Seminars in July 2010

  • Jon-Lark Kim (김종락), On self-dual codes
    On self-dual codes
    Jon-Lark Kim (김종락)
    Department of Mathematics, University of Louisville, Louisville, KY, USA
    2010/7/29 Thu 4PM-5PM

    Self-dual codes have become one of the most active research areas in coding theory due to their rich mathematical theories. In this talk, we start with an introduction to coding theory. Then we describe some recent results on the constructions of self-dual codes over rings, and applications to lattices and network coding theory. We conclude the talk with some open problems.

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  • Lou Shapiro, The uplift principle and the Riordan group
    The uplift principle and the Riordan group
    Lou Shapiro
    Department of Mathematics, Howard University, Washington DC, USA
    2010/7/23 Fri 4PM-5PM

    The Riordan group is an easy yet powerful tool for looking at a large number of results in combinatorial enumeration. At the first level it provides quick proofs for many binomial identities as well as a systematic way to invert them. We will see how they arise naturally when looking at the uplift principle as applied to classes of ordered trees. We will also discuss some recent results including the Double Riordan group, summer – winter trees, spoiled child trees, and will mention a few open problems as well. The main tools involved are generating functions, matrix multiplication, and elementary group theory.

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  • June Huh (허준이), Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
    Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
    June Huh (허준이)
    Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
    2010/7/9 Fri 4PM-5PM

    The chromatic polynomial of a graph counts the number of proper colorings of the graph. We give an affirmative answer to the conjecture of Read (1968) and Welsh (1976) that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we show log-concavity of the sequence by answering a question of Trung and Verma on mixed multiplicities of ideals. The conjecture on the chromatic polynomial follows as a special case.

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