Analysis, Geometry, and Dynamical Systems Seminar

Past sessions

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Young measures - Basic ideas and recent applications

Young measures have been successfully applied to various mathematical problems in material science and other areas. They are used as a tool to study problems which include different length scales, e.g., when microscopical structures affect the global behavior of a material.
In this talk we will give an introduction to Young measures, filling the abstract definitions with some intuitive ideas. We will apply this concept to equations modelling shape memory alloys (one-dimensional thermoelasticity with nonconvex energy density) and will present a recent existence result (in collaboration with Johannes Zimmer, MPI Leipzig).

Nonuniformly hyperbolic dynamics and Lorenz attractors

We consider the statistical properties of Lorenz-like strange attractors which arise out of two-dimensional return maps for the flow. The questions we ask are: Does the system admit an ergodic invariant measure which is mixing? How fast is the mixing? Is the measure physical: ie does it describe the statistical behaviour of many points in the basin? To answer these questions we use a "Markov extension" or "Young tower" to reduce the dynamics to that of an expanding map over a hyperbolic base set, with infinitely many branches having variable return times. This talk is based on ongoing work with S. Luzzatto. I hope to make the talk accessible to nonspecialists in ergodic theory.

Global Rough Solutions for Nonlinear Dispersive Equations

This will be a PDE talk aimed at a general audience, describing some recent work with J. Colliander, G. Staffilani, H. Takaoka, and T. Tao. We will focus on new results which give basic descriptions of the long time behavior of semilinear Schroedinger equations, but some of the techniques employed are applicable to other PDE's. In particular, the talk will describe how so-called almost-conservation laws and interactaction Morawetz-type inequalities help understand the regularity properties of these equations.

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. After a short explanation of this result, we investigate the question to know how particular such initial data are. The first step consists in finding a "linearizability condition": when is the nonlinear term relevant in the semi-classical limit? This conditions reduces the problem to the study of a linear equation. For this equation, we use a "profile decomposition" technique, introduced by P. Gerard. This shows that according to the nonlinearity considered, quadratic oscillations are the only relevant initial data, or are not. We also present some applications to a nonlinear superposition principle, and to the study of finite time blow up.

Combining logic systems: Why, how, what for?

Motivated by applications in artificial intelligence and software engineering that require the joint use of different deduction formalisms, the interest in combination of logic systems has recently been growing, but the topic is also of interest on purely theoretical grounds. Several forms of combination have been studied, like product, fusion, temporalization, parameterization, synchronization and, more recently, fibring. In this guided tour of the issues raised by the combination of logics, we define fibring (the most general form of combination) in a very simple (yet useful) context, discuss some examples and establish some interesting transference results, namely preservation of strong completeness and nonpreservation of congruence. We end the tour with a brief reference to some open problems. The talk is based on a recent overview paper (together with C. Sernadas) available at http://www.cs.math.ist.utl.pt/ftp/pub/SernadasA/03-SS-fiblog22.pdf to appear in the CIM Bulletin.

Hofer-Zehnder sensitive capacity of cotangent bundles and symplectic submanifolds

We will consider a modified Hofer-Zehnder capacity sensitive to the homotopy class of the periodic orbits and show that if a symplectic manifold admits a free Hamiltonian circle action then it has bounded Hofer-Zehnder (sensitive) capacity. We give two applications of this result. Firstly, we prove that every closed symplectic submanifold has a neighborhood with finite Hofer-Zehnder capacity. Secondly, consider a closed manifold with an effective circle action whose fixed point set has trivial normal bundle. Then, its standard cotangent bundle has bounded Hofer-Zehnder capacity.

Existence and nonexistence results for anisotropic quasilinear elliptic equations

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