Merthyr Tydfil, Wales

Publications and preprints

2021
  • Slide polynomials and subword complexes (with Anna Tutubalina)
    Sbornik: Mathematics, 2021, 212:10, 1471–1490
    [arXiv] [PDF] [PDF in Russian] [Short version (2 pages, in Russian)]

    Subword complexes were defined by A. Knutson and E. Miller in 2004 for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S. Assaf and D. Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres.

2020
  • Pipe dreams for Schubert polynomials of the classical groups (with Anna Tutubalina)
    Preprint, 29 pages
    [arXiv] [PDF]

    Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety of a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for the Weyl groups of the classical types.

  • Multiple flag varieties
    Journal of Mathematical Sciences 248, 338-373 (2020)
    [PDF] [PDF (in Russian)]

    This is a survey of results on multiple flag varieties, i.e. varieties of the form G/P1 x…x G/Pk. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of G-orbits and existence of an open G-orbits on a multiple flag variety.

2017
  • Singularities of divisors on flag varieties via Hwang's product theorem
    Bulletin of the Korean Mathematical Society, 54 (2017), vol. 5, 1773–1778.
    [arXiv] [PDF]

    We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety X=G/P and an effective Q-divisor D stable with respect to a Borel subgroup the pair (X, D) is Kawamata log terminal if and only if [D]=0.

  • 2016
  • Determinantal identities for flagged Schur and Schubert polynomials (with Grigory Merzon)
    European Journal of Mathematics 2:1, 227–245 (2016)
    [arXiv] [PDF]

    We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.

  • Grassmannians, flag varieties, and Gelfand-Zetlin polytopes
    Recent Developments in Representation Theory. Proceedings of Maurice Auslander Distinguished Lectures and International Conference, Contemporary Mathematics series, 673, 179–226. American Mathematical Society, 2016.
    [arXiv] [PDF]

    The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics of Gelfand-Zetlin polytopes.
    These are extended notes of a talk given at Maurice Auslander Distinguished Lectures and International Conference in April 2013.

  • Ribbon graphs and bialgebra of Lagrangian subspaces (with Victor Kleptsyn)
    Journal of Knot Theory and its Ramifications, 26 (2016), 1642006, 26 pages.
    [arXiv] [PDF]

    To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of L-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

  • 2012
  • Schubert calculus and Gelfand-Zetlin polytopes (joint with V. Kiritchenko and V. Timorin)
    Russian Mathematical Surveys, 64:4, 685–719 (2012)
    [arXiv] [PDF] [PDF (in Russian)]

    We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.

  • Springer fiber components in the two columns case for types A and D are normal (joint with N. Perrin)
    Bulletin de la Société Mathématique de France, 140:3, 309-333 (2012)
    [arXiv] [PDF]

    We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N2=0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen-Macaulay, and have rational singularities.

  • 2008
  • Desingularizations of Schubert varieties in double Grassmannians
    Functional Analysis and its Applications 42:2, 126-134 (2008)
    [arXiv] [PDF] [PDF (in Russian)]

    Let X be the direct product of two Grassmann varieties of k- and l-planes in a finite-dimensional vector space V, and let B be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog to Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott--Samelson desingularizations for Schubert varieties.

    2007
  • Bruhat order for two subspaces and a flag
    Preprint, 30 pages
    [arXiv] [PDF]

    The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a finite-dimensional vector space V; or, equivalently, the closure of an orbit of the group GL(V) acting on the direct product of two full flag varieties. We obtain a similar result for triples consisting of two subspaces and a partial flag in V; this is equivalent to describing the closure of a GL(V)-orbit in the product of two Grassmannians and one flag variety. We give a rank criterion to check whether such a triple can be degenerated to another one, and we classify the minimal degenerations. Our methods involve only elementary linear algebra and combinatorics of graphs (originating in Auslander-Reiten quivers).

  • Orbites d'un sous-groupe de Borel dans le produit de deux grassmanniennes (Orbits of a Borel subgroup on the direct product of two Grassmannians)
    Ph. D. thesis (in English, with an introduction in French)
    [PDF, 1,7 Mb]

  • 2006
  • On generating sets of ideals defining S-varieties
    Moscow Univ. Math. Bull. 60 (2005), no. 3, 1-4 (2006)
    [PDF] [PDF (in Russian)].

    Let G be a semisimple algebraic group. A theorem due to Kostant describes the ideal defining the G-orbit closure of the sum of highest-weight vectors in a (reducible) G-module, such that the corresponding highest weights are linearly independent. This ideal is generated by quadratic polynomials. In this paper we generalize this result, assuming that the highest weights can be linearly dependent. In this case the equations defining these varieties are not necessary quadratic.

