The handle number of a link in the three sphere is the least number of disjoint proper arcs needed to be attached to the link so that resulting (possibly disconnected) graph is unknotted. In particular, the exteriors of handle number one links admit genus 2 Heegaard splittings. In this talk, some results on nonhyperbolic handle number links will be presented. This is joint work with Abby Thompson (UC Davis).

Given a Riemann surface C and a meromorphic quadratic differential f(z) dz^2 on C, one can naturally define a flat metric on C (with conical singularities). The geodesics in this metric are well-studied objects which appear in various contexts: for example, they appear in Teichmuller theory and in the WKB analysis of second-order linear ODEs. In joint work with Gaiotto and Moore we recently showed that these geodesics are also part of the story of "wall-crossing" studied by algebraic geometers. I will explain some aspects of this connection and its geometric applications.

Ideal magnetohydrodynamics (MHD) equations arise in many areas such as astrophysics and plasma physics, and they consist of a set of nonlinear conservation laws. As a non-strictly hyperbolic system, the solutions display rich features. In this talk, I will present our developments in designing high order discontinuous Galerkin methods to reliably simulate this system, with the main focus on the numerical treatments for the divergence-free constraint on the magnetic field, and the positivity of density and pressure.

Given a finite group G of order n, let the group matrix be an nxn matrix with x_{gh^{-1}} as (g,h) entry. The study of the factorization of the determinant over C led to the definition of character and development of representation theory of finite group. It turns out that there is a one-to-one correspondence between the irreducible factors and equivalence classes of irreducible representations. As a special case when G is abelian, the group determinant factor completely into linear forms.

We investigate a one-dimensional model describing the motion of liquid drops sliding down an inclined plane (the so-called quasi-static approximation model). We prove existence and uniqueness of a solution and investigate its long time behavior for both homogeneous and inhomogeneous medium (i.e. constant and non-constant contact angle).

We investigate the existence, uniqueness, and long-time behavior of solutions to a new coagulation model arising in the theory of branching processes. Informally, the equation models a coalescence process with multiple interactions, where the size and number of interacting clusters are sampled randomly. It is well-known that solutions of the equation correspond to L\'{e}vy measures for an associated continuous-state branching process, and we give an existence proof independent of the latter theory. Also, under a suitable regular variation assumption on the sampling measure, we characterize all nontrivial scaling limits of solutions. Our results include, as a special case, the scaling limits for the classical Smoluchowski model with constant rate kernel , previously studied by Menon and Pego, among others. This is joint work with Gautam Iyer and Bob Pego.

In this talk we will define what it means for a group to admit a left- or bi-invariant ordering; we will explore some of the properties of groups that admit such orderings; and we will construct bi-invariant total orders on free groups and on the pure braid groups.

In preparation for Professor Gordana Matic's visit, this talk will give an introduction to contact topology and some of the important tools and theorems that have lead to significant advances in this field over the last two decades. We will first provide some working examples of contact structures on 3-manifolds, and some essential theorems. Then we will get into a discussion of open books and how these relate to contact structures. This surprising connection has lead to many new techniques allowing topologists to further understand properties of contact 3-manifolds.

I will recall the Fock-Goncharov definition of cluster varieties, reformulating the theory in terms of birational geometry of Calabi-Yau manifolds, and explain how one of the main open questions of the subject, the Fock-Goncharov dual basis conjecture, is a very special case of the general conjecture of Gross, Hacking, and me (a theorem in dimension two) on mirrors to CY manifolds with maximal boundary. In particular I will deduce from our theorem the Fock-Goncharov conjecture for skew symmetric matrices of rank two. No previous knowledge of cluster varieties, or mirror symmetry, required.

In joint work with Ko Honda and WIll Kazez, we have defined an invariant of a contact 3-manifold with convex boundary as an element of Juhász’s sutured Floer homology. This invariant generalizes the contact invariant of Ozsvath and Szabo in Heegaard Floer homology in the closed case. We describe a natural gluing map on sutured Floer homology and show how it produces a (1+1)-dimensional TQFT and how this leads to obstructions to fillability.

We will continue our discussion about obstacles with implicit constraints. In particular we will talk about the regularity of solutions and some generalizations to obstacles with a general modulus of semi-concavity. The hope will be to show how tools from Nonlinear Elliptic PDE theory can be used to show regularity when the obstacle is a priori a very general semi-concave function.

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in a price impact model. These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments. The presentation is based on a joint paper with Peter Bank, see http://arxiv.org/abs/1110.3250

Following da Silva's Lecture Notes on Symplectic Geometry, we'll provide an elementary introduction to symplectic manifolds. In this talk, we'll define a symplectic vector space and prove that all symplectic vector spaces of the same dimension are symplectomorphic. We'll conclude with an introduction to symplectic manifolds and discuss the relevance of our work with vector spaces. No prerequisites are required - this talk will be accessible to anybody interested.