Onstring( Jointpolytop.SI

Eunjeongl.eewithyeong Dong-,

Yoo s i k Kim,andPark )YunhyungcholsungkyunkwanUri.ve.

)

Mirror symme.my forX

FameSmooth:

FamewassuntW:

N→GQ屹刈 르 Joe W

StminorE.g

X= P.

'QA씨=

이x.9%Eg

W :,→Z sdw

SYZmir orsymmetryicanbeconstmctedby.gluingPnono('sminorcharts

partialY ) Compactify.mg

it=0e
Et

여올,

Z근 9.

More

Specificany,

eachowgrangiantomsLCCM.co?assignsLaurenTpoy(Special )n=mi (ET → eM : F대0ThenMID
DE 1

d im L( F000' sd.sc potential )mi(uncorectedmiwrisobtained.by gwmg thoseton-

Kx1

cwseva.ir?YidS.teechgwimgmapsat5Miisfiesmuta+ion
→5m,

Fan .

var

---

(10g CY (Y i TE

flag-von'→--TE--=,

To ricFoosphere al )T

doublebruhat.eu
GHKK ).

( Cluster var )
.

WL.

of

Synthetic geometry
De fin( M W)
.

:

l.CM

(l)

symp.me.

toricof dim M=27

ishagrangianif.d.im L.

(2)moni Cold s.

=naol.todiscwitwc.pl/Istrj.-Jialm.cplx.str.onM.

D

리Uniti(D:e.g,JD )-7T.lt(M.

L).

L=Gf(j , 5 )
아( D)

MisCR

-to1if.

Simply not,Jo de=

의n

duo j.

duall.ie

E727( M.co )Example:cpt.symp.la.io

mfd.

µ: M

with

→ △ £112dg.

*二卄

momentpolytope.li
MirLagrang.am) For

(2)

Fix

re int ㅨ)

XE맾,.

LiThenis)=M :<=F0I,

TorresaHamilton ion

isµ x>,for The Stationgen.by

Take

an

Flow(II

Nu''

Fact :II

1'•.

gradient vec.la FieldMM )of Mx.

t.tk!
ifhol.discsboundedtatle.at#Facetsmanyhol.discs
Tby<-bounded.by

LrXp

Smooth.

wt ( p)=:Il,(j J)-,Lv

he n0,.

Color

Orbit

doing JIX.

*IfL.

redatedbyasymplectomorphism.the.nl#h0I.discs

L.

andarebounded.bynrrennnnnnope.rs
GW=시卄一 hol.discsbounded.by L2

in various tHow
o0toEachFor?

Counthol.d.sc

rePresents

anelementof El M L),,defmemodulispace.MIL{
M.LI/U:h0l. }BE TIM L),p),,u : ( D 2D) → (iofhol.discsboundedbyL.ws
B.

=M. ( Li.

{(U , z):ut M ( L ,p),

ZE Da] = p*와i

Mi ( L f )
,

(U.

Z)

-7tt

L
U(Z)

Evaluation

map}

mode "
withSpace ofOne

markethol.d.sc s

Point

Thm.

d.im M ( L , p )d.im ( L)=trap)-2WhereM:Masterof p∴Countryis( degreeofMeaninghee=p*µ ( p):( calledMaster(=yden# hol.discsbdd.by

cannot= i N

on'OpenM
L= 2.

.

pasng

ThroughGromou-Witeninvai.ci )Index

relativeChain

numberofaIndexdi s e)

ageneric ptinL)Potential function

monotonelegrangiantoms.tn
ForLiHorn ( H . ( LA)LETet:,→E

Complex due tone
ondefomationspacef.in,

orP'→듦in,

Me Space=PE파사)
calledisafunction

potentialofLE2PNEZ(IE세Li피싱 ).

.

wtdcountsGMaslovindexz.gg
Xp(→ToricFooCase△ = Pi NAexit

Thehol.d.is < s.

Central fiber

Lo :=ricaisTheUnique)Monotone

LogfiberThin ( Cho-WuOh )=EAll

ducwithIF,facetpexpitnta.jwiy.it

areM 2=Of:

Forwardregister andprimitive NormalF.

E"

x式訓

vector toF_

Computing

potential Ans"

fornon-torica.si~~

WFS

w

Def

sm

Givenooa.

proj.ua尤 ( £ 1 { o} )Tico) = Xo=X

xcalledaXa,flat

PNxest.it
股km

maphimAct고eto ricva玭一) c{t}( Normal )Toricpwj

is.

t.

Generation ofdeXWEF의

to Xo,E' (t)←와surject.ve

Tim( Hanada-Karen)

ㅋ Continuous

0 issyrplectomosphis.nu

onmap

Evili.

一一

Wo

0 : X→ x.

Xos.tl f

EYE ToIisa쏗옽ㅇㅿoopendeuse.com
oflegrtgiant.ru
Every generic
fiber

Ns.

Example.

X=ㅋSL3이B

toricdegeneratims.lt

X -7 N!

•.

onEast )玔一t

玉人入

issymple.to0o.

Gelfont-C대inP아78

항*In

Caseof( partial )5mg dar fiber of I

flag variety

withGo toricdesisotopicsubmfd.tnon-tomslegfibersappe.esisaSmooth.

,

every

Gorensteininotos.iedeX

Generation:tonede

Generation

WhereXo

isGreenstem Fan0-7

Xo

Est△o ㅋ cPopCor,Ek) is,X

MonotoneinX.

shouldFan 0he.

Thm ( Nishinow-Nohaa-U.de )IfX.

admitsasmallresolut.cn
non

then

Wu=EOFZ

F : feet

If Xohaswith,EX c

Coding Z 2.

locus

String polytopes.cio.

GSet=SheandX =G1B

arationdconexpolytopedeterminedbytuodataiili.ir

String polytope.isden ded.in. ) E wiIINE

드 wtlat.eeㅍ <다-.

--,△by유(NIn >.

nnsetfdomiuetwtshopentieshreacw.EE.

(71 09 (N n I I(2)A : # → a.

and

⇒(3)

Let

=

入irr.rep.ru/hishestwt
dimva.lk:

deermines

itextend.es

Ln i GXB0=to

X : B→1g z),,charactera.

b : = ( ybX유 itoricva.assoc.to( Alexear-Brian )

Xo is
가-m

△i

Fan o.

,i:

T→ E,

£0 B
제n z )"P

一伐 2 £ ( 2,2in )( Set △ 유 :△.

.

요=△ i 12.2 ,.

...

if)

containsaui.ge int

laticept.Guremte.ir( Xo is Foo)a isintegral )

Thin

InI

tone

degene ration

땀(2)IFWii→PWi*WhenandXq

adm itsWi

con b.

09.

(eWon

X9

earea.

→:

comes

Ireland

Small

Superpot.

Superpotential

byte Solution

=,

then

""

Clustermutat.org.

,

then

potentialAn