Title: Eigenvalue statistics in the absolutely continuous spectrum
Abstract: In this work we present the statistics of eigenvalues for points in the ac spectrum of some Anderson type models with decaying randomness. The statistics agrees almost everywhere with respect to the random parameter with the free eigenvalue statistics.
Title: Informatics and Modeling Platform for Stable Isotope-Resolve Metabolomics
Abstract: Recent advances in stable isotope-resolved metabolomics (SIRM) are enabling orders-of-magnitude increase in the number of observable metabolic traits (a metabolic phenotype) for a given organism or community of organisms. Analytical experiments that take only a few minutes to perform can detect stable isotope-labeled variants of thousands of metabolites. Thus, unique metabolic phenotypes may be observable for almost all significant biological states, biological processes, and perturbations. Currently, the major bottleneck is the lack of data analysis that can properly organize and interpret this mountain of phenotypic data as highly insightful biochemical and biological information for a wide range of biological research applications. To address this limitation, we are developing bioinformatic, biostatistical, and systems biochemical tools, implemented in an integrated data analysis platform, that will directly model metabolic networks as complex inverse problems that are optimized and verified by experimental metabolomics data. This integrated data analysis platform will enable a broad application of SIRM from the discovery of specific metabolic phenotypes representing biological states of interest to a mechanism-based understanding of a wide range of biological processes with particular metabolic phenotypes.
Title: Some progresses on two-dimensional Riemann problems in gas dynamics
Abstract: Two dimensional Riemann problems for compressible fluid flows assume the simplest piecewise sectorial initial state but provide the most fundamental wave configurations, including the reflection of oblique shocks and vortex-shock interaction etc. In this talk I will show many fascinating pictures, based on 2D Riemann solutions, to disclose the mysteries of compressible fluid world both through analytical tools (in the form of mathematical theorems) and computational techniques (in the form of simulations). The analysis is based on the characteristic decomposition theory we developed recently, while the simulations are obtained using the generalized Riemann problem (GRP) scheme that is equipped with a highly accurate solver in the construction of numerical fluxes by a way of tracking singularities analytically and keeping entropy exactly computed.
Title: Higher-order analogues of the exterior derivative complex
Abstract: I will discuss some earlier joint work with E. M. Stein concerning div-curl type inequalities for the exterior derivative operator and its adjoint in Euclidean space R^n. I will then present various higher-order generalizations of the notion of exterior derivative, and discuss some recent div-curl type estimates for such operators. Part of this work is joint with A. Raich.
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
