mr -th676:dg÷clayton/teaching/m676f16/lectures/day1.pdfpotent far project kvrietes in park n or spa...
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![Page 1: Mr -th676:Dg÷clayton/teaching/m676f16/lectures/day1.pdfpotent far project kvrietes In park N or spa 155% soc}) corresponds to 1h GIT qbtut XP 642. e), which is a prtalrobie f apnlifatk](https://reader033.vdocument.in/reader033/viewer/2022060821/6099ddd0ba23591a77656e90/html5/thumbnails/1.jpg)
Mr
-th676:Dg÷he main object in this case : 154%5013) .
Wht is th } thing ?
It B the sympkotr reduction f the toricsympkclc manifold 5x5xn÷5 by the diagml
Hamiltonian aothf So 131, redwd at the f.be or 8 in the moment polytope fthe akin .
It B (alma ) a Hric syapkdn manifold, mean git his ath - angle coordinates & a moment
mp
ii. 151¥ sds ) → PER"3
whelk push formal more an the aux pdytpe P
B a anstt multiple f hebegne mere .
The main god f the case D to unlisted wht the hide is gay on in the previous pwgph .
why sheld guard.
( aka .
random polygons ).
@ Th , space try at to be the moduli space f n. step dend random walks in
1123,
dih provide
He ksn trade model for ring polymers like batnsl DNA .
Applied mthemtzins wt to be able to mkgrte or th space and to sample points
uniformly fm it Hid D base smelts The of way to mtgrk B to do Monte Carlo
irtgnkh).
Th ,} area has alot of impressive numerical experiments & hewBlidphysrs
"
proofs : bt rbkef
little rigorous mthemkl jslifwk or real theses .
mmh f It is rigorous in the field his been peed us ,y synthetic geometry .
�2� synthetic redueth B the html potent ophkn in He Sppkdxatgoy . By the Kirwan - Kempf .Ness
them,there B a correspondence bt fat topology ) to the Geometric Invariant they 161T)
![Page 2: Mr -th676:Dg÷clayton/teaching/m676f16/lectures/day1.pdfpotent far project kvrietes In park N or spa 155% soc}) corresponds to 1h GIT qbtut XP 642. e), which is a prtalrobie f apnlifatk](https://reader033.vdocument.in/reader033/viewer/2022060821/6099ddd0ba23591a77656e90/html5/thumbnails/2.jpg)
potent far projectkvrietes
.
In park N .
or spa 155%soc } ) corresponds to
1h GIT qbtut XP' [ 642 . e),
which is a prtalrobie f apnlifatk f the
moduli spare f n - pointed ones f guy 0,
a ok.a . Man .
Thean reek Hw lakebrogeonhnz) moduli spaces & radar wdb seems to be completely unexplored
,
uhth obviously peak an opportunity for the mhwtdstht .
÷mdomWa÷tdgsid random walk in RdB an around co Hath funit sgmb whose directors are chosen
independents & unit rmf a the mit sphere Sd' 'E Rd
>
n
ta Tuon-
, ,5
For mddj polyus , he are ntrgtd in the the f the wlb & net so much their loath,so its cmmiut
to take the quotient J the to Hhgrp It
. Egninlwtf, Think f the edges as vats.
then the onfjmth span or mdnti she fn step lgniktnl) random w.lk in Rd is jft Chee!This B a opt
,
Riemannian,
nldt ) . dinnsid mitld.
It B my sole to mtgte ar th } space ( use
sphil cowdiwts ! ) & to same powb uniformly from it ( hai. ) .
In fat , in prone one wnlf dost are it the orient th of the random w.lk either.
so the
spare yoire refitted in D oftn 1st ')%o (d) ,
the onfignklmohli space of n step radar was
in Rd up to trans HR & attn.
![Page 3: Mr -th676:Dg÷clayton/teaching/m676f16/lectures/day1.pdfpotent far project kvrietes In park N or spa 155% soc}) corresponds to 1h GIT qbtut XP 642. e), which is a prtalrobie f apnlifatk](https://reader033.vdocument.in/reader033/viewer/2022060821/6099ddd0ba23591a77656e90/html5/thumbnails/3.jpg)
HomeO Its hrdr to one up
wl nice words.
on th space &
�2� This is nt netlf a manifold I why ? )
so usually jst deal wl HI.
=nd wdbs / polygons.
suppose we rw require or n ' step robin walks to fm loops, meenytyretm to their stay port after eat
n steps.
this D obvif one subset of 15" ) ' or 15%0 (d) .
hat how a not
orhands
on it ?
Wbt is the done wdith algebraically ?
£, + ..
. ten = E.
Notice thtths B really depths,one freak gut flhe rector.
To mhe it fry , define n : 5"
→ Rd yule ,.
. in ) te ,+
. . ten.
then the loop rnkn
walks unit to ). Tf 0 3 a rglr uk fu ,
then this shld by a smooth sbmifldf C. ') " of
diners: on nld - i ) - d = nd - n . d.
We might also he intend in nillolbo (d) whih Shoed be a manifold /or something ) of dimension
nd - n - d - din ( Sold )) = nd - n . d - dH¥.nld - I ) . dtdst' I
Far polymers u are motf in listed in the used =3,in MM are lhse m th spy & their ( real ) dims ins ;
Sporkin(g)
"
2 n
lush'fg°' ' |2n- 3
2 n - 3
ni'to Vsol 3) 2 n - 6
![Page 4: Mr -th676:Dg÷clayton/teaching/m676f16/lectures/day1.pdfpotent far project kvrietes In park N or spa 155% soc}) corresponds to 1h GIT qbtut XP 642. e), which is a prtalrobie f apnlifatk](https://reader033.vdocument.in/reader033/viewer/2022060821/6099ddd0ba23591a77656e90/html5/thumbnails/4.jpg)
Tahj fw grated the cannon to lopbx algebraic geometry , You attain shed nkf fog on the
spares with em red dinars,
sue these at test here a choice f bey Copley algebra variety .
Indeed,she 5 = EP
'
,
he her a nkl ihlitwk Hutu well : spa fmh mis & the pgok wief #'T.
Much mere subtle , the ft tut ii' to Vsa } ) is Also a project variety .
Bet he stthegtbh,we need to Hk .
Sit akt manifolds. . .