We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.
The Oxford Mathematics Colloquia are generously sponsored by Oxford University Press.
The puzzling phenonenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating medium. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials. Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.
The World population is growing at about 80 million per year. As time goes by, there is necessarily less space per person. Perhaps this is why the scientific community seems to be obsessed with folding things. In this lecture Dick James presents a mathematical approach to “rigid folding” inspired by the way atomistic structures form naturally - their features at a molecular level imply desirable features for macroscopic structures as well, especially 4D structures. Origami structures even suggest an unusual way to look at the Periodic Table.
Richard D. James is Distinguished McKnight University Professor at the University of Minnesota.
Please email @email to register.
The Oxford Mathematics Public Lectures are generously supported by XTX Markets.
The Constantin-Lax-Majda model is a 1d system which shares certain features (related to vortex stretching) with the 3d Euler equation. The model is explicitly solvable and exhibits finite-time blow-up for an open subset of smooth initial data. In 1990s De Gregorio suggested adding a transport term to the system, which is analogous to the transport term in the Euler equation. It turns out the transport term has some regularizing effects, which we will discuss in the lecture.
We will discuss 2d Euler and Boussinesq (incompressible) flows related to a possible boundary blow-up scenario for the 3d axi-symmetric case suggested by G. Luo and T. Hou, together with some easier model problems relevant for that situation.
The optimal function spaces for the local-in-time well-posedness theory of the Navier-Stokes equations are closely related to the scaling symmetry of the equations. This might appear to be tied to particular methods used in the proofs, but in this talk we will raise the possibility that the equations are actually ill-posed for finite-energy initial data just at the borderline of some of the most benign scale-invariant spaces. This is related to debates about the adequacy of the Leray-Hopf weak solutions for predicting the time evolution of the system. (Joint work with Hao Jia.)
We describe an invariant manifold of the equations of molecular dynamics associated to a given discrete group of isometries. It is a time-dependent manifold, but its dependence on time is explicit. In the case of the translation group, it has dimension 6N, where N is an assignable positive integer. The manifold is independent of the description of the atomic forces within a general framework. Most of continuum mechanics inherits some version of this manifold, as do theories in-between molecular dynamics and continuum mechanics, even though they do not inherit the time reversibility of molecular dynamics on this manifold. The manifold implies a natural statistics of molecular motion, which suggests a simplifying ansatz for the Boltzmann equation which, in turn, leads to new explicit far-from-equilibrium solutions of this equation. In some way the manifold underlies experimental science, i.e., the viscometric flows of fluids and the bending and twisting of beams in solids and the procedures commonly used to measure constitutive relations, this being related to the fact that the form of the manifold can be prescribed independent of the atomic forces.