, optional) 20%

- scientific reports, symposium etc 0%

- other activities (to be specified) ………………… 0%

Final evaluation methods, E/V. Written exam

Minimal requirements for mark 5 Requirements for mark 10


( 10 point scale) (10 point scale)

Correct solutions to indicated subjects (for mark 5) in final exam

Average results to periodic/continuous testing.



CS II dr. Cezar Joiţa


Ob.14 Introduction to Algebraic Topology

Name Algebraic Topology Code Ob.14 Year of study I Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 175

Teacher(s) Prof. Liviu Ornea, Conf. V. Vuletescu




Exact Sciences




56 42 14


Required Algebra I, II; Point Set Topology (as in introductory courses of Mathematical Analysis or Manifolds)

Recommended Differential Geometry on Manifolds













1.Knowledge and understanding 3. Knowledge and understanding of the interplay between algebra,

geometry and topology. 4. Understanding algebraic invariants of topological spaces. 5. Ability to use algebraic tools in geometry problems.


courses for solving specific problems. 6. Ability to decide classification problems in geometry by using

algebraic tools. 3. Attitudinal

9. to develop an interest for the field; 10. to realize the importance of the field of algebraic topology in

contemporary mathematics; 11. to assume an ethical conduct in scientific research; 12. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 12. Review of point set topology. 13. Homotopy of maps and paths. 14. The foundamental group of a topological space. 15. Covering spaces. 16. Singular homology. 17. The exact sequence of the pair. 18. Excision. The Mayer Vietoris sequence. 19. Classification of surfaces. 20. Review of differential forms on manifolds. 21. De Rham cohomology. 22. Kuenneth formula for products and bundles. 23. Poincaré duality. 24. Introduction to sheaves and de Rham theorem (sketch of proof).

Tutorials : 11. Commutativity of the foundamental group of Lie groups. 12. Explicit computations of fundamental group for various

topological spaces. 13. Explicit computations of integral homology using simplicial

decompositions. 14. Homology of spheres and applications (Brouwer, Borsuk-Ulam

etc.) 15. Explicit computations of homology groups using Mayer-Vietoris

sequence. 16. Explicit examples of integration on manifolds, use of Stokes

theorem. 17. Vector bundles, homotopy invariance of pull-back. 18. Explicit computations of de Rham cohomology using Mayer-

Vietoris sequence.



Bibliography

1. R. Bott, L. Tu: Differential formas in algebraic topology, Springer, 1982.

2. M. Greenberg, J. Harper Algebraic Topology. A first course, Benjamin/Cummings Pub. Co., 1981

3. A. Hatcher: Algebraic Topology, Cambridge Univ. Press, 2002. 4. T. Dieck, Algebraic Topology, European Math. Society, 2008. 5. S. Matveev, Lectures on algebraic topology, European

Matematical Society, 2006. Necessary scientific infrastructure




Written exam





Date: Lecturer(s) signature(s): Prof. Liviu Ornea

Assoc. Prof. Victor Vuletescu


Ob.15 Riemannian Geometry

Name Riemannian Geometry Code Ob.14 Year of study I Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 150

Teacher(s) Prof. Ion Mihai




Exact Sciences




42 28 14


Required Differential Geometry on Manifolds Differential Geometry of Curves and Surfaces

Recommended

Mathematical Analysis Differential (ordinary and partial) equations.











General competences (mentioned in MSc program sheet) Specific competences

1.Knowledge and understanding 6. Knowledge and understanding of the interplay between geometry,

topology, algebra and analysis. 7. Understanding the meaning of geodesic, curvature, complete

manifolds, (Riemannian) submanifold. 2. Instrumental

7. Ability to use mathematical methods and tools studied in various previous courses for solving specific geometrical problems.

8. Ability to describe and solve problems by using specific geometrical tools..

3. Attitudinal 13. To develop an interest for the field. 14. To realize the importance of the field of Riemannian geometry in

contemporary mathematics. 15. To assume an ethical conduct in scientific research. 16. To optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 1. Riemannian manifolds. Levi-Civita connection. 2. The geodesics of a Riemannian manifold. 3. Complete Riemannian manifolds. Hopf-Rinow theoerm. 4. The sectional curvature, Manifolds with constant sectional

curvature, Schur’s theorem. Einstein manifolds. 5. Laplace-Beltrami operator on Riemannian manifolds. Hodge

theorem. Poicare duality. 6. Hermitian manifolds. 7. Riemannian submanifolds. Gauss and Weingarten formulas. Gauss-

Codazzi-Mainardi equations. 8. Minimal submanifolds. Totally geodesic and totally umbilical

submanifolds Tutorials :

19. Examples of Riemannian manifolds. 20. Explicit computations of geodesics. 21. Applications of Hopf-Rinow theorem. 22. Explicit computations of curvature. 23. Examples of Hermitian manifolds 24. Explicit computations on submanifolds.


Bibliography

1. A. Besse, Einstein Manifolds, Springer, 2008. 2. M. Do Carmo, Riemannian Geometry, Birkhauser, 14th edition,

2013.


3. M.Spivak, A Comprehensive Introduction to Differential Geometry (Vol.1, 2, 3), Publish or Perish, 1999.

4. P.Petersen, Riemannian Geometry, Springer, 2006.


Library


{Total=100%} - final exam results 60% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 20% - results to mid-term examination (oral, optional) 20% - scientific reports, symposium etc 0% - other activities 0% Final evaluation methods, E/V.

Written exam






Prof. Mihai Ion


Ob.21 Homological Algebra

Name Homological Algebra Code Ob.21 Year of study I Semester 2 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 150

Teacher(s) Prof. Dragoș Ștefan




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Algebra I, II; Rings Recommended













8. Knowledge and understanding 1. Understanding the importance of homological algebra in other

fields (e.g. group theory, algebraic topology, algebraic geometry, etc.)

2. Understanding of basic concepts of homological algebra. 3. Understanding the homological invariants associated to algebraic

structures. 4. Ability to use homological methods in Algebra.

2. Instrumental 9. Ability to use formal mathematical language. 10. Ability to analyse and communicate mathematical methods and

models. 11. To seek new sources of mathematical knowledge. 12. To be able to outline a problem in the field of study independently,

to develop a solution method, to solve and to evaluate the results. 3. Attitudinal

17. To develop an interest for the field; 18. To realize the importance of the field of homological algebra in

contemporary mathematics 19. To assume an ethical conduct in scientific research; 20. To valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 25. Basic notions of category theory. 26. (Co)chain complexes and their (co)homolgy. Long exact sequences 27. Projective and injective resolutions. 28. Derived functors. Tor and Ext. Tor and Ext for nice rings 29. Tor and flatness 30. Ext and extensions 31. Dimension in homological algebra. Rings of small dimension

Tutorials : 25. Explicit computations of (co)homology groups. 26. Explicit computations of Tor and Ext for various modules. 27. Computation of projective and injective dimension of various

modules. 28. Applications of group (co)homology and Hochschild (co)homology.


Bibliography

6. Joseph J. Rotman, An Introduction to Homological Algebra, Springer, 2008.

7. H. Cartan și S. Eilenberg, Homological Algebra, Princeton University Press, retipărită 2016.


8. S.I. Gelfand and Y.I. Manin, Methods of Homological Algebra, Springer, retipărită 2013.

9. P. Hilton și U. Stambach, A course in homological algebra, Springer Science & Business Media, 2012.


Library

Final mark is given by: Weight, in % {Total=100%}

- final exam results 70% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 20% - results to mid-term examination (oral, optional) 0% - scientific reports, symposium etc 10% - other activities (to be specified) 0% Final evaluation methods, E/V.

Written exam






Prof. Dragoș Ștefan


Ob.22 Introduction to Algebraic Geometry

Name Introduction to Algebraic Geometry

Code Ob.22


DF



semester 150

Teacher(s) Prof. Marian Aprodu




Exact Sciences




42 28 14


Required Algebra I, II; Galois Theory Geometry I, II; Analysis I, II.

Recommended

Homological Algebra













1.Knowledge and understanding 9. Understanding the main concepts and notions of algebraic

geometry. 10. Ability to recognize properties of given algebraic varieties

(irreducibility, dimension, smoothness etc). 11. Knowledge and understanding of the interplay between algebra

and geometry.


courses for solving specific problems.

3. Attitudinal 21. to develop an interest for the field; 22. to realize the importance of the field of algebraic geometry and its

central role in contemporary mathematics 23. to assume an ethical conduct in scientific research; 24. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lectures : 1. Spaces with functions, definitions, elementary properties and

examples from various fields (topology, differential geometry, algebra).

