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Series: PDE Seminar

The usual boundary condition adjoined to a second order elliptic equation is the Dirichlet problem, which prescribes the values of the solution on the boundary. In many applications, this is not the natural boundary condition. Instead, the value of some directional derivative is given at each point of the boundary. Such problems are usually considered a minor variation of the Dirichlet condition, but this talk will show that this problem has a life of its own. For example, if the direction changes continuously, then it is possible for the solution to be continuously differentiable up to a merely Lipschitz boundary. In addition, it's possible to get smooth solutions when the direction changes discontinuously as well.

Series: Geometry Topology Seminar

A broken fibration is a map from a smooth 4-manifold to S^2 with isolated Lefschetz singularities and isolated fold singularities along circles. These structures provide a new framework for studying the topology of 4-manifolds and a new way of studying Floer theoretical invariants of low dimensional manifolds. In this talk, we will first talk about topological constructions of broken Lefschetz fibrations. Then, we will describe Perutz's 4-manifold invariants associated with broken fibrations and a TQFT-like structure corresponding to these invariants. The main goal of this talk is to sketch a program for relating these invariants to Ozsváth-Szabó invariants.

Series: Geometry Topology Seminar

We will begin with an overview of the Burau representation of the braid group. This will be followed by an introduction to a contact category on 3-manifolds, with a brief discussion of its relation to the braid group.

Series: Analysis Seminar

In this talk we will discuss a generalization of monotone sequences/functions as well as of those of bounded variation. Some applications to various problems of analysis (the Lp-convergence of trigonometric series, the Boas-type problem for the Fourier transforms, the Jackson and Bernstein inequalities in approximation, etc.) will be considered.

Series: Analysis Seminar

Note time change.

Let I_\alpha be the fractional integral operator. The Olsen inequality, useful in certain PDEs, concerns multiplication operators and fractional integrals in the L^p-norm, or more generally, the Morrey norm. We strenghten this inequality from the one given by Olsen.

Series: ACO Student Seminar

Two independent proofs of the polyhedrality of the split closure of Mixed Integer Linear Program have been previously presented. Unfortunately neither of these proofs is constructive. In this paper, we present a constructive version of this proof. We also show that split cuts dominate a family of inequalities introduced by Koppe and Weismantel.

Series: PDE Seminar

In this talk we will consider three different numerical methods for solving nonlinear PDEs:

- A class of Godunov-type second order schemes for nonlinear conservation laws, starting from the Nessyahu-Tadmor scheme;
- A class of L1 -based minimization methods for solving linear transport equations and stationary Hamilton- Jacobi equations;
- Entropy-viscosity methods for nonlinear conservation laws.

All of the above methods are based on high-order approximations of the corresponding nonlinear PDE and respect a weak form of an entropy condition. Theoretical results and numerical examples for the performance of each of the three methods will be presented.

Series: Stochastics Seminar

We will introduce the Dunkl derivative as well as the Dunkl process and some of its properties. We will treat its radial part called the radial Dunkl process and light the connection to the eigenvalues of some matrix valued processes and to the so called Brownian motions in Weyl chambers. Some open problems will be discussed at the end.

Series: Analysis Seminar

In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994), K. Astala proved that a compact set E of Hausdorff dimension d is mapped under a K-quasiconformal map f to a set fE of Hausdorff dimension at most d' = \frac{2Kd}{2+(K-1)d}, and he proved that this result is sharp. He conjectured (Question 4.4) that if the Hausdorff measure \mathcal{H}^d (E)=0, then \mathcal{H}^{d'} (fE)=0. This conjecture was known to be true if d'=0 (obvious), d'=2 (Ahlfors), and more recently d'=1 (Astala, Clop, Mateu, Orobitg and UT, Duke 2008.) The approach in the last mentioned paper does not generalize to other dimensions. Astala's conjecture was shown to be sharp (if it was true) in the class of all Hausdorff gauge functions in work of UT (IMRN, 2008). Finally, we (Lacey, Sawyer and UT) jointly proved completely Astala's conjecture in all dimensions. The ingredients of the proof come from Astala's original approach, geometric measure theory, and some new weighted norm inequalities for Calderon-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt A_p theory. These results are intimately related to (not yet fully understood) removability problems for various classes of quasiregular maps. The talk will be self-contained.

Series: Geometry Topology Seminar

Cannon: "A f.g. negatively curved group with boundary homeomorphic to the round two sphere is Kleinian". We shall outline a combinatorial (complex analysis motivated) approach to this interesting conjecture (following Cannon, Cannon-Floyd-Parry). If time allows we will hint on another approach (Bonk-Kleiner) (as well as ours). The talk should be accessible to graduate students with solid background in: complex analysis, group theory and basic topology.