Myhill Lecture Series -Gopal Prasad
4:00PM, Thu Oct 20 2016, Math 250
Tarik Aougab (Brown)
Hyperbolic subspaces and super-polynomial divergence
4:00PM, Fri Oct 21 2016, Math 122
There are several useful properties satisfied by geodesics in negatively curved spaces, including the classical Morse lemma, which states that if p is any relatively efficient path with endpoints on the geodesic, then p must lie in a bounded neighborhood of the geodesic. In other words: there are no efficient paths connecting points on the geodesic that drift far away from the geodesic. We will discuss several related properties, including "super-linear divergence" for geodesics in negatively curved spaces, and we prove that all of these properties are actually equivalent in great generality: a geodesic in an arbitrary (not necessarily hyperbolic or negatively curved) geodesic metric space satisfies one of these properties if and only if it satisfies all of them. We use this equivalence to study hyperbolic extensions of surface groups, free groups, and certain hyperbolic subgroups of relatively hyperbolic groups. This represents joint work with Matthew Durham and Samuel Taylor.
Algebra Seminar- Gufang Zhao, University of Massachusetts, Amherst.
Cohomological Hall algebras and quantum groups
4:00PM, Mon Oct 24 2016, Math 250
This talk is based on my joint work with Yaping Yang. We use ideas from enumerative geometry and stable homotopy theory to study quantum groups and their representations. I will talk about a geometric construction of quantum group associated to each quiver Q and each oriented cohomology theory A. The construction uses cohomological Hall algebras of Nakajima quiver varieties. I will focus on two examples. 1. when A is the Morava K-theory, we obtain a family of new quantum groups, linking complex representations to modular representations of a Lie algebra. 2. when A is the equivariant elliptic cohomology, we establish a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra, rational sections of which gives the algebra of elliptic R-matices, and algebra structure of which is related to the factorization structure on affine Grassmannian over an elliptic curve.
Algebra Seminar- Chad Mangum, Niagara University
Fermionic Representations of Twisted Toroidal Lie Algebras
4:00PM, Mon Oct 31 2016, Math 250
Lie algebra representation theory has been significant in various areas of mathematics and physics for several decades. In this talk, we will discuss one instance of this theory, namely certain representations of twisted (2-)toroidal (Lie) algebras, which we view as universal central extensions of twisted multi-loop algebras. The usual loop algebra realization generalizes the familiar realization of affine Kac-Moody algebras. To facilitate our study of the representation theory, however, we will discuss a new realization given by generators and relations; this is similar to a realization by Moody, Rao, and Yokonuma in the untwisted case. Subsequently, we will discuss an application, namely fermionic free field representations, which are similar to those of Feingold and Frenkel in the case of affine algebras. If time permits, we will compare these results to other recent work by the authors. This is joint work with Kailash Misra and Naihuan Jing.
Tenured Faculty Meeting
4:00PM, Thu Nov 3 2016, Math 250
Bulent Tosun (Alabama)
Obstructing pseudo-convex embeddings of Brieskorn spheres into complex 2-space
4:00PM, Fri Nov 4 2016, Math 122
A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. An important question that has been circulating among contact and symplectic topologist for some time asks: whether every contractible smooth 4-manifold admits a Stein structure? In this talk we will provide examples that answer this question negatively. Moreover, along the way we will provide new evidence to a closely related conjecture of Gompf, which asserts that a nontrivial Brieskorn homology sphere, with either orientation, cannot be embedded in complex 2-space as the boundary of a Stein submanifold (This is joint work with Tom Mark.)