2. Morphisms of spaces with functions, products, separated spaces with functions, tangent spaces.

3. Affine varieties, Zariski topology, the correspondence between ideals and varieties.

4. Regular functions, coordinate rings, description of the morphisms between affine varieties, the identification theorem of affine varieties.

5. Abstract algebraic varieties. 6. Affine morphisms and finite morphisms, the geometric

interpretation of the Noether normalization Lemma. 7. The dimension theory for algebraic varieties, the geometric

interpretation of Krull’s principal ideal, the algebraic interpretation of the dimension.

8. The dimensions of the fibres of a morphism and applications. 9. Differentials and tangent vectors, smooth and singular points. 10. Etale morphisms. 11. Projective varieties, Zariski topology, the zeroes theorem and

the correspondence between projective varieties and homogeneous ideals.

12. Homogenization and de-homogenization, projective varieties as algebraic varieties, homogeneous coordinate rings.

13. Examples of projective varieties: finite sets of points and their equations, Veronese varieties, Segre varieties, blow-ups in a point, Grassmannians.


14. Complete varieties. Tutorials :

29. Examples of spaces with functions, verification of the definitions and properties.

30. Prove that some sets are affine varieties, and find their equations. 31. Examples of sets that are not affine varieties. 32. Images of morphisms, examples of images that are not varieties,

determination of equations in some cases where images are affine varieties.

33. Decomposition of affine varieties into irreducible components; Description of specific coordinate rings.

34. Regularity of maps, determination of points where maps are regular and where they are not regular.

35. Determination of dimensions of various affine varieties. 36. Determination of singular and smooth points of specific varieties. 37. An irreducible algebraic variety is not a union of countably many

proper closed subset; other more advanced exercises on abstract algebraic varieties.

38. Points in the projective space, understanding the difference between working with affine coordinates and homogeneous coordinates.

39. Homogenization and de-homogenizations of particular equations. 40. Determination of singular points of given projective varieties,

verification of smoothness for projective varieties. 41. Projective transformations, identifying plane curves that are

projectively equivalent. 42. Intersections in the projective plane.

Bibliography

- I. V. Dolgachev. Classical Algebraic Geometry: A Modern View. Cambridge Univ. Press 2012.

- J. Harris. Algebraic Geometry: A first Course. Springer Graduate Texts in Math. 133, corrected edition 1995.

- R. Hartshorne. Algebraic Geometry. Springer Graduate Texts in Math. 52, 8th corrected edition 1997.

- G. R. Kempf. Algebraic Varieties. Cambridge University Press 1993.

- D. Mumford. The Red Book of Varieties and Schemes. Springer Lecture Notes in Math. 1358, 2nd expanded edition 1999.

- I. Shafarevich. Basic Algebraic Geometry I: Varieties in the Projective Space. Springer, 3rd edition 2013.


Library



- final exam results 60% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 20% - results to mid-term examination (oral, optional) 10% - scientific reports, symposium etc 10% - other activities (to be specified) ………………… 0% Final evaluation methods, E/V.

Written exam






Prof. Marian Aprodu


Ob.23 Groups and Representations

Name Groups and Representations Code Ob.23 Year of study I Semester 2 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF

Type{Ob – compulsory, Op- elective, F – optional} Op ECTS 6 Total hours in curriculum 42 Total hours for


semester 150

Teacher(s) Prof. Daniel Bulacu




Exact Sciences




42 28 14


Required Linear Algebra; Group Theory

Recommended

Module Theory; Algebra Representation Theory









7. Preparation for quizzes 8 14. Other activities…. 0 TOTAL hours of individual study (/semester)=108




1.Knowledge and understanding - Knowledge and understanding of the interplay between groups, algebras and their representations. - Understanding the influence of the character tables on representation theory. - Ability to use group representation tools in solving problems in classical group theory. 2. Instrumental - Ability to use mathematical methods studied in various previous courses for solving specific algebra problems. - Ability to decide classification problems in group theory by using group representation theory tools. 3. Attitudinal - to develop an interest for the field; - to realize the importance of the field of group representation theory in contemporary mathematics - to assume an ethical conduct in scientific research; - to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 1. Group representations. 2. Irreducible modules. 3. Maschke’s theorem. 4. Irreducible characters. 5. Characters tables and orthogonality relations. 6. Some elementary character tables. 7. Permutations and characters. 8. Applications to group theory. 9. Burnside’s theorem.

Tutorials :

1. Examples and constructions (representations, irreducible representations, characters, etc.)

2. Schur’s Lemma and applications. 3. Explicit computations of some character tables. 4. Algebraic integers. 5. Induced characters.


Bibliography

[M] C. Musili, Representations of Finite Groups, Texts and Readings in Mathematics, AMS, 2011.

[GL] Gordon James, Martin Liebeck, Representations and Characters of Groups, Cambridge University Press, 2009..


[FH] W. Fulton, J. Harris, Representation Theory A First Course, GTM 129, Springer, 2004.


Library



Written exam






Prof. Daniel Bulacu


Ob. 24 Number Theory

Name Number Theory Code Ob24 Year of study I Semester 2 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Conf. Alexandru Gica




Exact Sciences




42 28 14


Required Elementary Number Theory; Calculus; Algebra

Recommended

Analytic Number Theory, Algebraic Number Theory














analysis, geometry and number theory. 13. Acquire the prerequisites for advanced courses in number theory. 14. Ability to use basic algebraic, analytical and geometrical tools in

number theory problems. 2. Instrumental

14. Ability to use mathematical methods studied in various previous courses for solving specific number theory problems.

15. Ability to decide classification problems in number theory by using algebraic, analytical and geometrical tools.

3. Attitudinal 25. to develop an interest for the field; 26. to realize the importance of the field of number theory in

contemporary mathematics 27. to assume an ethical conduct in scientific research; 28. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 1. Formal series. Generating functions. 2. Characters. Dirichlet characters. Primitive characters. 3. Gauss sums. Finite Fourier analysis. 4. Vinogradov’s theorem on the least quadratic nonresidue. 5. Prime numbers in arithmetical sequences. Dirichlet’s theorem. 6. Lattices. Convex sets. 7. Geometry of numbers. Minkowski’s theorem of the convex body. 8. Sum of three sqauares. 9. Quadratic forms, genus theory. 10. Class numbers for quadratic forms of a given discriminant.

Tutorials :

B. Playing with formal series and generating functions. C. Connection between Legendre symbol and characters. D. Fourier coefficients of Dirichlet characters. E. Computing values of L-functions. F. How to use Dirichlet’s theorem in various problems. G. Applications of Minkowski’s theorem. H. Working with quadratic forms.


Bibliography

I. Heng Huat Chan, Analytic Number Theory for Undergraduates, World Scientific, 2009


J. David Cox, Primes of the for x^2+ny^2, Wiley, 2013 K. I.E. Leonard, J.E. Lewis, Geometry of convex sets. Wiley, 2015


Library



Written exam






Assoc. Prof. Alexandru Gica

Ob.25 Complex Analysis and Rieman Surfaces

Name Complex Analysis and Rieman Surfaces

Code Ob25

Year of study I Semester 2 Assessment (E/V/C) E


DF

Type{Ob – compulsory, Op- elective, F – optional} Op ECTS 5



120


Teacher(s) C.S. II Cezar Joita


Computer Science

Department Mathematics Total hours per semester in curriculum

Main domain (sciences, art, culture)

Exact Sciences




42 28 14


Required Complex analysis

Recommended














1.Knowledge and understanding In depth study of complex analysis in one variable.

Knowledge and understanding of basic notions in the theory of Riemann Surfaces. Understanding the influence of the complex analysis in geometry, toplogy and algebraic geometry.



2. Instrumental Ability to use mathematical methods studied in various previous courses for solving specific geometric problems.

3. Attitudinal To develop an interest for the field;

To assume an ethical conduct in scientific research; To optimally capitalize one’s own potential in scientific activities.

SYLLABUS

Lecture: 1. Breef review of the notions and results studied in Complex Analysis I – the second year course in the License programm. 2. Cauchy-Pompeiu theorem. The d-bar equation for compactly supported functions. 3. Runge approximation theorem. Mittag-Leffler theorem. 4. The d-bar equation. Weierstrass theorem. 5. The Gamma Functiom. The zeta function. 6. Sheaves of abelian groups. 7. Analytic continuation. The monodromy theorem. 8. The notion of Riemann surface. Examples. Holomorphic and meromorphic functions on Riemann surfaces. 9. Differential forms on Riemann surfaces. 10. The first cohomology groups with values in a sheaf. Leray's theorem. 11. Morphisms of sheaves. The cohomology sequence associated to a short exact sequence of sheaves. 12. Divisors and vector bundles on Riemann surfaces. 13. Finitness theorems for compact Riemann surfaces. 14. Riemann-Roch theorem.

