February 8 (Wednesday 3:15pm)
Avron Douglis Lecture
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On the rigidity of black holes
Sergiu Klainerman
Mathematics Department -- Princeton University
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Abstract:
The rigidity conjecture states that all regular, stationary1 solutions of the Einstein field equations in vacuum are isometric to the Kerr solution. The simple motivation behind this conjecture is that one expects, due to gravitational radiation, that general, dynamic, solutions of the Einstein field equation s settle down, asymptotically, into a stationary regime. A well known result of Carter, Robinson and Hawking has settled the conjecture in the class of real analytic spacetimes. The assumption of real analyticity is however very problematic; there is simply no physical or mathematical justification for it. During the last five years I have developed, in collaboration with A. Ionescu and S. Alaxakis, A. Ionescu a strategy to dispense of it. In my lecture I will these results and concentrate on some recent results obtained in collaboration with A. Ionescu.
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February 9
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Non-uniqueness of Leray-Hopf type solutions to the Navier-Stokes
equation on a two dimensional negatively curved Riemannian manifold.
Chi Hin Chan
The Institute of Mathematical Sciences -- The Chinese University of Hong Kong
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Abstract:
In this talk, we will present a piece of joint work with
Magdalena Czubak in which we constructed, for a special choice of finite
energy initial datum, a family of non-unique Leray-Hopf type solutions to
the Navier-Stokes equation on a two dimensional negatively curved manifold.
The finite energy initial datum we take arises from the gradient of a
non-trivial bounded harmonic function on a general negatively curved
manifold. By the term Leray-Hopf type solution associated to a given finite
energy initial datum, we understand the one which has finite energy, finite
dissipation, and at the same time satisfies the global energy inequality.
This non-uniqueness result on a two dimensional negatively curved
Riemannian manifold is in sharp contrast with the classical well known
uniqueness result for Leray-Hopf solutions to the Navier-Stokes equation on
the 2 dimensional Euclidean space.
Since this work lies in the borderline of regularity theory of solutions
to the Navier-Stokes equations and geometric analysis, necessary background
in both areas will be presented.
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February 21, Appl. Math Colloquium
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Resent Advances in Fast Solvers for Variational Inequality Problems
Michael Hintermuller
Department of Mathematics -- Humbolt-Universitat, Berlin
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Abstract:
In this talk a generalized Newton framework for solving variational inequalities (VIs) of the first and second kind is presented. Depending on regularity issues related to dual variables a Moreau-Yosida approximation of the feasible set of the problem is considered. In cases of VIs of second kind, which exhibit a non-smooth part in the objective (which is not the indicator function of a convex set), an approach relying on the Fenchel dual is pursued. The talk also contains a report on the numerical behavior of the associated Newton-type solvers.
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February 22, Aziz Lecture
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Resent Advances in Fast Solvers for Variational Inequality Problems
Michael Hintermuller
Department of Mathematics -- Humbolt-Universitat, Berlin
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Abstract:
Non-smooth operator equations arise in many practical applications in biomedical or engineering sciences as well as mathematical imaging or finance. In this talk, for the numerical solution of such problems a generalized Newton framework in function space is discussed. Relying on the concept of semismoothness, locally superlinear convergence of the associated Newton iteraton is established and its mesh independent convergence behavior upon discretization is shown. The efficiency and wide applicability of the method is highlighted by considering constrained optimal control problems for fluid flow, contact problems with or without adhesion forces, phase separation phenomena relying on non-smooth homogeneous free energy densities and restoration tasks in mathematical image processing.
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February 23
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Blow up phenomena in high contrast composites
Yulia Gorb
Department of Mathematics -- University of Houston
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Abstract:
In this talk we present results of our resent study of blow up phenomena in high contrast two-phase disersed composites described by PDEs with rough coefficients. The gradients of soluions to such problems exhibit singular behavior - blow up - that we capture and fully characterize. Both linear and non-linear formulations will be explored in the talk.
