Title: Sub-Exponential Decay Estimates on Trace Norms of Localized Functions of SchrodingerOperators
Abstract: In 1973, Combes and Thomas discovered a general technique for showing exponential decay of eigenfunctions. The technique involved proving the exponential decay of the resolvent of the Schrodinger operator localized between two distant regions. Since then, the technique has been applied to several types of Schrodinger operators. Recent work has also shown the Combes–Thomas method works well with trace class and Hilbert–Schmidt type operators. In this talk, we build on those results by applying the Combes–Thomas method in the trace, Hilbert–Schmidt, and other trace-type norms to prove sub-exponential decay estimates on functions of Schrodinger operators localized between two distant regions.
Title: On the ground state of the magnetic Laplacian in corner domains
Abstract: I will present recent results about the first eigenvalue of the magnetic Laplacian in general 3D-corner domains with Neumann boundary condition in the semi-classical limit. The use of singular chains show that the asymptotics of the first eigenvalue is governed by a hierarchy of model problems on the tangent cones of the domain. We provide estimations of the remainder depending on the geometry and the variations of the magnetic field. This is a joint work with V. Bonnaillie-Nol and M. Dauge.
Featuring: Simón Sedillo (organizer and filmmaker); Geoff Boyce (Doctoral Candidate, School of Geography and Development, University of Arizona); Aviva Chomsky (Department of History, Salem State University); Vanessa Hall (Kentuckians for the Commonwealth); and Ann Kingsolver, (Department of Anthropology and Appalachian Studies Center, University of Kentucky)
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: Smoothness of isometries between subRiemannian manifolds
Abstract: In a joint work with Enrico Le Donne (Jyvaskyla, Finland), we show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of harmonic coordinates, establishing that in an arbitrary subRiemannian manifold there exists an open dense subset where all isometries are smooth.
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: 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.