Tutorials :

Examples and exercises related to the notions and results studied during the lectures. Student presentations of reports and small projects.

Bibliography

L. Greene, Robert E.; Krantz, Steven G.: Function theory of one complex variable. 3rd ed. Graduate Studies in Mathematics 40. American Mathematical Society, 2006.

M. Forster, Otto: Lectures on Riemann surfaces. Graduate Texts in Mathematics, New York - Heidelberg -Berlin: Springer-Verlag, 1981.

N. Freitag, Eberhard: Complex analysis 2. Universitext. Berlin: Springer, 2011.

O. Freitag, Eberhard; Busam, Rolf: Complex analysis. 2nd ed., Universitext. Berlin: Springer, 2009.


P. Rodriguez, Rubi, Kra, Irwin, Gilman, Jane P.: Complex Analysis, Springer, Graduate Texts in Mathematics, 2013

Q. Narasimhan, Raghavan: Compact Riemann surfaces. Lectures in Mathematics, ETH Zurich. Basel: Birkhauser, 1992.

R. Narasimhan, Raghavan; Nievergelt, Yves: Complex analysis in one variable. 2nd ed., Boston, MA: Birkhauser, 2001

S. Napier, Terrence; Ramachandran, Mohan: An introduction to Riemann surfaces., Cornerstones. New York, NY: Springer, 2011.


Library


{Total=100%}













C.S. II Cezar Joiţa


Op.31-35 Hopf Algebras

Name Hopf Algebras Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. Sorin Dascalescu




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Algebra I, II, Rings and categories of modules

Recommended

Groups and representations













1.Knowledge and understanding 15. Knowledge and understanding of the concepts of algebra and its

dual, coalgebra. 16. Understanding the relevance of representations. 17. Understanding the role of algebraic methods in geometry and

analysis. 2. Instrumental

16. Ability to use mathematical methods studied in various previous courses for solving specific problems.

17. Understanding the importance of classification problems. 3. Attitudinal

29. to develop an interest for the field; 30. to realize the importance of Hopf algebra theory in contemporary


SYLLABUS

Lecture : 9. Algebras and coalgebras. 10. Modules and comodules. 11. Bialgebras and Hopf algebras. 12. Hopf modules. 13. Integrals for Hopf algebras. 14. Semisimple Hopf algebras. 15. Actions and coactions of Hopf algebras.

.

Tutorials : 1. Computation of dual (co)algebras. 2. Understanding sigma notation. 3. Checking Hopf algebra structures. 4. Finding integrals on certain Hopf algebras.


Bibliography

1. M. Sweedler, Hopf Algebras, Benjamin, New-York, 1969. 2. E. Abe, Hopf Algebras, Cambridge Univ. Press., 1977. 3. S. Dăscălescu, C. Năstăsescu, Ş. Raianu, Hopf algebras: an

introduction, Marcel Dekker, 2000 4. D. E. Radford, Hopf Algebras, World Scientific, 2012 5. Tomasz Brzezinski, Robert Wisbauer, Corings and Comodules,

Series: London Mathematical Society Lecture Note Series (No. 309), Cambridge University Press, 2003.


6. Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. Algebras, rings and modules. Lie algebras and Hopf algebras. Mathematical Surveys and Monographs, 168. American Mathematical Society, Providence, RI, 2010.


Library



Written exam






Prof. Sorin Dăscălescu


Op.31-35 Lie Algebras

Name Lie Algebras Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. Dragoș Ștefan




Exact Sciences




42 28 14


Required Algebra I and Algebra II

Recommended








7 13. Other activities 0

7. Preparation for quizzes 4 14. Other activities 0 TOTAL hours of individual study (/semester) = 78




1.Knowledge and understanding 18. Understanding of basic concepts of the theory Lie algebras. 19. Knowledge of the main properties of Lie algebras, especially

nilpotence, solvability, and semisimplicity. 20. Knowledge of Lie algebra representations. 21. Knowledge and understanding of the Killing-Cartan classification of

the finite dimensional simple Lie algebras.

2. Instrumental 18. Ability to use formal mathematical language. 19. Ability to analyse and communicate mathematical methods and

models. 20. To seek new sources of mathematical knowledge. 21. To be able to outline a problem in the field of study independently,

to develop a solution method, to solve and to evaluate the results. 3. Attitudinal 33. To develop an interest for the field; 34. To develop new mathematical skills; 35. To assume an ethical conduct in scientific research; 36. To be able to evaluate critically the obtained results.

SYLLABUS

Lecture : 1. The basic notions of Lie Algebras Theory. 2. Modules and representations. The classification of representations of

sl(2, C). 3. Nilpotent Lie algebras, Engel Theorem. 4. Solvable Lie algebras, Lie Theorem. 5. Semisimple Lie algebras, the Killing form, Cartan's criteria. 6. Irreducible representations, Weyl Theorem. 7. Root systems, the root space decomposition. 8. The classification of semisimple Lie algebras (presentation of the main

results, without proofs). Tutorials : 43. Examples of Lie algebras. 44. Explicit computations (Killing form, root system, Cartan matrice,

etc.).


Bibliography

10. K. Erdmann, M. J. Wildon. Introduction to Lie Algebras. Springer (2006)

11. J.E. Humphreys. Introduction to Lie Algebras and Representation Theory, Graduate Texts în Mathematics. Springer (1997)


12. N. Jacobson. Lie algebras, Interscience Publishers (1961) 13. Ian M. Musson, Lie Superalgebras and Enveloping Algebras,

American Mathematical Society, Providence, RI, 2012. Necessary scientific infrastructure

Library


{Total=100%} - final exam results 70% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 20% - results to mid-term examination (oral, optional) 0% - scientific reports, symposium etc 10% - other activities (to be specified) 0% Final evaluation methods, E/V.

Written exam






Prof. Dragoș Ștefan


Op.31-35 Special Topics in Category Theory

Name Special Topics in Category Theory

Code Op.31-35

Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. dr. Daniel Bulacu




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Rings and categories of modules; Algebra I, II

Recommended

Braid Groups; Algebraic Topology













1.Knowledge and understanding Knowledge and understanding of the interplay between algebra,

analysis and category theory. Understanding the influence of category theory on Algebraic

Topology and Topological Quantum Filed Theory. Ability to use categorical tools in algebra problems.

2. Instrumental Ability to use mathematical methods studied in various previous

courses for solving specific categorical problems. Ability to generalize results in algebra to the categorical settings.

3. Attitudinal to develop an interest for the field; to realize the importance of the field of category theory in

contemporary mathematics to assume an ethical conduct in scientific research; to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 1. Categories, functors and natural transformations. 2. Initial and final objects. Equalizers and coequalizers. 3. Adjoint functors. 4. Monoidal categories. Mac Lane’s coherence theorem. 5. (Co)monads and (co)algebras in monoidal categories, and (co)

representations. 6. The Eilenberg-Moore and Kleisli constructions associated to a

monad. 7. Bimonads versus monoidal categories. Applications to categories of

(co)representations. 8. Braided monoidal categories and braided functors. 9. The left and right centre constructions.

Tutorials : 1. Examples and constructions ((co)limits, (co)products, etc.) 2. Adjoint functors versus (co)monads. 3. Comparison between Moerdijk and Lack and Virelizier definitions

for a bimonad. 4. Braided categories arising from different mathematical domains. 5. Explicit computations of some centers.


Bibliography

1. C. Kassel, Quantum Groups, Springer-Verlang 1995. 2. C. Kassel, V. Turaev, Braid Groups, Springer-Verlag 2008. 3. F. Borceux, Handbook of Categorical Algebra I, Cambridge Univ.

Press 2008.


4. S. Mac Lane, Categories for the Working Mathematician, second edition, Springer-Verlag 1998.

5. D. Bulacu, Algebras and Coalgebras in braided monoidal categories, Ed. Univ. Buc. 2009.

6. M. Kashiwara, P. Schapira, Categories and sheaves, Springer-Verlag 2005.

7. H. Simmons, An Introduction to Category Theory, Cambridge Univ. Press 2011.


Library



Written exam






Prof. Daniel Bulacu


Op. 31-35 Combinatorics in Commutative Algebra

Name Combinatorics in Commutative Algebra

Code Op. 31-35

Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU – humanities

DF

Type{Ob – compulsory, Op – elective, F – optional} Op ECTS 5 Total hours in curriculum 42 Total hours for


semester 120

Teacher(s) Assoc. Prof. Marius Vlădoiu




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Introduction to Commutative Algebra;

Homological Algebra

Recommended

Algebraic Curves; Introduction to Algebraic Topology









7. Preparation for quizzes 5 14. Other activities… 0 TOTAL hours of individual study (/semester) = 78




1. Knowledge and understanding 22. Knowledge and understanding of the interplay between

commutative algebra, combinatorics and topology. 23. Understanding the influence of combinatorics on the algebraic

properties. 24. Ability to use the computational methods in commutative algebra

problems. 2. Instrumental

22. Ability to use mathematical methods studied in various previous courses for solving specific commutative algebra problems.