Expository texts, refereed proceedings

    2011
  • Convex chains for Schubert varieties (joint with V. Kiritchenko and V. Timorin)
    Oberwolfach reports, 41/2011, 15-18.
    [PDF]

    This is an extended abstract of Valentina Kiritchenko's talk on the Oberwolfach meeting «New developments in Newton-Okounkov bodies» in August, 2011. We define analogs of Demazure operators acting on convex polytopes. This leads to a simple inductive construction of Gelfand-Zetlin polytopes and their generalizations.

  • Gelfand-Zetlin polytopes and Demazure characters (joint with V. Kiritchenko and V. Timorin)
    Proceedings of the International Conference «50 years of IITP», ISBN 978-5-901158-15-9, 5 pages
    [PDF]

    This is a short self-contained exposition of our formula for the Demazure characters of Schubert varieties in terms of the exponential sums over faces of Gelfand-Zetlin polytopes. All necessary definitions are given in the text.

Educational texts, popularization of mathematics

    2022
  • Friezes and continued fractions
    MCCME, Moscow (2022). 64 pages, in Russian.
    [PDF]

    We describe the notion of Conway-Coxeter's friezes and their relation to continued fractions, the Farey graph, the hyperbolic plane...

  • 2019
  • On quadratic residues
    Kvant No. 10, 2019, pp. 2–11, in Russian
    [PDF]

    We discuss the main properties of quadratic residues and give a proof of Gauss's quadratic reciprocity law due to Eisenstein.

  • 2015
  • Ideas of Newton-Okounkov bodies (joint with V. Kiritchenko and V. Timorin)
    Snapshots of Modern Mathematics from Oberwolfach, 8/2015
    [MFO website] [PDF]

    In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion.

  • Young diagrams and q-combinatorics
    Kvant No. 1, 2015, pp. 7–12, in Russian
    [PDF]

    In this paper for high school students we introduce Gaussian binomial coefficients and discuss their properties, comparing them to well-known theorems from ``ordinary'' combinatorics. We also obtain classical results due to Euler about the generating function for partitions.

  • Three glances on the Aztec diamond
    MCCME, Moscow (2015). 48 pages, in Russian
    [PDF]

    We discuss three proofs of the Aztec diamond theorem, stating that the number of domino tilings of an Aztec diamond of rank n equals 2n(n+1)/2.
    These are the notes of a mini-course given at the Summer school “Contemporary Mathematics” in Dubna, Russia, in July 2014.

  • 2014
  • Young diagrams, plane partitions and alternating sign matrices
    MCCME, Moscow (2014). 64 pages, in Russian
    [PDF]

    How many ways are there to decompose a natural number into several summands, if two sums that differ only in the order of their summands are considered the same partition? It turns out that there is no simple answer to this seemingly elementary question. However, this question gives rise to a very interesting theory, and the results of this theory have applications in various branches of mathematics and mathematical physics.
    These are the notes of a mini-course given at the Summer school “Contemporary Mathematics” in Dubna, Russia, in July 2013.

  • 2009
  • Reflection groups and regular polyhedra
    MCCME, Moscow (2009). 48 pages, in Russian
    [PDF]

    Notes of my mini-course for undergraduate students given at the Summer school “Contemporary Mathematics” in Dubna, Russia, in July 2008. The main goal of these notes is to prove Schläfli's theorem on the classification of regular polyhedra in an n-dimensional space. As a byproduct of our proof, we obtain a classification of finite reflection groups in a Euclidean space.

Selected presentations

    2015
  • Schubert polynomials and rc-graphs
    [PDF]
    Meeting of Generations (Dynasty Foundation conference), Moscow, June 9–11, 2015
  • Partitions and Young diagrams: from Euler to our days
    Slides [PDF] Problem set [PDF]
    « Junior Data Analysis School» (Malyi ShAD), Yandex, Moscow, Russia, February 14, 2015, in Russian.
    An expository talk for high school students on combinatorics of Young diagrams and related objects.
  • 2014
  • Schubert polynomials and pipe dreams
    [PDF]
    Topology of Torus Actions and Applications to Geometry and Combinatorics (ICM Satellite Conference), Daejeon, Republic of Korea, August 7, 2014
  • 2013
  • Symmetry in mathematics and beyond
    [PDF]
    Polytechnic Museum, Moscow, Russia, October 8, 2013, in Russian.
    This is a lecture for general audience on various appearances of symmetry in nature, art and mathematics.
  • 2012
  • Schubert polynomials, pipe dreams and associahedra
    [PDF]
    Algebra and Geometry: International conference in honor of Askold Khovanskii's 65th birthday. Moscow, Russia, June 4, 2012
  • 2010
  • Schubert calculus and Gelfand-Zetlin polytopes
    [PDF]
    International Conference on Representations of Algebras (ICRA XIV), Tokyo, August 12, 2010