Georgia Institute of Technology College of Sciences

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.

We present an algorithm that generates sets of size equal to the degree of a given projective variety. The steps of this "CCAR" algorithm are individually well-known, but we argue that when combined they form a versatile and under-used method for studying problems in computational algebraic geometry. The latter part of the talk will focus on applying the CCAR algorithm to examples from Schubert calculus.

Oriented matroids are matroids with extra sign data, and they are useful in the tropical study of real algebraic geometry. In order to study the topology of real algebraic hypersurfaces constructed from patchworking, Renaudineau and Shaw introduced an algebraically defined filtration of the tope space of an oriented matroid based on Quillen filtration. We will prove the equality between their filtration (together with the induced maps), the topologically defined Kalinin filtration, and the combinatorially defined Varchenko-Gelfand dual degree filtration over Z/2Z. We will also explain how the dual degree filtration can serve as a Z-coefficient version of the other two in this setting. This is joint work with Kris Shaw.