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Dynasty Fellowship Report 2011
Valentina Kiritchenko 1.

Main results

In 2011, my main results are a construction of convex geometric divided dierence operators (joint with Evgeny Smirnov and Vladlen Timorin) and a description of equivariant algebraic cobordism of complete ag varieties and of minimal rank wonderful symmetric varieties (joint with Amalendu Krishna). I also continued the joint pro ject with Klaus Altmann and Lars Petersen on a relation between colored fans on one side and polyhedral divisors on wonderful compactications on the other side. Divided dierence operators (or Demazure operators) play a key role in Schubert calculus and representation theory. We constructed convex geometric analogs of Demazure operators. Geometric Demazure operators act on polytopes and take a polytope to a polytope of dimension one greater. For instance, Gelfand-Zetlin polytopes can be obtained by applying a suitable composition of geometric Demazure operators to a point. In contrast with the classical Demazure operators, the convex geometric Demazure operators can be dened not only for the root system of a reductive group but for a more general combinatorial datum. The construction and main results are outlined in [KST3]. This can be further developed in three directions. First, the geometric Demazure operators can be used to give a convenient description of NewtonOkounkov polytopes for BottSamelson resolutions of Schubert varieties (for GLn such a description can be deduced from the iterative construction of these polytopes given by Dave Anderson). In particular, an elementary description of string polytopes (generalizations of GelfandZetlin polytopes to arbitrary refductive groups) can be obtained this way. Second, the results of [KST1] (presentation of Schubert cycles by faces of the GelfandZetlin polytopes and formulas for the Demazure characters via exponential sums over integer points in these faces) can be extended to arbitrary reductive groups. In particular, a general version of mitosis (a combinatorial procedure for computing Schubert polynomials in the case of GLn in terms of pipe-dreams introduced by Knutson and Miller) can be constructed this way. Third, the geometric Demazure operators can be studied in a more general setting (not related to the classical root systems). For instance, Newton polytopes of Bott towers (toric varieties obtained from a point by successive pro jectivizations of rank two vector bundles) admit a simple description via geometric Demazure operators. Equivariant algebraic cobordism is dened for algebraic varieties (over a eld k of zero characteristic) with an action of an algebraic group G. The denition reminds of the denition of equivariant Chow rings, namely, instead of considering the universal classifying space B G (which is not algebraic) one takes a sequence of algebraic varieties approximating B G and then takes the inverse limit of their Chow (or cobordism) rings. The only dierence is that for cobordisms this sequence should be chosen more carefully (since cobordism can be non-zero for arbitrary negative degrees). The resulting equivariant cobordism ring can be non-trivial in arbitrary positive as well as negative degrees. For
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Dynasty Fellowship Report 2011

instance, if G is a torus of dimension n then the equivariant cobordism ring of a point is isomorphic to the ring of graded power series in n variables with coeents in the Lazard ring (that is, coincides with the complex cobordism ring of B G). For a connected reductive group G split over k , we obtained a Borel type presentation for the equivariant algebraic as well as complex cobordism of the variety of complete ags G/B . We also computed the equivariant algebraic cobordism ring for the symmetric varieties of minimal rank by extending the methods of Brion and Joshua from equivariant Chow rings to cobordisms. We relate the language of colored fans (combinatorial ob jects describing spherical varieties) and the language of polyhedral divisors (describing varieties with a torus action). Colored fans are usual fans together with an additional combinatorial data (colors), namely, some rays in the fan can be colored. Polyhedral divisors are linear combinations of usual divisors with coecients being polyhedral fans (that is, decompositions of an ane space into polyhedra). Each spherical variety can be described by a colored fan as well as by a polyhedral divisor on a suitable wonderful compactication (the latter are equivariant compactications of homogeneous spaces with nice properties, in particular, they have a unique closed orbit). Our plan is to describe the polyhedral divisor in terms of the colored fan and vice versa. Previously, we did this for horospherical varieties but recently realized that the same approach works for arbitrary spherical varieties.

Publications and preprints [KST1] joint with Evgeny Smirnov and Vladlen Timorin, Schubert calculus and Gelfand-Zetlin polytopes, 33 pages, arXiv:1101.0278v2 [math.AG] [KST2] joint with Evgeny Smirnov and Vladlen Timorin, Gelfand-Zetlin
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polytopes and Demazure characters, Proceedings of the International Conference 50 years
of IITP, 5 pages, Moscow, IITP RAS, 2011 [KST3] joint with with , Convex chains for Schubert varieties, Oberwolfach reports, 41/2011, 15-18 [KK] joint with , Equivariant Cobordism of Flag Varieties and of Symmetric Varieties,18 pages, arXiv:1104.1089v1 [math.AG] [HK] joint with , Schubert calculus for algebraic cobordism, Journal fur die reine und angewandte Mathematik (Crelle), Volume 2011, no. 656, 5985

Evgeny Smirnov and Vladlen Timorin Amalendu Krishna Jens Hornbostel
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Talks

Conference talks
July August International Conference 50 years of IITP, Moscow Oberwolfach mini-workshop New developments in NewtonOkounkov bodies, Oberwolfach, Germany

Seminar talks
February Seminar of the Laboratory of algebraic geometry, Higher School of Economics, Moscow


Valentina Kiritchenko April

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Dynasty Fellowship Report 2011

Seminar Riemann surfaces, Lie algebras and Mathematical physics, Independent University of Moscow November Lie groups and Invariant theory Seminar, Moscow State University 4. Freie Universit Berlin at

International collaboration
pro ject Spherical varieties and polyhedral divisors on wonderful compactications, joint with Klaus Altmann and Lars Petersen (visited Moscow in November 2011) pro ject Equivariant cobordism of spherical varieties, joint with Amalendu Krishna 5.

Tata Institute, Mumbai

Teaching

I teach on a regular basis at the Faculty of Mathematics, Higher School of Economics. Together with Alexander Kolesnikov, I teach a course Calculus of variations and optimal control:

http://www.hse.ru/edu/courses/34463194.html I participate in problem solving sessions for the 1st year undergraduate students: courses Calculus I, Algebra I, Geometry I and Logic. Together with Alexey Gorodentsev, I run an undegraduate learning seminar Toric geometry, Grassmannians, ags and symmetric functions. http://vyshka.math.ru/1112/sem_gor-kir.html In February 2011, I gave a lecture Polytopes and equations at the winter mathematical school for university students organized by the HSE: http://www.hse.ru/news/recent/26949846.html In summer 2011, I taught a course Schubert calculus and 3264 conics for high school students at the summer school Contemporary mathematics in Dubna: http://www.mccme.ru/dubna/2011/courses/kirich.htm I supervise a 4th year student (Diploma f-vectors of GelfandZetlin polytopes) and two 2d year students (course pro jects Schubert polynomials and FominKirillov theorem). At the HSE, I coordinate the PhD program in Mathematics. In particular, I was responsible for the admission exams and for licensing of a new PhD program in Geometry and Topology in 2011. I've also prepared a proposal for an academic PhD program in Mathematics (the academic or full-time PhD program already exists at the HSE for some sciences and provides scholarships worth 25000 Rubles per month).