Title:  Uniform estimates in homogenization and applications

Abstract:  In a seminal paper of 1987, M. Avellaneda and F.H. Lin have introduced a powerfull method to show uniform Hölder and Lipschitz estimates for elliptic systems with oscillating coefficients. In this talk, I will investigate some consequences of these estimates for the large scale behavior of potentials and the asymptotics of boundary layers in homogenization. I will also address a generalization of the Lipschitz estimate to domains with oscillating boundary. The latter is a joint work with C. Kenig.

Title:  Eigenvalue Multiplicities of the Hodge Laplacian on Coexact 2-Forms for Generic Metrics on 5-Manifolds

Abstract:  In 1976, Uhlenbeck used transversality theory to show that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator are all simple for a residual set of C^r metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of C^r metrics such that the nonzero eigenvalues of the Hodge Laplacian on k forms are all simple.  We continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms by proving that for a residual set of C^r metrics, the nonzero eigenvalues of the Hodge Laplacian acting on coexact 2-forms on a closed 5-manifold have multiplicity 2.  We structure our argument around a study of the Beltrami operator, using techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics.  We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n=4m+1 and k=2m.

Title:  DeBranges' Proof of the Bieberbach Conjecture

Abstract:  See Bulletin Board on 7th Floor- POT for abstract

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:  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.