Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/dfc/2009/reports/Kiritchenko2010report.pdf Дата изменения: Tue Dec 14 10:27:00 2010 Дата индексирования: Mon Feb 14 00:21:42 2011 Кодировка: Windows-1251

Dynasty Fellowship Report 2010
lentin uirithenko

IF

Main results

sn PHIHD s mostly workedD together with ivgeny mirnov nd ldlen imorinD on new pproh to huert lulus on the vrieties of omplete )gs in Cn using the volume polynomil ssoited with qelfnd!etlin polytopesF his pproh llows us to ompute the intersetion produts of huert yles y interseting fes of polytopeF en exmple for n = 3 is disussed in uID some of the results were nnouned in uPD nd full version of ll results with proofs ppers in the preprint uF he min results re desried elowF o mke the exposition lererD s (rst review some previously known results tht we usedF yur pproh uses polytope rings introdued y uhovnskii nd ukhlikovF ith eh onvex polytope P D they ssoited grded ommuttive ring RP tht lives in degrees up to dim(P ) nd stis(es oinr? dulityF ht isD polytope ring ehves like the ghow e ring @or ohomology ringA of smooth vrietyF sn ftD they proved tht for n integral ly simple polytope P @simple mens tht there re extly d = dim(P ) edges meeting t eh vertexD nd integrlly simple mens tht primitive integer vetors prllel to the edges generte the lttie Zd A the ring RP is isomorphi to the ghow ring of the orresponding smooth tori vriety XP F ingle fes of P give rise to ertin elements of RP @representing yles given y the losures of the torus orits in XP AD whih generte RP s n dditive groupF sf [F ] is the element of RP orresponding to fe F D then [F ] ћ [G] = [F G] in RP D provided tht F nd G re trnsverseF sf [F ] nd [G] re not trnsverse one n lwys reple [F ] @using liner reltions in RP A y liner omintion of fes tht re trnsverse to G @there is wellEknown lgorithm for thisAF ht hppens for nonEsimple P c uiumrs uveh hs relted the polytope rings of some nonEsimple polytopes to the ghow rings of smooth nonEtori spheril vrietiesF sn prtiulrD he oserved tht the ring RP for the GelfandZetlin polytope P = P @whih is not simpleA ssoited with stritly dominnt weight = (1 , . . . , n ) Zn of the group GLn (C) is isomorphi to the ghow ring of the vriety X of omplete )gs in Cn F his ws the strting point of our investigtionF pirstD we developed tehniques for multiplying two elements in the ring of ny nonEsimple polytope P y multiplying lifts of these elements to the ring of simple polytope Q tht resolves P etion PD uF his result is used sustntilly in our omputtions in the ring of qelfndEetlin polytope @whih is not simpleAD nd might e used for more generl polytopes ssoited with other spheril vrietiesF sn prtiulrD this llows one to ssoite eh element of RP with liner omintion of fes of P @though not every fe of P orresponds to n element of RP AF his is hieved y onstruting n intermedite ZEmodule MP,Q suh tht there re funtoril injetion RP MP,Q nd surjetion RQ MP,Q @unfortuntelyD there is no funtoril homomorphism etween RP nd RQ for nonEsimple P AF
1


