Seminars in April 2014

  • Hiu-Fai Law, F-Matchings in a tree
    F-Matchings in a tree
    Hiu-Fai Law
    University of Hamburg, Germany
    2014/04/28 Monday 4PM-5PM
    Room 1409
    Given trees F and T, a F-matching is a collection of disjoint copies F in T. Generalizing results from Wagner (2009), Alon et al (2011) proved that the number of F-matchings for fixed F in a random tree T whose size tends to infinity is a.a.s. a multiple of any given integer m>0. We will discuss inverse and extremal problems on the number of F-matchings.
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  • Andrew Goodall, Graph polynomials from simple graph sequences
    Graph polynomials from simple graph sequences
    Andrew Goodall
    Charles University, Prague
    2014/04/14 Monday 4PM-5PM
    Room 1409
    The chromatic polynomial evaluated at a positive integer n is equal to the number of homomorphisms to the complete graph on n vertices. Many important graph polynomials are likewise determined by counting homomorphisms to a sequence of (multi) graphs, such as the Tutte polynomial and the independence polynomial. We give a powerful construction that produces graph polynomials in this way from sequences of simple graphs. It uses just two fundamental operations (disjoint union and interpretations of relational structures) and starts from a simple set of building blocks (coloured transitive tournaments).This will be illustrated by many examples, some of whose combinatorial properties have been well explored (such as that of the chromatic polynomial), but most of which are new. This is joint work with Jarik Nesetril and Patrice Ossona de Mendez.</p>
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  • Tomoo Matsumura, Pfaffian Sum formula of the double Schubert polynomials for the symplectic Grassmannians
    Pfaffian Sum formula of the double Schubert polynomials for the symplectic Grassmannians
    Tomoo Matsumura
    KAIST
    2014/04/08 Tuesday 4PM-5PM
    Room 1409
    It is well-known that the ring of symmetric polynomials has a special basis, consisting of Schur polynomials and being indexed by Young diagrams. There underlies the representation theory of general linear groups and the combinatorics of Young diagrams. The geometry behind it is the geometry of Schubert varieties of Grassmannians of subspaces in a complex vector space. If we change the geometric setup to the case of Grassmannians of isotropic subspaces of a symplectic vector space, we have analogous functions called Schur Q-functions and their slight generalization. I will explain an explicit closed formula to express those functions as sums of Pfaffians and also the geometry and combinatorics behind it. If time allows, I will mention the technique of how we obtained the formula, that involves the so-called divided difference operators. This is a joint work with Takeshi Ikeda.
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