23. Ability to use the computer packages for solving problems in commutative algebra.

3. Attitudinal 37. to develop an interest for the field; 38. to realize the importance of the field of commutative algebra in


SYLLABUS

Lectures 16. Regular sequences. Grade and depth. 17. Cohen-Macaulay rings and modules. 18. Systems of parameters. 19. Graded algebras over a field. 20. Hilbert series of graded algebras over a field. 21. Minimal graded free resolutions and graded Betti numbers. 22. Groebner bases. 23. Simplicial complexes. 24. Stanley-Reisner rings.

Tutorials 45. Examples and conterexamples (regular sequences, Cohen-

Macaulay rings, etc) 46. Computation of the Hilbert series of monomial ideals (with and

without using a computer). 47. Explicit computations of minimal graded free resolutions, and

finding the graded Betti numbers. 48. Computation of the Groebner bases of (graded) ideals (with and

without using a computer). 49. Finding the f-vectors and h-vectors of simplicial complexes.


Bibliography

1. C. Băeţica, Combinatorics of Determinantal Ideals, Nova Science Publisher, 2006.

2. W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998.


3. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, 1995.

4. E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Springer, 2005.


Library


{Total=100%} - final exam results 50% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homework 30% - results to mid-term examination (oral, optional) 20% - scientific reports, symposium etc 0% - other activities (to be specified) ………………… 0% Final evaluation methods, E/V.

Written exam






Assoc. Prof. Marius Vlădoiu


Op.31-35 Analytic Methods in Number Theory

Name Analytic Methods in Number Theory

Code Op.31-35


DF



semester 120

Teacher(s) Conf. Alexandru Gica




Exact Sciences




42 28 14


Required Elementary Number Theory; Calculus; Complex analysis

Recommended

Algebraic Number Theory













1.Knowledge and understanding 25. Knowledge and understanding of the interplay between analysis,

algebra and number theory. 26. Understanding the influence of analyticity on number theory

problems. 27. Ability to use analytic tools in number theory problems.

2. Instrumental


25. Ability to decide classification problems in number theory by using analytical tools.

3. Attitudinal 41. to develop an interest for the field; 42. to realize the importance of the field of analytic number theory in


SYLLABUS

Lecture : 11. Prime Number Theorem. 12. The Riemann zeta function. 13. Zero-free region for zeta function. 14. Riemann Hypothesis. 15. Prime numbers in arithmetical sequences. Dirichlet’s theorem. 16. Minkowski’s theorem of the convex body. 17. Schnirelmann density. 18. Circle method. 19. Analytical formulas for the class numbers

Tutorials : 50. Bernoulli numbers. 51. Connection with the values of zeta Riemann function. 52. Sums of three squares (Gauss’s theorem). 53. Application to Waring’s problem. 54. Vinogradov’s theorem.


Bibliography

14. Gica A., Panaitopol L., Probleme celebre de teoria numerelor, Editura Universităţii Bucureşti, 1998.

15. Jean-Marie De Koninck, Florian Luca, Analytic Number Theory: Exploring the Anatomy of Integers (Graduate Studies in Mathematics), AMS, 2012.


16. Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, American Mathematical Society, Providence, RI, 2004.


Library



Written exam






Assoc. Prof. Alexandru Gica


Op.31-35 Complex Geometry

Name Complex Geometry Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. dr. Liviu Ornea, CS1 Sergiu Moroianu




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Differential Geometry on Manifolds; Complex

analysis Recommended Algebraic Topology














geometry and topology. 29. Understanding the influence of analyticity on geometric properties. 30. Ability to use complex analytic tools in geometry problems.

2. Instrumental

26. Ability to use mathematical methods studied in various previous courses for solving specific geometric problems.

27. Ability to decide classification problems in geometry by using complex analytic tools.

3. Attitudinal 45. to develop an interest for the field; 46. to realize the importance of the field of complex geometry in


SYLLABUS

Lecture : 25. Complex manifolds. 26. Complex and holomorphic fibre bundles. 27. Hermitian manifolds. 28. The Chern connection. 29. Chern-Weil theory. 30. Kaehler manifolds. 31. Introduction in Hodge theory. 32. Calabi-Yau and Kaehler-Einstein manifolds. 33. Submanifolds.

Tutorials :

55. Examples and constructions (products, holomorphic actions etc.) 56. Projective and Grassmann spaces, tautological bundles. 57. Explicit computations of Chern classes. 58. Hopf manifolds. 59. Holonomy. Hyperkaehler manifolds.


Bibliography

17. P. Griffith, J. Harris, Principles of algebraic geometry, Wiley, 1994.

18. A. Moroianu, Lectures on Kaehler manifolds, Cambridge Univ. Press, 2007

19. W. Ballmann, Lectures on Kaehler manifolds, E.M.S., 2006



Library



Written exam






Prof. Liviu Ornea

CS1 Sergiu Moroianu


Op.31-35 Algebraic Geometry

Name Algebraic Geometry Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. dr. Marian Aprodu




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Differential Geometry on Manifolds; Complex

analysis Recommended Sheaf Theory; Algebraic Topology














geometry and topology. 32. Understanding the influence of analyticity on geometric properties. 33. Ability to use complex analytic tools in geometry problems.


courses for solving specific geometric problems. 29. Ability to decide classification problems in geometry by using

algebraic tools. 3. Attitudinal

49. to develop an interest for the field; 50. to realize the importance of the field of complex geometry in


SYLLABUS

Lecture : 1. Affine and quasi-affine varieties: regular functions, rational

functions, the birational viewpoint. 2. Local theory: the Krull dimension, regularity, consequences of the

factoriality of regular local rings. 3. Projective varieties; the Segre and Veronese maps. Projective

varieties are proper. 4. Projections and correspondences. 5. Divisors and the canonical class. 6. Coherent and quasi-coherent sheaves. The cohomology of projective

varieties. 7. Intersection theory on surfaces; the Riemann-Roch theorem,

blowing-up, the intersection form of the blown-up surface. 8. The language of schemes. Varities over finite fields.

Tutorials :

60. Examples and constructions of varieties, regular and rational functions.

61. Explicit dimension computations, exercises involving factoriality and regularity

62. Examples of projective varieties, consequences of Veronese embeddings.

63. Explicit characterisations of linear projections. 64. Divisorial computations. 65. Verification of coherence in specific examples. Cohomology of

particular examples of projective varieties. 66. Applications of intersection theory on surfaces. 67. Explicit examples of schemes, the ring of dual numbers, tangent

vectors.



Bibliography

1. J. Harris, Algebraic Geometry, Springer, 1992. 2. R. Hartshorne, Algebraic Geometry, Springer 1977. 3. G. Kempf, Algebraic Varieties, Cambridge, 1993. 4. Igor V. Dolgachev, Classical Algebraic Geometry: A Modern View,

Cambridge University Press, 2012 5. David A. Cox, John B. Little, Donald O'Shea, Using

Algebraic Geometry, Springer, 2005 6. Brendan Hassett, Introduction to Algebraic Geometry,

Cambridge University Press, 2007 7. Thomas A. Garrity, Richard G. Belshoff, Lynette Boos,

J. Ryan Brown, Carl Lienert, Algebraic Geometry: A Problem Solving Approach, AMS, 2013


Library



Written exam







Prof. dr. Marian Aprodu


Op. 31-35 Elements of Free Probability

Name Elements of Free Probability Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E


DF

Type{Ob – compulsory, Op- elective, F – optional} Ob ECTS 5



120

Teacher(s) Prof. dr. Lucian Beznea


Computer Science


Department Mathematics Main domain

(sciences, art, culture)

Exact Sciences



Program name 42 28 14 ** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project

Prerequisites

Required Algebra I, II; Point Set Topology, Analysis, Probability

Recommended

Functional Analysis













1.Knowledge and understanding 3. Knowledge and understanding of basic elements of matrices,

algebras and the concept of freeness 4. Manipulation of free random variables and applications to

analysis, combinatorics and random matrices

2. Instrumental 3. Ability to use mathematical methods studied in other fields, as for

instance counting techniques, formal power series and basic elements of probability

3. Attitudinal 10. to develop an interest for the field; 11. to realize the importance of freeness

12. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : o Review of elements of probability, distributions and their

moments, independence at the algebraic level o Voiculescu’s concept of freeness o Random matrices and the convergence of distribution of

eigenvalues to the Wigner’s semicircular law

o The free central limit theorem

o Asymptotic freeness for several random matrices o The combinatorics of freeness, the non-crossing partitions o Applications to the classification of free factors

Tutorials :

o The tutorials will be centered on the review of concepts needed in the lectures and specic computations for each of the topisc disucssed in the lectures


Bibliography

Dykema, Voiculescu and Nica, Free Random Variables, CRM monographs 1992

Nica, Speicher, Lectures on the combinatorics of free probability, Cambridge Univ. 2006

Anderson, Guionnet and Zeitouni, An introduction to random matrices, Cambridge Univ. 2009

Mingo and Speicher, Free probability and random matrices, Springer, 2017





{Total=100%}








Minimal requirements for mark 5

( 10 point scale)


Correct solutions to indicated subjects (for mark 5) in final exam

Average results to periodic/continuous testing.



