# Manchester Geometry Seminar 2009/2010

**Thursday 10 December 2009. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
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A New Class of Polynomials: *Composition Polynomials*

Victor Buchstaber (Steklov Mathematical Institute and University of Manchester)

`Victor.Buchstaber@manchester.ac.uk`

The talk is devoted to a new class of functions, which we call *composition functions* because they arise in the problem of composition of operators. Composition functions form a ring containing the ring of classical symmetric functions. Composition functions are indexed by compositions of positive integers in the way similar to how symmetric functions are indexed by partitions. We show how composition functions arise naturally in diverse areas of mathematics such as combinatorics, algebraic topology and quantum groups. As an application we construct a homomorphism from the ring of all convex polytopes to the ring of composition functions and give an explicit description of its image. The talk is based on recent results obtained jointly with N. Erochovets.