Title:  On a thermodynamically consisted Stefan problem with variable surface energy

Abstract:  Given a filtration of a simplicial complex we can construct a series of invariants called the persistent homology groups of the filtration. In this talk we will give a basic introduction to the theory of persistence and explain how these ideas can be used in data analysis.

Title:  Universal wave patterns

Abstract:  A feature of solutions of a (generally nonlinear) field

theory can be called "universal" if it is independent of side conditions like initial data. I will explain this phenomenon in some detail and then illustrate it in the context of the sine-Gordon equation, a fundamental relativistic nonlinear wave equation. In particular I will describe some recent results (joint work with R. Buckingham) concerning a universal wave pattern that appears for all initial data that crosses the separatrix in the phase portrait of the simple pendulum.  The pattern is fantastically complex and beautiful to look at but not hard to describe in terms of elementary solutions of the sine-Gordon equation and the collection of rational solutions of the famous inhomogeneous Painlev\'e-II equation.

Title:  Automating and Stabilizing the Discrete Empirical Interpolation Method for Nonlinear Model Reduction

Abstract:  The Discrete Empirical Interpolation Method (DEIM) is a technique for model reduction of nonlinear dynamical systems.  It is based upon a modification to proper orthogonal decomposition which is designed to reduce the computational complexity for evaluating reduced order nonlinear terms.  The DEIM approach is based upon an interpolatory projection and only requires evaluation of a few selected components of the original nonlinear term.  Thus, implementation of the reduced order nonlinear term requires a new code to be derived from the original code for evaluating the nonlinearity.  I will describe a methodology for automatically deriving a code for the reduced order nonlinearity directly from the original nonlinear code.  Although DEIM has been effective on some very difficult problems, it can under certain conditions introduce instabilities in the reduced model.  I will present a problem that has proved helpful in developing a method for stabilizing DEIM reduced models.

Any formulation of the Divergence Formula involves two sets of regularity assumptions, one of geometric nature (regarding the underlying domain) and one of analytic nature (pertaining to the vector field involved).  The celebrated version proved by  De Giorgi and Federer, while allowing the domain to be rough, requires the intervening vector field to be smooth in the entire space. For many applications the latter condition is unreasonably restrictive, and the question arises as to what are the optimal assumptions on the vector field and domain for the Divergence Formula to hold in the case when the vector field in question may lack continuity. In this talk I will discuss recent progress on this topic.

The remainder term $R(\lambda)$ for the spectral counting function $N(\lambda)$ likely encodes a great deal of dynamical information for the system at hand. For $\Omega \subset \mathbb{R}^n$, a piecewise smooth bounded domain, we prove an omega bound that depends on the dimension of the fixed point set of the billiard map; the approach taken is through boundary trace expansions. This is the first dynamical lower bound established in settings with boundary, at least to the knowledge of the authors. As a corollary, $R(\lambda)$ for the Bunimovich stadium is $\Omega(\lambda^{1/2})$, hence confirming a conjecture of Sarnak.

Please visit the Analysis and PDE Seminar webpage at http://www.ms.uky.edu/~kott/seminars13

The role of a biological membrane is to act as a barrier between ionic solutions. One way life controls ionic solution is through ion channels. A second more drastic way is by introducing a hole in the membrane itself. For example, in hemolysis, the osmotic swelling and rupture of a red-blood cell, a single hole forms in the membrane leading to the leak out of the contents of the cell. Similarly, in exocytosis a hole is formed by joining two membrane bilayers. These processes are mathematically challenging to study because they involve physical forces in the bulk and on surfaces with varying topology and predicting the time course is more consequential than the equilibrium end states. This talk will show how such complicated fluid mechanical problems yield to quantitative modeling and simulation when using the diffusive interface and energetic variational approach.  This is joint work with Fredric Cohen and Robert Eisenberg at the Rush University Medical Center in Chicago, IL

In this talk, I will first give a brief survey on Harnack inequalities for solutions of second-order elliptic and parabolic equations. Then I will describe my contribution on Harnack inequalities for non-divergent elliptic and parabolic equations on non-compact Riemannian manifolds.

It is well-known that the p-Hardy inequality is valid in a domain if the complement of the domain is uniformly p-fat. The same is true for the so-called pointwise Hardy inequalities as well, but for these the uniform fatness of the complement is also necessary, and so there is an equivalence between the two concepts. I will discuss this result and related Hausdorff content density conditions, and also some generalizations for weighted Hardy inequalities.

We discuss recent work with Jason Metcalfe and Daniel Tataru on local well posedness results for quasilinear Schrödinger equations. We will discuss both a natural functional framework, as well as the local smoothing, energy estimates and multilinear estimates required.