Prof. Dr. Lucian Beznea

Op.31-35 Several Complex Variables

Name Several Complex Variables Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E


DF




120

Teacher(s) C.S. II Cezar Joita



Computer Science



Exact Sciences



Program name Advanced studies in Mathematics

42 28 14


Required Complex analysis Complex analysis and Riemann surfaces

Recommended

Commutative algebra Differential geometry














1.Knowledge and understanding In depth study of complex analysis in several variables.

Understanding the influence of the complex analysis in geometry, topology and algebraic geometry.






SYLLABUS

Lecture: 1. The notion of holomorphic function of several complex variables. Cauchy integral formula. 2. Power series in several complex variables. 3. Analyticity of holomorphic functions of several complex variables. Elementary properties of holomorphic functions. 4. The Hartogs phenomenon. 5. The notion of a domain of holomorphy. 6. Cartan - Thulen theory, part I. 7. Cartan - Thulen theory, part II. 8. Plurisubharmonic functions. 9. Pseudoconvexity. 10. The ring of germs of holomorphic functions. 11. The preparation and division theorems of Weierstrass. 12. Analytic sets: definition, examples, elementary properties. 13. Local properties of analytic sets, part I. 14. Local properties of analytic sets, part II.

Tutorials :

Examples and exercises related to the notions and results studied during the lectures. Student presentations of reports and small projects.

Bibliography

13. E.M. Chirka: Complex analytic sets. Kluwer Academic Publishers, 1989.

14. J. P. Demailly: Complex Analytic and Differential Geometry. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

15. F. Forstneric: Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis. 2nd edition. Springer, 2017.

16. K. Fritzsche, H. Grauert: From holomorphic functions to complex manifolds. Springer, 2002.


17. R. Gunning: Introduction to holomorphic functions of several variables. Volume I: Function theory. Volume II: Local theory. Volume III: Homological theory. Wadsworth & Brooks/Cole Mathematics Series, 1990.

18. R. Gunning, H. Rossi: Analytic functions of several complex variables. Reprint of the 1965 original. AMS Chelsea Publishing, 2009.

19. L. Hormander: An introduction to complex analysis in several variables. 3rd revised ed. North-Holland, 1990.

20. L. Kaup, B. Kaup: Holomorphic functions of several variables. An introduction to the fundamental theory. Walter de Gruyter, 1983.

21. R. Narasimhan: Several complex variables. Chicago Lectures in Mathematics. The University of Chicago Press, 1971.

22. T. Ohsawa: Analysis of several complex variables. American Mathematical Society, 2002.

23. E. L. Stout: Polynomial convexity. Birkhauser, 2007.


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C.S. II Cezar Joiţa


Op.31-35 Submanifolds of Riemannian Manifolds

Name Submanifolds of Riemannian Manifolds

Code Op.31-35


DF



semester 120

Teacher(s) Prof. Ion Mihai




Exact Sciences




42 28 14

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Differential Geometry on Manifolds;

Riemannian geometry Recommended Complex geometry













1.Knowledge and understanding 34. Knowledge and understanding of the relation between the

geometry of submanifolds and the geometry of the ambient space. 35. Understanding how Riemannian invariants obstruct the existence of

certain type of submanifolds. 2. Instrumental

30. Ability to use mathematical and topological methods studied in various previous courses for solving specific Riemannian geometric problems.

31. Ability to decide classification problems in geometry by using Riemannian invariants.

3. Attitudinal 53. to develop an interest for the field; 54. to realize the importance of the field of Riemannian geometry in


SYLLABUS

Lecture : 1. Submanifolds in Riemannian manifolds. Gauss and Weingarten

formulae. Gauss, Codazzi and Ricci equations. 2. Totally geodesic submanifolds, minimal submanifolds, totally

umbilical submanifolds. 3. Invariants of Riemannian manifolds: scalar curvature, Ricci

curvature, Chen invariants. 4. Optimal geometric inequalities for Riemannian invariants of

submanifolds in constant curvature spaces. 5. Applications: obstructions to the existence of minimal submanifolds,

classification theorems of ideal submanifolds. 6. Special classes of submanifolds in Hermitian manifolds: complex

submanifolds, Lagrangian submanifolds, slant submanifolds, CR-submanifolds..

Tutorials : 68. Examples and constructions in real space forms. 69. Explicit computations of Chen invariants. 70. Examples of CR submanifolds 71. Examples of slant submanifolds. 72. Other metric structures (Sasakian, cosymplectic etc.) and their

submanifolds. N.B. The above is a description of the topics to be covered and are not in 1-1 correspondence with the 14 lectures/tutorials.

Bibliography

1. D. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, 2010.

2. B.Y. Chen, Geometry of Submanifolds, M. Dekker, 1973.


3. B.Y. Chen, Pseudo-Riemannian Geometry, Delta Invariants and Applications, World Scientific, 2011.

4. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. II, Wiley-Interscience, 1969.

5. I. Mihai, Geometria Subvarietăților în Varietăți Complexe, Editura Universității din București, 2002.

6. I. Mihai, A. Mihai, V. Ghișoiu, Culegere de Probleme de Geometrie Diferențială, Editura Universității din București, 2012.


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Written exam






Prof. Ion Mihai


Op.31-35 Algebraic Number Theory

Name Algebraic Number Theory Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Assoc. Prof. Alexandru Gica




Exact Sciences




42 28 14


Required Basic Algebra; Elementary Number Theory

Recommended

Analytic Number Theory; Class Field Theory













1.Knowledge and understanding 36. Knowledge and understanding of the interplay between algebra

and number theory. 37. Ability to use algebraic tools in number theory problems. 38. Understanding the most important factorization algorithm,

Number Field Sieve. 2. Instrumental


33. Ability to decide classification problems in number theory by using algebraic tools.

3. Attitudinal 57. to develop an interest for the field; 58. to realize the importance of the field of algebraic number theory in


SYLLABUS

Lecture :

34. Algebraic Number Fields; 35. Quadratic and cyclotomic fields; other examples. 36. Rings of Algebraic Integers. 37. Integral basis and discriminant. 38. Dedekind’s domains. 39. The group of the class ideals is finite. 40. Units. Dirichlet’s theorem. 41. Applications in solving Diophantine equations. 42. NFS (Number Field Sieve)

Tutorials :

73. Norm, trace, discriminant. 74. How to find the integral basis. 75. Algorithms for computing class numbers. 76. Algorithms for finding the units. 77. How to implement NFS.


Bibliography

20. G.J. Janusz, Algebraic number fields, AMS, 1996.


21. J.W.S Cassels, A. Frohlich, Algebraic Number Theory, London Mathematical Society, 2010

22. K. Kato, Number Theory 2, Introduction to Class Field Theory, AMS, 2011


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Written exam






Assoc. Prof. dr. Alexandru Gica


Op.31-35 Differential Topology

Name Differential Topology Code Op.31-35 Year of study II Semester 1 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. Liviu Ornea




Exact Sciences



Program name Algebra, Geometry and Cryptography

42 28 14


Required Differential Geometry on Manifolds; Algebraic Topology

Recommended

Sheaf Theory; Riemannian Geometry














geometry and topology. 40. Understanding the influence of topology on differential and

geometric properties.

2. Instrumental

34. Ability to use differential forms in topology. 35. Ability to decide classification problems in differential geometry

by using topological tools. 3. Attitudinal

61. to develop an interest for the field; 62. to realize the importance of the field of algebraic topology in

contemporary mathematics; 63. to realize the unity of algebra, geometry and analysis; 64. to assume an ethical conduct in scientific research; 65. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 78. Differentiable manifolds (with boundary). 79. Sard theorem. 80. Orientation. Brouwer degree. 81. Transversality. Whitney embedding theorem. 82. Intersection theory. Applications. 83. Lefschetz fixed point theorem. 84. Vector fields. Index. Poincaré-Hopf theorem.

