philadelphia-2017-abstracts

MAAGC 2017 ABSTRACTS

Sara Billey (Washington): 

Title: Reduced words and a formula of Macdonald

Abstract: Macdonald gave a remarkable formula connecting a weighted sum of reduced words for a permutation with the number of terms in a Schubert polynomial. We will review some of the fascinating results on the set of reduced words in order to put our main results in context. Then we will discuss a new bijective proof of Macdonald's formula based on Little's bumping algorithm. We will also discuss some generalizations of this formula based on work of Fomin, Kirillov, Stanley and Wachs. This project extends earlier work by Benjamin Young on a Markov process for reduced words of the longest permutation. This is joint work with Ben Young and Alexander Holroyd.

Melody Chan (Brown): 

Title: Young tableaux and the geometry of Brill-Noether varieties

Abstract: A theorem from the 1980s of Eisenbud-Harris and Pirola computes the genera of Brill Noether curves. In joint work with Alberto Lopez Martin, Nathan Pflueger, and Montserrat Teixidor i Bigas, we found a new proof and extension of this result via degenerations and combinatorics. I'll tell you about this work, and related work on enumerating set-valued Young tableaux.

Sergey Fomin (Michigan): 

Title: Noncommutative Schur functions

Abstract: The problem of expanding various families of symmetric functions in the basis of Schur functions arises in many mathematical contexts such as combinatorial representation theory and Schubert calculus. I will discuss an approach to this problem that employs noncommutative analogues of Schur functions. The talk is based on joint work with Jonah Blasiak and Curtis Greene.

Lauren Williams (Berkeley and IAS): 

Title: Combinatorics of the Amplituhedron

Abstract: The tree amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. In this talk I'll start with a gentle introduction to the amplituhedron, then give an equivalent "orthogonal" description of it. I'll then describe what the amplituhedron looks like in various special cases. For example, one can use the theory of sign variation and matroids to show that the amplituhedron A(n,k,1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement (and hence is homeomorphic to a closed ball). This is joint work with Steven Karp. If time permits, I may describe some related work on the m=4 amplituhedron which is joint with Karp and Yan Zhang.