Mathematics

 Coxeter-Knuth Graphs and a signed Little map

Sara Billey (University of Washington)

Abstract: We propose an analog of the Little map for reduced expressions for signed permutations. We show that this map respects the transition equations derived from Chevellay's formula on Schubert classes. We discuss many nice properties of the signed Little map which generalize recent work of Hamaker and Young in type A where they proved Lam's conjecture.   As a key step in this work, we define shifted dual equivalence graphs building on work of Assaf and Haiman and prove they can be characterized by axioms.   These graphs are closely related to both the signed Little map and to the Coxeter-Knuth relations of type B due to Kraskiewicz. (Organizer note: this is a special AGC seminar that will be held as part of a Math 595 class.)

Fibrations and polynomial invariants for free-by-cyclic groups

Spencer Dowdall (UIUC Math)

Abstract: The beautiful theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle. Namely, there are distinguished convex cones in the first cohomology M^1(M;R) whose integral points all correspond to fibrations of M, and the dynamical features of these fibrations are all encoded by McMullen's "Teichmuller polynomial." This talk will describe recent work developing aspects of this picture in the setting of a free-by-cyclic group G. Specifically, I will introduce a polynomial invariant that determines a convex polygonal cone C in the first cohomology of G whose integral points all correspond to algebraically and dynamically interesting splittings of G. The polynomial invariant additionally provides a wealth of dynamical information about these splittings. This is joint work with Ilya Kapovich and Christopher J. Leininger.

Beyond virtual nilpotency of fundamental groups of quasireguarly elliptic manifolds

Pekka Pankka (University of Jyväskylä)

Abstract: By Varopoulos' theorem, the fundamental group of a closed quasiregularly elliptic n-manifold has polynomial order of growth at most n. Thus, by Gromov's theorem, these groups are virtually nilpotent. Having this a priori knowledge, the quasiregular mapping can, however, be used to obtain sharper results on these fundamental groups. I will discuss two recent results: (1) a result with Rami Luisto showing that maximal growth of the group implies the group to be virtually abelian; (2) a result with Enrico Le Donne showing that fundamental groups of closed BLD-elliptic manifolds are (in fact) always virtually abelian. Interestingly, the proofs use completely different geometric methods: the quasiregular result uses Loewner-spaces of Heinonen and Koskela, and the BLD-result Pansu's theorems on Carnot groups.

Partial-Theta Identities Involving Squares

Albert Tamazyan (UIUC Math)

Abstract: Many q-series identities have nice partition theoretic interpretations. The well-known pentagonal number theorem is a good example. Three partial theta identities involving squares will be presented in this talk. The first of them is due to Ramanujan, the second one is due to Andrews and the third one (which is a special case of the Rogers-Fine identity) is due to Krishnaswami Alladi. The weighted partition forms of these identities will be shown and the q-hypergeometric proof of one of them will be presented.

Dynamics of some piecewise smooth Fermi-Ulam models

Jacop De Simoi (Toronto)

Abstract: Fermi-Ulam models are simple one-and-a-half degree of freedom mechanical systems which describe the dynamics of a ball bouncing freely between two oscillating walls. KAM theory implies, if the motion of the walls is sufficiently smooth, existence of invariant tori which prevent any form of diffusion to high energies. In a joint ongoing project with D. Dolgopyat we describe the dynamics of such systems assuming only piecewise smoothness of the wall motions. We are able to give an essentially complete description of the high energy dynamics which turns out to be either hyperbolic (i.e. diffusive) or dominated by elliptic islands. Time permitting I will also explain some work in progress regarding so-called dispersing Fermi-Ulam models and our strategy to attack the problem of ergodicity of this and related models.

Fingerprint databases for theorems

Sara Billey (University of Washington)

Abstract: Suppose that M is a mathematician, and that M has just proved theorem T. How is M to know if her result is truly new, or if T (or perhaps some equivalent reformulation of T) already exists in the literature? In general, answering this question is a nontrivial feat, and mistakes sometimes occur. We will discuss several existing databases of theorems which assign a small, language free, searchable "fingerprint" canonically to their theorems. We will also address the question: "How can we canonically fingerprint all theorems and formulas?" which is currently an unsolved problem. Some of the motivation for this discussion came from recent research at the intersection of combinatorics and probability pertaining to peak sets of permutations and statistical processes on graphs called meteors, earthworms and WIMPS. Some of these results will be explained as examples of the main theme. This talk is based on joint work with Chris Burdzy, Soumik Pal, Bruce Sagan, and Bridget Tenner.