2

Dynasty Fellowship Report 2010

xextD we nswered the following nturl questionX how to express huert yles s liner omintions of fes of the qelfnd!etlin polytope P c qiven two huert yles X w nd X w D we n represent X w nd X w s sums of expliitly desried fes heorem RFQD gorollry RFSD u so tht every fe ourring in the deomposition of X w is trnsverse to every fe ourring in the deomposition of X w gorollry RFUD uF his llows us to represent the intersetion of ny two huert yles y liner omintions of fes with nonnegtive oe0ients whih might led to trnsprent vittlewood! ihrdson rule for the vrieties of omplete )gsF yne of our presenttions for huert yles is formlly similr to the pomin!uirillov theorem on huert polynomils in terms of pipe-dreamsD ut n not e dedued from the ltter euse the elements [F ] will usully not elong to RP @only to MP A nd hene nnot e identi(ed with monomils in the orresponding huert polynomilF ipeEdrems re simple omintoril o jets whose reltion to fes of qelfndEetlin polytopes ws (rst notied y uognF e lso otined expliit formuls for the hemzure hrters @orresponding to the weight A of huert vrieties in terms of exponentil sums over lttie points in unions of fes in P heorem RFVD uD whih implyD in prtiulrD formuls for the degrees of huert vrieties vi sums of volumes of fes nd formuls for rilert funtions of huert vrieties vi irkhrt polynomilsF ell these formuls re in the spirit of the theory of xewton polytopes @qelfnd!etlin polytopes nturlly orrespond to pro jetive emeddings of the )g vrietyAF pormuls for the hemzure hrters nd for the degrees of huert vrieties generlize reent results of ostnikov nd tnley from very speil huert vrietiesD nmely uempf @or (1 3 2)EvoidingA vrietiesD to ll huert vrietiesF fesidesD s yprodut of our proof we found geometri reliztion of mitosis @ omintoril proedure for omputing huert polynomils in terms of pipeEdrems introdued y unutson nd willerA nd miniml reliztion of simplex s ui omplex di'erent from the previously known reliztionsF sn PHIHD tens rornostel nd s revised our pper on huert lulus in the lgeri oordism of omplete )g vrieties for ritrry redutive groups over (elds of zero hrteristi ruF por vrieties with lgeri ellulr deompositionD we dded the isomorphism theorem etween their lgeri nd omplex oordism eppendixD ruF PF

Publications and preprints

uI GelfandZetlin polytopes and ag varietiesD snterntionl wthemtis eserh xotiesD PHIHD noF IQD PSIP!PSQI uP From moment poytopes to string bodiesD yerwolfh eportsD IWGPHIHD QH!QQ ru joint with Jens HornbostelD Schubert calculus for algebraic cobordismD tournl fur die reine und ngewndte wthemtik @grelleAD PU pgesD in press ? u joint with Evgeny Smirnov and Vladlen TimorinD Schubert calculus on GelfandZetlin polytopesD http://www.mccme.ru/ valya/Schubert.pdf uQ Исчислительная геометрия: метод Шаля и ШубертаD sumitted to uvnt


lentin uirithenko

! Q! QF

hynsty pellowship eport PHIH

Talks

Invited conference talks
eugust epril sgwPHIH tellite gonferene on gomplex qeometryD qroup tions nd woduli spesD ryderdD sndi yerwolfh workshop elgeri groupsD yerwolfhD qermny

Seminar talks
xovemer FsF ernold seminrD wosow tte niversity ytoer wosow wthemtil oiety meeting eugust t snstitute of pundmentl eserhD wumiD sndi RF preie niversit? ferlin t t snstituteD wumi

International collaboration
pro jet rorospheril vrieties nd polyhedrl divisorsD joint with ulus eltmnn nd vrs etersen pro jet iquivrint oordism of spheril vrietiesD joint with emlendu urishn SF

Teaching

sn winter PHIHD s tught grdute topis miniEourse pheril vrieties t the snstitute for wthemtisD preie niversit? ferlinF ine eptemerD s hve een working t t the pulty of wthemtisD righer hool of ionomisF s ondut prolem solving sessions for the Pd yer undergrdute ourse glulus ss nd help with prolem solving sessions for elger ssD opology ss nd qeometry sF s m lso one of the orgnizers of the undegrdute lerning seminr ymmetri funtionsD qrssmnnins nd )gsF s help to oordinte our hh progrm in wthemtis tht strted in xovemerF sn prtiulrD s ws responsile for the dmission exmsF s supervise the following undergrdute ourse pro jetsX vel qusev @Qd yerA fEvetors of qelfnd!etlin polytopes veonid nushevih @Pd yerA gontinued frtion of e hmitry rifyn @Pd yerAD wonodromy of quss hypergeometri funtion sn PHIHD s wrote populrizing rtile for high shool students on the huert method for solving enumertive geometry prolems uQF s lso revised the (rst prt of the notes to my ourse qeometry of spheril vrietiesF