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Jacob Matherne, IAS, Princeton
Kazhdan-Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids.
I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Examples involving the geometry of hyperplane arrangements will be given throughout. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.
Samuel Cogar, University of Delaware
In this talk I will introduce a new modified transmission eigenvalue problem for scattering by a partially coated crack. Rather than study this problem in isolation, I will present a generalized Robin eigenvalue problem depending on a bounded linear operator that encodes the information for a given scattering medium. Results obtained in this general setting will then be applied to the case of scattering by a partially coated crack, including a new proof that finitely many eigenvalues exist when the surface impedance of the crack is sufficiently small. I will conclude with some numerical examples that both verify the theoretical results and demonstrate the sensitivit
Mark Hagen, University of Bristol
Abstract: Masur and Minsky's work on the geometry of mapping class groups, combined with more recent results about the geometry of CAT(0) cube complexes, motivated the introduction of the class of hierarchically hyperbolic spaces. A metric space \(X\) is hierarchically hyperbolic if there is a set of (uniformly) Gromov-hyperbolic spaces \(U\), each equipped with a projection from \(X\) to \(U\), satisfying various axioms that amount to saying that the geometry of \(X\) is recoverable, up to quasi-isometry, from this projection data. Working in this context often allows one to promote facts about hyperbolic spaces to conclusions about highly non-hyperbolic spaces: mapping class groups, Teichmuller space, "most" 3-manifold groups, etc. In particular, many CAT(0) cube complexes -- including those associated to right-angled Artin and Coxeter groups -- are hierarchically hyperbolic.
The relationship between CAT(0) cube complexes and hierarchically hyperbolic spaces is intriguing. Just as, in a hyperbolic space, a collection of n points has quasiconvex hull quasi-isometric to a finite tree (i.e. 1-dimensional CAT(0) cube complex), in a hierarchically hyperbolic space, there is a natural notion of the quasiconvex hull of a set of n points, and it is quasi-isometric to a CAT(0) cube complex, by a result of Behrstock-Hagen-Sisto. The quasi-isometry constants depend on n in general. However, when each hyperbolic space U is quasi-isometric to a tree, it turns out that this dependence disappears. From this one deduces that, if \(X\) is a metric space that is hierarchically hyperbolic with respect to quasi-trees, then \(X\) is quasi-isometric to a CAT(0) cube complex. I will discuss this theorem and some of its group-theoretic consequences. This is joint work with Harry Petyt.
Sharon Di, Columbia University, Civil Engineering
There are no conferences this week.