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March 1
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Advances in multiscale modeling of coarsening in materials
Maria Emelianenko
Department of Mathematical Sciences -- George Mason University
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Abstract:
This talk will survey recent developments in the field of multiscale modeling of polycrystalline materials, where the main challenge lies in understanding how statistical distributions develop in process of coarsening of material microstructure and how these distributions in turn relate to material properties. What do topological reconfigurations during grain growth have in common with particle collisions in gas dynamics? What is the reason the stationar distribution of grain boundaries has a Boltzmann shape? These and other questions will be discussed in the context of stochastic Levy theory, thermodynamics and nonlocal diffusion, and sometimes unexpected connections with other fields of science will be revealed. Preliminary results of applying mesoscale modeling to practical materials will also be shown.
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March 8
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Growth of Sobolev norms for the cubic defocusing nonlinear Schoedinger equation in polynomial time
Marcel Guardia
Departament de Matemàtica Aplicada -- Universitat Politecnica de Catalunya
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Abstract:
We consider the cubic defocusing nonlinear Schroedinger equation in the two dimensional torus. Fix s>1. Colliander, Keel, Staffilani, Tao and Takaoka (2010) proved existence of solutions with s-Sobolev norm growing in time by any given factor R. Refining their methods in several aspects we find solutions with s-Sobolev norm growing in polynomial time in R. This is a joint work with V. Kaloshin.
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March 14 (Wednesday 3:15pm)
Colloquium
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De Giorgi methods applied to regularity issues in Fluid Mechanics
Alexis Vasseur
Department of Mathematics -- University of Texas
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Abstract:
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March 28 (Wednesday 11am MTH 3206)
Joint Probability/PDE seminar
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TBA
Jim Nolen
Department of Matematics -- Duke University
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Abstract:
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March 29
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Efficient Computation of the Semi-Classical Limit of the Schroedinger Equation
Peter Smereka
Dept. of Mathematics -- University of Michigan
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Abstract:
An efficient method for simulating of the propagation of a localized solution of the Schroedinger equation near semiclassical limit is presented. The method is based on a time dependent transformation of the independent variables, closely related to Gaussian wave packets and yields a Schroedinger type equation that is very amenable to numerical solution in the semiclassical limit. The wavefunction can be constructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in the dimensions of up to 5 or 6 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.
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April 5
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Mean field limits for many particles' systems
Pierre-Emmanuel Jabin
Department of Mathematics, CSCAMM-- UMD
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Abstract:
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April 19
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The thin one-phase free boundary problem.
Daniela De Silva
Department of Mathematics -- Columbia University
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Abstract: We describe some regularity results for "thin" free boundaries, that is free boundaries of a one-phase problem which occur on a lower dimensional subspace. This problem appears for example in the context of free boundary problems involving the fractional Laplacian $(-\Delta)^{1/2}.$
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May 3
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TBA
Misha Perepelitsa
Department of Mathematics -- University of Houston
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Abstract:
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Special seminar May 17
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A Better Diffusion Monte Carlo
Jonathan Weare
Department of Mathematics -- University of Chicago
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Abstract:
Diffusion Monte Carlo was developed forty years ago within the Quantum Monte Carlo community to compute ground state energies of the Schrodinger operator. Since then the basic birth/death strategy of DMC has found its way into a wide variety of application areas. For example efficient resampling strategies used in sequential importance sampling algorithms (e.g. particle filters) are based on DMC. As I will demonstrate, some tempting generalizations of the basic DMC framework lead to an instability in the time discretization parameter. This instability has important consequences in, for example, applications of DMC in sequential importance sampling and rare event simulation. We suggest a modification of the basic DMC algorithm that eliminates this instability. In fact, the new algorithm is more efficient than DMC under any condition (parameter regime). We show numerically and analytically that the modified algorithm is stable in unstable regimes for DMC.
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Special accomodations for individuals with disabilities can be made
by calling in advance (301) 405-5048. It would be appreciated if we are
notified at least one week in advance.
