University of Minnesota

The Landau equation stands as one of the fundamental equations in kinetic theory and plays a key role in plasma physics. However, computing it presents significant challenges due to the complexity of the Landau operator,  the dimensionality, and the need to preserve the physical properties of the solution. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges. These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. 

It has long been appreciated that in QCD-like theories without fundamental matter, confinement can be given a sharp characterization in terms of symmetry. More recently, such symmetries have been identified as 1-form symmetries, which fit into the broader category of generalized global symmetries.  In this talk I will discuss obstructions to the existence of a 1-form symmetry in large N QCD, where confinement is a sharp notion. I give general arguments for this disconnect between 1-form symmetries and confinement, and use 2d scalar QCD on the lattice as an explicit example.  

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 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.

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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 find exact solutions of Maxwell's equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination. We test the method by doing theoretical structure determination on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea is that the structure is the orbit of a group, and this group is a subgroup of the invariance group of the differential equations. Joint work with Dominik Juestel and Gero Friesecke. (Acta Crystallographica A72 and SIAM J. Appl Math).

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.)

Cloaking recently attracts a lot of attention from the scientific community due to the progress of advanced technology. There are several ways to do cloaking. Two of them are based on transformation optics and negative index materials. Cloaking based on transformation optics was suggested by Pendry and Leonhardt using transformations which blow up a point into the cloaked regions. The same transformations had previously used by Greenleaf et al. to establish the non-uniqueness for Calderon's inverse problem. These transformations are singular and hence create a lot of difficulty in analysis and practical applications. The second method of cloaking is based on the peculiar properties of negative index materials. It was proposed by Lai et al. and inspired from the concept of complementary media due to Pendry and Ramakrishna. In this talk, I will discuss approximate cloaking using these two methods. Concerning the first one, I will consider the situation, first proposed in the work of Kohn et al., where one uses transformations which blow up a small ball (instead of a point) into cloaked regions. Many interesting issues such as finite energy and resonance will be mentioned. Concerning the second method, I provide the (first) rigorous analysis for cloaking using negative index materials by investigating the situation where the loss (damping) parameter goes to 0. I will also explain how the arguments can be used not only to establish the rigor for other interesting related phenomena using negative index materials such as superlenses and illusion optics but also to enlighten the mechanism of these phenomena.