Tutorials : 1. Morse functions. Examples. 2. Review of Stokes theorem. Computations. 3. Cohomology. Duality. 4. Fibre bundles. Chern classes. The splitting principle.


Bibliography

1. V. Guillemin, A. Pollack, Differential topology, Chelsea Publ., 2010

2. M.W. Hirsch, Differential topology, GTM 33, Springer, 1994. 3. J. Milnor, Topology from the differentiable viewpoint, Princeton

Univ. Press, 1997 4. A.R. Shastri, Elements of differential topology, CRC Press, Boca

Raton, 2011. 5. A. Wallace, Differential topology. First steps, Dover Publications,

2006.



Library



Written exam






Prof. Liviu Ornea


Op.41-45 An Introduction to Quantum Group Theory

Name An Introduction to Quantum Group Theory

Code Op.41-45


DF



semester 120

Teacher(s) Prof. Daniel Bulacu




Exact Sciences




30 20 10

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Hopf algebras; Category theory

Recommended

Lie algebras; Algebraic topology













1.Knowledge and understanding 41. Knowledge and understanding of the interplay between category

theory, Hopf algebra theory and algebraic topology. 42. Understanding the influence of quantum groups in solving

mathematical-physics problems. 43. Ability to use Hopf algebra and Lie algebra tools in quantum

group problems. 2. Instrumental

36. Ability to use mathematical methods studied in various previous courses for solving specific algebraic problems.

37. Ability to make connections between quantum group theory and Hopf algebra and Lie algebra theory, respectively.

3. Attitudinal 66. to develop an interest for the field; 67. to realize the importance of the field of quantum groups in


SYLLABUS

Lecture : 43. The quantum plane and its symmetries. 44. The quantum groups GLq(2) and SLq(2). 45. The quantum enveloping algebra of sl(2). 46. (Co)Quasitriangular Hopf algebras. 47. Categories of Yetter-Drinfeld modules. 48. Drinfeld’s quantum double. 49. The FRT construction. Applications. 50. Quantum groups defined by actions and coactions of a Hopf algebra

on an algebra, respectively on a coalgebra.

Tutorials : 85. Examples and constructions (bialgebra structures, actions,

coactions, etc.) 86. Lie algebras and enveloping algebras. 87. Explicit computations of some Harish-Chandra homomorphisms. 88. Representation-theoretic interpretation of the quantum double. 89. Bicrossed products of groups and bialgebras.


Bibliography

23. C. Kassel, Quantum Groups, Springer-Verlang 1995. 24. Lambe şi D. Radford, Introduction to the quantum Yang-Baxter

equation and quantum groups: an algebraic approach, Kluwer Academic Publishers 1997.


25. S. Majid, Foundations of quantum groups theory, Cambridge Univ. Press 1995.

26. D. E. Radford, Hopf Algebras, World Scientific 2012 27. D. Bulacu, Algebras and coalgebras in braided monoidal

categories, Ed. Univ. Buc. 2009. 28. Susumu Ariki, Hiraku Nakajima, Yoshihisa Saito, Algebraic

Groups and Quantum Groups, AMS 2012. Necessary scientific infrastructure

Library



Written exam






Prof. Daniel Bulacu


Op.41-45 Vector Bundles and Applications

Name Vector Bundles and Applications

Code Op.41-45


DF



semester 120

Teacher(s) Assoc. Prof. Victor Vuletescu




Exact Sciences




30 20 10


Required Differential Geometry on Manifolds; Sheaf Theory;

Recommended

Complex analysis; Algebraic Topology















1.Knowledge and understanding 44. Knowledge and understanding of the interplay between geometry

and topology. 45. Understanding the importance of the study of vector bundles on

geometric properties. 46. Ability to use topological and differential tools in geometry



39. Ability to decide classification problems in geometry by using complex analytic tools.

3. Attitudinal 70. to develop an interest for the field; 71. to realize the importance of the field of complex geometry in


SYLLABUS

Lecture : 1. Basic definitions, examples; the tautological bundle on projective

spaces and on grassmanians. 2. Operations with vector bundles. 3. The projectivisation of a vector bundle: splitting manifolds. 4. Classification of line bundles (real and complex). 5. Characteristic classes (Chern and Stiefel-Whitney) defined

topologically 6. Chern classes defined using differential forms. 7. Classifying spaces for vector bundles. 8. Applications of characteristic classes: obstructions to embeddings,

vector fields on spheres. 9. Positive line bundles on complex manifolds.

Tutorials : 90. Examples of vector bundles; tangent bundle, tensor bundles, etc. 91. Examples of non-trivial vector bundles. 92. Computation of Chern and Stiefel-Whitney classes of the tangent

bundle for some explicit manifolds. 93. Examples of applications of characteristic classes


Bibliography 1. R. Bott, L. Tu; Differential forms în algebraic topology. Graduate


Texts în Mathematics, 82. Springer-Verlag, New York-Berlin, 1982 2. R.O. Wells, Differential analysis on complex manifolds.

Third edition. With a new appendix by Oscar Garcia-Prada. Graduate Texts in Mathematics, 65. Springer, New York, 2008.

3. R. Lazarsfeld; Positivity în algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys în Mathematics 49. Springer-Verlag, Berlin, 2004

4. Ch. Okonek, M. Schneider, H. Spindler Vector bundles on complex projective spaces. Corrected reprint of the 1988 edition. With an appendix by S. I. Gelfand. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2011


Library



Written exam






Assoc. Prof. Victor Vuletescu


Op. 41-45 Hyperbolic geometry

Name Hyperbolic geometry Code


DF


individual study


120

Teacher(s) C.S. I Sergiu Moroianu




Exact Sciences





30 20 10


Required Group Theory (Algebra); Complex Analysis; Curves and surfaces; Differential Geometry

Recommended

Riemannian Geometry












TOTAL hours of individual study (/semester) = 90 General competences (mentioned in MSc program sheet) Specific competences

1.Knowledge and understanding In-depth understanding of the foundations of hyperbolic geometry with an emphasis on dimension 2

Understanding the influence of hyperbolic geometry in complex analysis, algebraic geometry, and the toplogy of surfaces and 3-manifolds.




Lecture:


SYLLABUS

1. Review of Euclidean Geometry; Postulate V; definition of the hyperbolic plane. 2. Review of differential and Riemannian geometry: vector fields, differential forms, metrics, covariant derivative, geodesics. 3. Covering spaces, the fundamental group. 4. Actions of Moebius transformations on the Riemann sphere; homographies; compact hyperbolic surfaces; action of the group of homographies on the ideal boundary. 5. Closed geodesics and conjugacy classes in the fundamental group. Simple geodesics. 6. Pants decompositions, Fenchel-Nielsen coordinates and Teichmuller space 7. The symplectic form on the Theichmuller space 9. Hyeprbolic 3-manifolds; rigidity under deformations. 10. Survey of the hyperbolization theorem. Hyperbolization of compact Riemann surfaces.

Tutorials : Computations, examples and exercises related to the notions and results studied during the lectures. Student presentations of reports; projects in teams.

Bibliography

o Ratcliffe, J: Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer , 2006.

o Guillarmou, C & Moroianu, S: Surfaces. Online course notes http://www.imar.ro/%7Esergium/fisiere/rs.pdf

o Chang, S-Y A, Nonlinear elliptic equations in conformal geometry.

Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (2004).


Library, internet access


{Total=100%} - final exam results 80% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 10% - results to mid-term examination (oral, optional) 0% - scientific reports, symposium etc 10% - other activities (to be specified) ………………… 0%








CSI dr. Sergiu Moroianu


Op.41-45 Representation Theory of Algebras

Name Representation Theory of Algebras

Code Op.41-45


DF



semester 120

Teacher(s) Prof. Sorin Dascalescu




Exact Sciences




30 20 10


Required Algebra I, II, Rings and categories of modules, Groups and representations

Recommended

Lie Algebras













1.Knowledge and understanding 47. Knowledge and understanding of the concept of representation. 48. Understanding certain classification results and their relevance.


courses for solving specific problems. 41. Understanding the importance of knowing the representations of

an object. 3. Attitudinal

74. to develop an interest for the field; 75. to realize the importance of Representation Theory in contemporary


SYLLABUS

Lecture : 1. Projective modules over artinian algebras. The structure of artinian

algebras. 2. Algebras of finite representation type. The Brauer-Thrall conjectures. 3. Representations of quivers. The quiver algebra. 4. Gabriel’s Theorem.

.

