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)

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.

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

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. . .