Int G eHs wean

ref Fulton Padhaoipa.de 1997Costa KentsetD modulispace of feral maps

Enumerativegeometry quantum 06 logySolutions to c calf multipl tableen un geom of g c ring associativeIII

ation between coefficients

solveproblem ingeneral

Apply idea to fed Nd rationalplane wives g Dpassingthrough Sd 1 general pro b

Outline modulispace compactification

1Katsevich op of stable maps

1Gromov With the

dQuantum 66

dcompute ad 9 c

T Lt theory on Kortseuidspace

Talk 2 Compactification ofmodulispaces

Mg prog ironing curves ofgenesgpig prey G n nodal curves ofgoing µ1eNg n Lprog noising cu we ofgene g u L n distinct points

MIN L program nodal caves genus_g wi n distinctnonsing me bedptsstability condition finite atomgrasps

g o tho u treesof prog lines meetingtransversely each uldistinct

tread a soso.ir's D

E ai nos Mo ImoL.µ

Zn 4 Me P'von 3

Mo.AE IP ndry g s 19

AIB hi h3t T get denser DCA 1B C II n32pts

II

I DHforgetfulMorphisms Fgm Fgmh 7 m

forgets n n pointsEx MT T Fo a

i.sesieuksk i

i j k e E I n3

Fo n Fo si j ieE3 IP

i e PCi j k l

In IP Phi I k e P Ci k Ij l P ie l j kl ly equivalent drones difference corresponds

to tape 6 of a 45ThPullback 2 DCAID a knotALIBIED

kontseuichspe.ee of stablemaps diskequiv's2

X smoothpegvariety p E Ag X

Mgm X pGR sp pCprog norsing Ves g

C pi sp pi n C pi pi ap piif 7 C C

ont trx

MTCXp L C p r pp stable s V E E C irred o p

47 EE IP getsmapped to a ptKe E a 33 specialpls

l E genes I mappedto aptHer E F 31 specialpt

EI CcFg n Cpt o Fgm

MT CP 1 E G Ci nGrassmancan

b Agn X ok Mg.mx X

properties a X to make ill n Xp nice

e x X is convex ifVp IP X H Ctp p't TX o

makes MT n X p smooth

r ko generous C Liegp actingtransitively on X

TX glob ger X convex have naturalbasis of Schubertclassesereted

for AI X

When X prog nonsang cux have th s 1 3

te n Xpnormalprog war whose sing ee quot of re Sig va by

Lutegroup

tonCxp dim

din Xt Sp GTX th 3 if o

the openlocus ofstable mapswith no untrue ant MI Xp is smooth

Bdry of them X p i a divisor w normal

Boundary of Tom Xp34 A I D En pitpre p E A Xhave divisor D CAD pype

Toaol.ycx.p.lyTo Due.zCXpn

0

it Fi j k ELI

Dci liee 2 DCA B Pi PsAI En

iY Yepaper p

Forgetful nor the n X p MI si ke3 P

PC lieDci j 16 e

keel relations

ie ants

is i En haveevaluation maps pgpMT X p X

Ec pi Pn µ ftCpGiven 2 K E A X

J pics u offerMTn X p Il

Ip Cri K GronerWriterinvariants

Properties of Ip h rn

a Symmetry in Vib if 8 homogeneous then Ipcr rn possiblynonzero

dem ri din ATc en k between GW and

enumerativegeometry

Learmonth Ti Tn pur din ab var of X corresp r 2n CA X

I calm IT dm ti G 2g g general elts

They 1 The scheme theoretic intersect

pi g h n n pi g Ris a fr of redused pts Supp n KMonEsp121 Iplo Rn

Gse9u c Ipcr m pfdmaps µ P X nap

Alpi cg pclan p c ArXst

14 4 R2 p de my Ti IP

n 7 d I 2n zt Jci TR th 3diene

Id Ep Ep NdId I h 3d i

TalkhIm Aix i cycles

A X Ane X Cn il cycles

fix 14 properget f A X A Y

VT 1 degCv fcv far if din D far0 otherwise

Properties of Ipcr NnI p e Mo.ua p o Mo.nxX

6 Ctcp pi HCGp pap Ip n

l Xpipit feel

pi Cr u upicrnp Chu urn

Ipcr K f p't no um I rib urnat nxX p MIX

0 En EZ the o

no the n positive din

p has positive den fibers

p Cm xx 0

3 MIn tpt

Ip Cr nm r.ururg

Case

II r I C A X

Pto Ip Cr rn p u upicraMT Cxp

4 mgncx.pi M n icx.pl L e Cpicrew up CraMonXp

pick u upickMT Xpo

has poi den fibersp o

Io Cr rib 1.8209

II o C A X p c XThen Eph rn K 8 Ip K K

Leanna K Ipcr rn pouted naps µ C X witha CECIp sit

Nataraj Itslogan There are 4pm choices forpcp

Q gy Assume X homogeneous so we havea basisof SchubertclassesTo L C A X T Tp C A'X Tpa Tna E A X Cte rest

for each n o wedefine theumber NCnp i nm p Ip Tpi TinThis is non zoo adin sumsto de MI Xpin whichcase it is plantedraked men meeting n gene npr

µof T Kpel E i E m

o Ei m Si's J T UT

Ki Schubert basisVi F j gg fo inwhich case gy I

define Cg S to be the nurse of gigNew we have

saT.utj.gegfxTiuTjUTe gefTfE eFfIo.TiTjTegefyzgefgef JTeUTf

Tf caff of TNT usualahemringIT UT off TyUJ u Te Te u Ie Fmit bydeferring

A awayfrom 0

For 2 C A X defne thepotential function

Crs E Z s Ipcreven info

Lemma15 th F only fnifeg nay effectivePEA X set Ip Cr foconsequence 8 y T

Go ym ter Ip Cto Tin 7is aformal powerseries in Eye yid QEEyD

O E i j le ein define

us loishygy Z th 8 T DThe

dfne quantumproduct by Ti Tj ti hegetTfExtend QLTyD 1nearly t Q 4777 nod A X oQKyD

Commutative4 it is To L Fellows from I 145

AssociativeCT y Tie Egg Edt get loftedSldTdT

THEE Ee t geeegeetdDeEkIEEEY3

Finally define QH X quantum cohomology rug

Grp A X Q LTV D Gra Gto ofthegraded polyr

Y.toosymCv xoQatunique wax ideal

defies care rug structure on Lee TEVTTnoduleLIV D V vector fields on A xIl

Big quantumcocoa rugi X vo QFv D

compute Nd using g c r

yo Ym lodassical y quantumx Xpto

11 f o

n.E.tn ftioo utiY IIreplace loguanfuncy by Tey 2 I Nhp ii am pis40put Tam o

e Die GTPYPYp

i IamCate X lP2 To L Ti Tene Te EptT Tj Pigott t.ge'T IKIT T It Tin T t Tiz ToT HI Tiz T T T 122 To

T Tu TWiT t 5222To

CT Ti Te T CT Tu Tere TE Therwhen f dike N Cn p 40 only when n ydin which case it is Nd

t g D and edy yid 1

compute Tigle and use 158d

Nd D.IQ Nd.NdzLdidEPgd z d dz 3ddI

59