Tutorials : 5. Description of projectives over certain algebras. 6. Explicit description of representations of certain quivers. 7. Discussing the representation type of certain algebras.


Bibliography

1. P. Etingof et al., Introduction to representation theory, AMS, 2011.

2. C. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, AMS, 2006.

3. R. S. Pierce, Associative algebras, Springer Verlag, 1982.


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{Total=100%} - final exam results 60%


- hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 20% - results to mid-term examination (oral, optional) 20% - scientific reports, symposium etc 0% - other activities (to be specified) ………………… 0% Final evaluation methods, E/V.

Written exam






Prof. Sorin Dascalescu


Op.41-45 Algebraic Groups

Name Algebraic Groups Code Op.41-45 Year of study II Semester 2 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. dr Marian Aprodu, Prof. dr. Dragoş Ştefan




Exact Sciences




30 20 10

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Algebraic geometry: Sheaf theory; Commutative

algebra; Recommended

Differential Geometry on Manifolds; Algebraic Topology; Lie groups
















geometry and topology. 50. Understanding the influence of algebraic context on geometric

properties. 51. Ability to use complex algebraic tools in geometry problems.


courses for solving specific geometric problems. 43. Ability to decide classification problems in geometry by using

complex analytic tools. 3. Attitudinal

78. to develop an interest for the field; 79. to realize the importance of the field of complex geometry in


SYLLABUS

Lecture : 10. Introduction. Algebraic varieties, definitions, examples. Projective

algebraic varieties. The homogenous coordinates ring. 11. Affine groups; definitions, examples. Actions of algebraic groups

on varieties. 12. Semidirect products, linearization of affine groups. 13. The Lie algebra associated to an algebraic group. 14. Representations, the adjoint representation. Reductive linear

groups, examples. 15. Elements of the theory of invariants: Hilbert's theorem of finiteness

of invariants. 16. Hilbert series of algebraic groups. 17. Classical invariant theory, the Cayley-Sylvester formula.

Tutorials : 1. Examples of algebraic varieties (affine and projective).

Computations of homogeneous coordinate rings. 2. Explicit examples of affine groups and actions. 3. Computations of Lie algebras. 4. Examples and constructions of reductive groups. 5. Applications of Hilbert’s finiteness theorem. 6. Computations of Hilbert series. 7. Applications of the Cayley-Sylvester formula.


Bibliography

1. S. Mukai; An introduction to invariants and moduli, Cambridge Studies în Advanced Mathematics 81 (2012).


2. D. Mumford; Abelian varieties, Tata Institute Bombay, Hindustan .2008Book Agency, New Delhi,

3. D. Mumford, J. Fogarty, J. Kirwan; Geometric invariant theory. Third edition. Springer-Verlag, Berlin,1994.

4. J. Humphreys; Linear algebraic Groups, Graduate Texts în Math. 21, Springer-Verlag, 1980


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Written exam






Prof. dr. Marian Aprodu

Prof. dr. Dragoş Ştefan

Op.41-45 Introduction to quasi-regular Dirichlet forms

Name Introduction to quasi-regular Dirichlet forms

Code Op.41-45

Year of study

II Semester 2 Assessment (E/V/C) E


DF





120

Teacher(s) Prof. Dr. Lucian Beznea


Computer Science



Exact Sciences



Program name ** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project

Prerequisites Required Calculus, measure theory, partial differential equations (basic knowledge)

Recommended











1.Knowledge and understanding In depth study of complex analysis in one variable.

Knowledge and understanding of basic notions in the theory of Riemann Surfaces. Understanding the influence of the complex analysis in geometry, toplogy and algebraic geometry.






SYLLABUS

Introduction. The course intends to be an introduction to the theory of (non-symmetric) Dirichlet forms, presenting geometric aspects and connections with the infinite dimensional analysis. Lecture: 1) Coercive closed forms on L2 –spaces, sector condition, Stampacchia projection 2) The associated resolvent and co-resolvent 3) The generator and the associated semigroup of contractions, closable forms 4) Examples: the classical case, the general case in Euclidean spaces 5) Contraction, Markovian operators, Dirichlet forms 6) Symmetric sub-Markovian operators, energy measure, ”carre du champ” operator, intrinsic metric 7) Potentials: properties, convergence, ”reduite” on an open set 8) The induced Choquet capacities, exceptional sets 9) Quasi-continuity, quasi-continuous versions: properties, convergence 10) Quasi-regular Dirichlet forms Tutorials : • Examples and counterexamples: 1-dimensional classical case. • Selection of kernels • Student presentations of reports and small projects.

Bibliography

[1] Ma, Z.M., Rockner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, 1992. [2] Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes . Walter de Gruyter, 2011. [3] Beznea, L., Boboc, N.: Potential Theory and Right Processes . Kluwer/Springer, 2004. [4] Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space . Walter de Gryter, 1991. [5] Oshima, Y. : Semi-Dirichlet forms and Markov processes. De Gruyter, Studies in Mathematics 48, Berlin, 2013.


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Date: Lecturer signature:

Prof. dr. Lucian Beznea


Op.41-45 Valuation Theory and Local Fields

Name Valuation Theory and Local Fields

Code Op.41-45


DF



semester 120

Teacher(s) Assoc. Prof. Tiberiu Dumitrescu




Exact Sciences




30 20 10


Required Field Theory, Elementary Number Theory

Recommended

Galois Theory












1.Knowledge and understanding 52. Knowledge and understanding of the interplay between arithmetic,

algebraic and topological properties of valued fields. 53. Knowledge and understanding connections between local and

global aspects of Algebra and Number Theory. 2. Instrumental

44. Ability to use the methods of valuation theory inpure and applied mathematics.

3. Attitudinal 82. to develop an interest for the field; 83. to realize the importance of the valuation theory in contemporary

mathematics.

SYLLABUS

Lecture : 51. Valuations and absolute values on fields. 52. Algebraic and topological properties of valued fields. 53. Complete fields. Hensel Lemma and its consequences. 54. Prolongations of valuations and ramification theory. 55. The field of Newton-Puiseux series. 56. Local fields and their classification. 57. Division algebras over local fields.

Tutorials : 94. P-adic numbers. 95. Local-global principle in Number Theory. 96. P-adic analogue of the complex number field C. 97. Elements of p-adic analysis. 98. Irreducible polynomials over local fields.


Bibliography

29. Z.I. Borevitch, I.R. Shafarevitch, Number Theory 30. .E. Artin, Algebraic Bunbers and Algebraic Functions. 31. I. Reiner, Maximal orders, Academic Press 1975. 32. F. Q. Gouvea p-adic Numbers-an Introduction, Springer 1997. 33. J.P.Serre, Local Fields, Springer 1979. 34. M. Vajaitu, A. Zaharescu Non-archimedean Integration and

Applications, Romanian Academic Press 2007. Necessary scientific infrastructure

Library



- final exam results 70% - hands-on lab test&quiz 0% - results to periodic tests/quizzes/homeworks 10% - results to mid-term examination (oral, optional) 0% - scientific reports, symposium etc 20% - other activities (to be specified) ………………… 0% Final evaluation methods, E/V.

Written exam



Correct solutions to indicated subjects (for mark 5) in final exam Average results to periodic tests/quizzes/homeworks

Correct solutions to all subjects in final exam. Successful presentations of scientific reports. Good results to all periodic tests/quizzes/homeworks


Assoc. Prof. Tiberiu Dumitrescu


Op.41-45 Variational Methods in Riemannian Geometry

Name Variational Methods in Riemannian Geometry

Code Op.41-45


DF



semester 120

Teacher Assoc. Prof. Catalin Gherghe




Exact Sciences




30 20 10

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Differential Geometry on Manifolds, Real and

Complex analysis Recommended Differential (ordinary and partial) equations.














geometry and topology. 55. Understanding the meaning of geodesic, minimal submanifold,

harmonic map. 56. Ability to use complex analytic tools in geometry problems.

