Title:  Using the method of layer potentials to solve a mixed boundary value problem

Abstract:  Following the exposition by William McLean in his book Strongly Elliptic Systems and Boundary Integral Equations, we use the method of layer potentials to show that on a bounded Lipschitz domain, the mixed problem for Laplace’s equation is equivalent to a 2 × 2 system of boundary integral equations.

Title: Self-improvement properties for nonlocal equations

Abstract:  I will present some results related to generalization of Meyers result to nonlocal equation. It happens that any weak solution of a nonlocal equation with data in L2 is automatically better at the integrability AND differentiability scale. This is a completely new phenomenon relying on the nonlocality of the operator. The proof is based on a new stopping time argument and a suitable generalization of Gehring lemma.

Title:  Extremal functions in modules of systems of measures

Abstract:  We study Fuglede’s p-modules of systems of measures in condensers in the Euclidean spaces. First, we generalize the result by Rodin that provides a way to compute the extremal function and the 2-module of a family of curves in the plane to a variety of other settings. More specifically, in the Euclidean space we compute the p-module of images of families of connecting curves and families of separating sets with respect to the plates of a condenser under homeomorphisms with some assumed regularity. Then we calculate the module and find the extremal measures for the spherical ring domain on polarizable Carnot groups and extend Rodin’s theorem to the spherical ring domain on the Heisenberg group. Applications to special functions and examples will be provided. Joint work with Melkana Brakalova and Irina Markina.

Title:  Sub-Exponential Decay Estimates on Trace Norms of Localized Functions of Schrodinger Operators

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.

Title:  Compressible Navier-Stokes equations with temperature dependent dissipation

Abstract:  From its physical origin, the viscosity and heat conductivity coe!cients in compressible fluids depend on absolute temperature through power laws. The mathematical theory on the well-posedness and regularity on this setting is widely open. I will report some recent progress on this direction, with emphasis on the lower bound of temperature, and global existence of solutions in one or multiple dimensions. The relation between thermodynamics laws and Naiver-Stokes equations will also be discussed. This talk is based on joint works with Weizhe Zhang.

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.