Georgia Institute of Technology College of Sciences

A neural code C on n neurons is a collection of subsets of {1,2,...,n} which is used to encode the intersections of subsets U_1, U_2,...,U_n of some topological space. The study of neural codes reveals the ways in which geometric or topological properties can be encoded combinatorially. A prominent example is the property of max-intersection completeness: if a code C contains every possible intersection of its maximal codewords, then one can always find a collection of open convex U_1, U_2,..., U_n for which C is the code. In this talk I will answer a question posed by Curto et al. (2018), which asks if there is a way of determining max-intersection completeness from examination of the neural ideal, an algebraic counterpart to the neural code.

Chebyshev polynomials offer a natural basis for solving polynomial equations. When we switch from monomials to Chebyshev polynomials, we can replace toric varieties with Chebyshev varieties. We will introduce these objects and discuss their main properties, including equations, dimension, and degree. This is an ongoing project with Zaïneb Bel-Afia and Simon Telen.

A toric vector bundle is a vector bundle over a toric variety equipped with a linear action by the torus of the base. Toric vector bundles pf rank r were famously classified by Klyachko (1989) using certain combinatorial data of compatible filtrations in an r-dimensional vector space E. This data can be thought of as a higher rank generalization of an (integer-valued) piecewise linear function. In this talk, we give an interpretation of Klyachko data as a "piecewise linear map" to a tropical linear space. This point of view leads us to introduce the notion of a "matroidal vector bundle", a generalization of toric vector bundles to (possibly non-representable) matroids. As a special case and by-product of this construction, one recovers the tautological classes of matroids introduced by Berget, Eur, Spink and Tseng. This is a work in progress with Chris Manon (Kentucky).

The connections between representations of complex semisimple Lie algebras and the geometry of the corresponding flag manifolds have a long history. Moreover, combinatorics plays an important role in the related computations. My talk is devoted to new aspects of this story. On the Lie algebra side, I consider certain modules for quantum affine algebras. I discuss their relationship with Macdonald polynomials, which generalize the irreducible characters of simple Lie algebras. On the geometric side, I consider the quantum K-theory of flag manifolds, which is a K-theoretic generalization of quantum cohomology. A new combinatorial model, known as the quantum alcove model, is also presented. The talk is based on joint work with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.