2. Instrumental 45. Ability to use mathematical methods and tools studied in various

previous courses for solving specific geometrical problems. 46. Ability to describe and solve problems in geometry by using tools

from complex analysis. 3. Attitudinal

84. To develop an interest for the field. 85. To realize the importance of geometric variational theory in

contemporary mathematics. 86. To assume an ethical conduct in scientific research. 87. To optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 58. The Euler-Lagrange equations in the Euclidian space. 59. Geodesics on surfaces. The first variation of the arc length. Bonnet

theorem. 60. Jacobi fields. Conjugate points. 61. Minimal surfaces. Variational methods. 62. Area minimizers surfaces. Second variation formula. 63. Harmonnic maps between Riemannian manifolds. The first

variation formula. 64. The stability of harmonic maps. The second variation formula. 65. Holomorphic maps between complex manifolds

Tutorials : 99. Explicit computations of the solutions of the Euler-Lagrange

equations. Geodesics. 100. Examples of minimal surfaces. 101. Examples of harmonic maps. 102. Construction of harmonic maps


Bibliography

1. T. Colding, W.Minicozzi, A Course in Minimal Surfaces – AMS 2011.

2. John Oprea, Differential Geometry and its Applications, Math. Assoc. of America, 2007.

3. Hajime Urakawa, Calculus of Variations and Harmonic Maps, AMS, 1991.


4. Jurgen Jost, Riemannian Geometry and Geometry Analysis, Springer, 2005.


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Written exam





Date: Lecturer signature:

Assoc. Prof. Catalin Gherghe


Op.41-45 Multiplicative Ideal Theory

Name Multiplicative Ideal Theory Code Op.41-45 Year of study II Semester 2 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DS

Type{Ob – compulsory, Op- elective, F – optional} F ECTS 5 Total hours in curriculum 30 Total hours for


semester 120

Teacher(s) Assoc. Prof. Tiberiu Dumitrescu




Exact Sciences




30 20 10


Required Algebra I, II.

Recommended

Commutative Algebra, Algebraic Number Theory.













1.Knowledge and understanding 5. Knowledge and understanding of basic techniques of

Multiplicative Ideal Theory. 6. Knowledge and understanding of classes of examples. 7. Understanding connections with Number Theory and Algebraic

Geometry. 2. Instrumental

D. Ability to use the star operation machinery as a unifying tool for the whole theory.

E. Ability to compute class groups and to use them in solving specific problems (e.g. factorization problems).

3. Attitudinal to start reading recent papers in this area. to begin doing research in this area. to interact with the Multiplicative Ideal Theory community.

SYLLABUS

Lecture : 24. Star operations on integral domains. 25. Abstract elementary number theory. 26. Fractional divisorial ideals. 27. Invertible ideals and class groups. 28. Valuation, Prufer and Bezout domains. 29. Krull domains and generalizations. 30. Almost Dedekind domains.

Tutorials : o Applications and illustrations. o Solving homework exercises. o Student short papers presentations.

Bibliography

o R. Gilmer, Multiplicative ideal theory (Queen's Papers în Pure and Applied Mathematics 90), Kingston, 1992.

o F. Halter-Koch, Ideal systems: An introduction to multiplicative ideal theory (Monographs and Textbooks în Pure and Applied Mathematics 211), Dekker, 1998.

o C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules (London Mathematical Society Lecture Note Series 336), Cambridge University Press 2006.


o M. Fontana, J. Huckaba and I. Papick, Prufer domains, Dekker 1998.

o M. Fontana, S. Kabbaj, B. Olberding and I. Swanson (Editors), Commutative Algebra, Noetherian and Non-Noetherian Perspectives, Springer, 2010.


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Written exam



Correct solutions to indicated subjects (for mark 5) in final exam Average results at homeworks.

Correct solutions to all subjects in final exam. Good results at homeworks or successful presentations of scientific reports.


Assoc. Prof. Tiberiu Dumitrescu


Op.41-45 Kaehler Manifolds

Name Kaehler Manifolds Code Op.41-45 Year of study II Semester 2 Assessment (E/V/C) E Formative category: DF – fundamental, DG – general, DS – special, DE – economics/managerial, DU- humanities

DF



semester 120

Teacher(s) Prof. Liviu Ornea, CSI Sergiu Moroianu




Exact Sciences




30 20 10

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Riemannian geometry; Complex analysis

Recommended

Algebraic Geometry; Algebraic Topology; Differential Topology













1.Knowledge and understanding 57. Knowledge and understanding of the interplay between algebraic

and Riemannian geometry 58. Ability to use complex analytic tools in Riemannian geometry



48. Ability to decide classification problems in Riemannian geometry by using complex analytic tools.

49. Ability to attack more advanced and sophisticated problems in algebraic and Riemannian geometry.

3. Attitudinal 88. to develop an interest for the field; 89. to realize the importance of the fields of algebraic and Riemannian

geometry in contemporary mathematics and in theoretical physics; 90. to assume an ethical conduct in scientific research; 91. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lecture : 1. Complex and Hermitian manifolds. Canonical connexion. 2. Kaehler manifolds. Curvature. 3. Hodge theory. Topological consequences. 4. Chern classes. Calabi-Yau theorem. 5. Kaehler-Einstein manifolds. 6. Kodaira embedding theorem. 7. Vanishing theorems. 8. Bundles. Divisors. Adjunction formula.

Tutorials :

103. Examples and constructions of Kaehler manifolds. 104. Curvature of projective spaces. 105. Blow-up. Examples. 106. Weitzenboeck formula. 107. Quaternion and hyperkaehler manifolds.


Bibliography

35. W. Ballmann, Lectures on Kähler manifolds, European Math. Soc., 2006.

36. J.-P. Demailly, Complex analytic and differential geometry, disponibilă online la adresa http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf


37. D. Huybrechts, Complex geometry. An introduction, Springer, 2005.

38. A. Moroianu, Lectures on Kähler geometry, Cambridge Univ. Press, 2007

39. C. Voisin, Hodge theory and complex algebraic geometry, I, Cambridge Univ. Press, 2002.


Library



Written exam






Prof. Liviu Ornea,

CSI Sergiu Moroianu


Op.41-45 Computational Methods in Algebraic Geometry

Name Computational Methods in Algebraic Geometry

Code Op.41-45


DF



semester 120

Teacher(s) Assoc. Prof. Marius Vladoiu




Exact Sciences




30 20 10

** C-lecture, S-practicals/tutorials, L-laboratory practical activity, P-scientific project Prerequisites Required Algebra I,II, basic Algebraic Geometry Recommended Commutative Algebra













1.Knowledge and understanding 59. Knowledge and understanding of the interplay between

commutative algebra and combinatorics, and their computational aspects.

60. Knowledge and understanding of the algorithmic aspects of the computation of algebraic/homological invariants.

61. Ability to use mathematical software (Singular, CoCoA) for research.

2. Instrumental 50. Ability to use computer algebra software for learning, verifying

theorems/conjectures. 3. Attitudinal

92. to develop an interest for the field; 93. to realize the crucial importance of the computer algebra software

in the modern research; 94. to assume an ethical conduct in scientific research; 95. to optimally valorise one’s own potential in scientific activities.

SYLLABUS

Lectures: 66. Monomial orders, initial ideals, Gröbner bases, Macaulay’s theorem. 67. The division algorithm in the polynomial ring with n indeterminates,

Buchberger’s criterion for computing a Gröbner basis, Buchberger’s algorithm for computing a Gröbner basis.

68. Reduced Gröbner bases and their uniqueness. Elimination monomial orders, elimination’s theorem.

69. Algorithms in Algebraic Geometry (we work over the complex number field) applied to: the intersection of two varieties, projection of a variety, Zariski closure of difference of varieties.

70. Finite varieties algorithms. Sharply estimation of the number of points (without multiplicity) of the zero-set of a zero-dimensional ideal.

71. Irreducible varieties. Applied algorithms to determine the irreducible components of a variety and to compute the dimension of an irreducible variety.

72. Implicit equations of a rational parametrized variety and for the image of a regular morphism between projective varieties.

73. Multiplicity of the intersection point of two curves: methods of computation. Check whether a certain variety is smooth. Find the equations of the tangent cone of a variety at a point.

74. The degree and the arithmetic genus of a projective variety: applied algorithms.

75. The Betti diagram of the ideal of a collection of points in P^2. Tutorials: since each course comes with some specific algorithms, the tutorials are simply dealing with examples of the corresponding algorithms.


N.B. The above is a description of the topics to be covered and is intended to be in 1-1 correspondence with the 10 lectures.

Bibliography

1. D. Cox, J. Little, D. O’Shea - Ideals, Varieties and Algorithms, Springer, 3rd edition, 2007.

2. D. Cox, J. Little, D. O’Shea – Using Algebraic Geometry, 2nd edition, Springer, 2005.

3. G.M. Greuel, G. Pfister - A Singular Introduction to Commutative Algebra, Springer, 2nd edition, 2008.

4. V. Ene, J. Herzog - Grobner bases în Commutative Algebra, Amer. Math. Soc., vol. 130, 2011.

5. Jurgen Herzog, Takayuki Hibi - Monomial Ideals, Springer, 2010.


Library



Written exam



Correct solutions for all indicated subjects in final exam or correct solutions (for mark 5) and Average results to periodic/continuous testing.



Assoc. Prof. Marius Vlădoiu

advanced studies in mathematics-prezentare...

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