Abstract: A major problem in four-dimensional topology is to understand the difference between topological and smooth four-manifolds, e.g. four-manifolds which are homeomorphic but not diffeomorphic. Smooth manifolds are usually studied by considering invariants which count solutions to a PDE on the four-manifold, like the instanton or Seiberg-Witten equations. These invariants are well-behaved on manifolds with nice geometric properties, like positive scalar curvature or symplectic forms, but their use for closed manifolds has mostly plateaued. In this talk, we will discuss a slightly different perspective on invariants of four-manifolds and, if time, more topology-intrinsic constructions of exotic four-manifolds and exotically embedded surfaces in four-manifolds. This is joint work with Adam Levine and Lisa Piccirillo.
Abstract: I will give a brief introduction to decorated Teichmüller theory and the definition of the U-equations and how natural coordinates on decorated Teichmüller space solve these equations. I will end by showing how a hyperbolic structure on an annulus is related to a complex structure on a torus and how a natural generalisation of the u-equations on this annulus is related to the Jacobi triple product identity for Theta functions on the corresponding torus.
Abstract: Gromov initiated a program to prove statements of the following form: Suppose we are given two simplicial complexes X and Y, where X is "complicated" and Y is lower dimensional. Then any map f: X-> Y must have at least one "complicated" fiber. In this talk, I will describe various results of this kind for finite volume locally symmetric spaces, that are proved by bringing new tools into the picture from minimal surface theory and representation theory, including related recent work by Bader-Sauer. If time permits, I will also discuss some applications to systolic geometry, global fixed point statements for actions of lattices on contractible CAT(0) simplicial complexes, and/or non-abelian higher expansion and branched cover stability.  Based on joint work with Mikolaj Fraczyk.
Abstract: Since their introduction by Francois Labourie, Anosov subgroups of real semisimple Lie groups have come to be accepted as the "correct" generalization of convex cocompact subgroups in rank one. Convex cocompact subgroups of rank one groups may be characterized as holonomy groups of negatively curved locally symmetric Riemannian manifolds whose geodesic flows are Axiom A in the sense of Smale.  The naive generalization of this to the higher rank Anosov subgroup situation is false, but we will explain the construction of another locally homogeneous Axiom A flow (often with no interpretation as a geodesic flow) associated to any Anosov subgroup. Time permitting, we will discuss applications of this construction to the meromorphic continuation of certain zeta functions and questions of exponential mixing. All results are joint work with Benjamin Delarue and Daniel Monclair.
Abstract: The homology of Yang-Baxter operators were discovered/constructed by Victoria Lebed and the speaker in Spring of 2012. They generalize homology of racks, Â quandles, and set theoretic Yang-Baxter operators. A proper visualization of precubic modules gives the decisive hint into successful definition.
In the first part of my talk I will discuss the background of Yang-Baxter homology. In the second part, I will discuss how the homology of the HOMFLYPT Yang-Baxter operator can be (partially) computed by using a cascade of filtrations and their splittings. We speculate that Yang-Baxter homology is related to Khovanov homology (they both generalize Jones and HOMFLYPT polynomials) and can be used to analyze phase transition for some physical processes. The work described in my talk started with my student Xiao Wang, and continued with my students Anthony Christiana, Ben Clingenpeel, Ali Guo, and visiting students Jinseok Oh and Hongdae Yun.
Abstract: Thurston revolutionized the study of diffeomorphisms of Riemann surfaces with a myriad of tools: hyperbolic geometry, geometric group theory, Teichmüller theory, and much more. One goal of this work is to see how much of his vision persists in higher dimensions. In this talk, I will tell a few stories about 4-manifolds and their mapping class groups, with an eye towards the Nielsen realization problem for certain algebraic surfaces. Some of the results I plan to share are joint work with Seraphina Lee and Tudur Lewis.
Abstract: The systole of a 3-manifold is the length of the shortest closed geodesic. Given a closed 3-manifold M and link L such that M\L is hyperbolic, the essential systole of M\L is the length of the shortest closed geodesic which is not nullhomotopic in M. In this talk, we will discuss and motivate the study of essential systoles of hyperbolic link complements, including their application towards answering a question of Freedman and Krushkal about the existence of "filling links" in closed 3-manifolds. This is joint work with Chris Leininger.