Events

Abstract:  When you encounter an algebraic variety in the wild, you might ask: What do its components look like? Which components are smooth? How do the components intersect? For Springer fibres, answers to these questions are only known in some very special cases. This is particularly surprising because other aspects Springer fibres have been studied for the past 50 years and they appear throughout combinatorics and adjacent areas. For just one example: Hall–Littlewood polynomials can be obtained from the cohomology of Springer fibres by taking graded Frobenius characteristic.

One classical theorem says that the components of Springer fibers are indexed by standard Young tableaux. In this talk, we will discuss the benefits of instead using webs to index the components in two cases: the "two row" case, and our recent contributions in the "two column" case. We will see that in these cases, webs both characterize and describe the smooth components of Springer fibres, and give a geometric interpretation of rotation of webs.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: A biased graph is a pair (G,B) consisting of a graph G and a collection B of “balanced" cycles in G. Biased graphs arise naturally as the cycles that have identity weight in a group-labelling of arcs where opposite arcs have inverse weight. I present an excluded minor characterization for biased graphs that have such a group-labelling over Z₃.

Abstract: The totally asymmetric simple exclusion process (TASEP) is a finite Markov chain of particles hopping between adjacent sites on a one-dimensional lattice. The multispecies TASEP is a generalization in which particles have different types. These processes have interesting connections to algebraic combinatorics: the stationary distribution of the TASEP on a circle is connected to Macdonald polynomials at t=0, whereas the stationary distribution of the open-boundary TASEP is connected to Koorwinder polynomials at t=0.

Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the multispecies TASEP on a circle. It has been a long-standing open problem to find a combinatorial formula for the stationary distribution of the multispecies TASEP with open boundaries. Recently, we studied the combinatorics of Ferrari–Martin multiline queues using type A crystals. In this talk, we use crystals of type C to construct an analog of multiline queues and give a combinatorial formula for the stationary distribution of the multispecies open-boundary TASEP for a certain specialization of the boundary parameters. This is joint work with Olya Mandelshtam.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: Standard tableaux are certain grids of numbers that lead a double life in algebraic combinatorics, with distinct roles in geometry and in representation theory. Extending the geometry to K-theory led to a corresponding extension of the combinatorics to a theory of increasing tableaux. I will discuss a long and ongoing program to explicate the combinatorial dynamics of these tableaux, which seems to be revealing that they also have a mysterious second life in representation theory. Despite the algebraic connections, the core problem is fundamentally combinatorial: to give a sufficiently good bijection between tableaux and a set of planar diagrams. 

Abstract: Standard tableaux are certain grids of numbers that lead a double life in algebraic combinatorics, with distinct roles in geometry and in representation theory. Extending the geometry to K-theory led to a corresponding extension of the combinatorics to a theory of increasing tableaux. I will discuss a longstanding plot by such tableaux to prevent me from explicating their combinatorial dynamics. Despite their reticence, we seem to be uncovering that these tableaux also have a mysterious second life in representation theory.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: