UNDERGRADUATE TALKS AND PROJECTS

MICHAEL ANSHELEVICH
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Undergraduate-level research and talks

  • A characterization of ultraspherical distributions, 2011.

  • A write-up, aimed at an undergraduate audience, of a result about generating functions for orthogonal polynomials. Shows that the only orthogonal polynomials with generating functions of a special form are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. Defines all the words above.
  • Integration and Hermite polynomials, 2010.

  • Answers the question "Can we integrate x2 e- x2/2?", and related questions from calculus. The method uses Hermite polynomials. These polynomials have numerous other interesting and important properties, and appear in many other parts of mathematics, so a survey of these properties is included.

  • Derivatives and trees, 2005.

  • While rules for computing first derivatives of products, compositions, and inverses of functions are well-known, the rules for higher derivatives are more complicated, and not easy to write down. In this talk, I explain how to use combinatorics to find formulas for the higher-order product, chain, and inverse function rules in calculus. Many pictures of combinatorial structures, such as partitions and trees.

  • Orthogonal polynomials and counting permutations, 2006.

  • A talk on how to count moments using combinatorics. A number of examples of orthogonal polynomials are given. For all these examples, the corresponding moments are positive integers, and so count some combinatorial structures: pairings, partitions, and permutations. More complicated moments involve combinatorial structures with crossings. The last part of the talk defines linearization coefficients, which, miraculously, count exactly the same combinatorial structures when one only inhomogeneous objects are included.

Expository talks aimed at graduate students

  • Lattice paths and orthogonal polynomials, 2007.

  • Moments, orthogonal polynomials, partitions, cumulants, Motzkin paths, etc., etc. Similar to the last undergraduate talk above, but somewhat more complicated. Ends with a discussion of the moment calculations for non-commutative polynomials.

  • Free Brownian motion (print version), 2009.

  • Different ways of looking at the free Brownian motion. In Combinatorics, it is related to counting Dyck paths by Catalan numbers. In Operator Theory, it is defined using free creation and annihilation operators. In Random Matrix theory, it appears as the limit of matrix-valued Brownian motions. It also has a realization as a limit in the group algebra of the Permutation Group.