MenuMathematical Physics and Harmonic Analysis Seminar
Spring 2023
| Date: | January 27, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Gaik Ambartsoumian, University of Texas at Arlington |
| Title: | On integral geometry using objects with corners |
| Abstract: | Integral geometry is dedicated to the study of integral transforms mapping a function (or more generally, a tensor field) defined on a manifold to a family of its integrals over certain submanifolds. A classical example of such an operator is the Radon transform, mapping a function to its integrals over hyperplanes. Generalizations of that transform integrating along smooth curves and surfaces (circles, ellipses, spheres, etc) have been studied at great length for decades, but relatively little attention has been paid to transforms integrating along non-smooth trajectories. This talks will discuss some recent results about Radon-type transforms that have a “corner” in their paths of integration (broken rays, cones, and stars) and their relation to imaging techniques based on physics of scattered particles (Compton camera imaging, single scattering tomography, etc). |
| Date: | February 3, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Patricia Ning, TAMU |
| Title: | Mosco Convergence of Dirichlet Forms on Machine Learning Gibbs Measures |
| Abstract: | The Metropolis-Hastings (MH) algorithm as the most classical MCMC algorithm, has had a great influence on the development and practice of science and engineering. The behavior of the MH algorithm in high-dimensional problems is typically investigated through a weak convergence result of diffusion processes. In this paper, we introduce Mosco convergence of Dirichlet forms in analyzing the MH algorithm on large graphs, whose target distribution is the Gibbs measure that includes any probability measure satisfying a Markov property. The abstract and powerful theory of Dirichlet forms allows us to work directly and naturally on the infinite-dimensional space, and our notion of Mosco convergence allows Dirichlet forms associated with the MH Markov chains to lie on changing Hilbert spaces. Through the optimal scaling problem, we demonstrate the impressive strengths of the Dirichlet form approach over the standard diffusion approach. |
| Date: | February 24, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Eitan Tadmor, University of Maryland |
| Title: | TALK CANCELLED: Emergent Behavior in Collective Dynamics |
| Abstract: | A fascinating aspect of collective dynamics is self-organization of small scale interactions into high-order structures with larger-scale patterns. It is a characteristic feature of “social particles” which actively probe the environment and emerge in various types of clusters. In different contexts these clusters take the form of flocks, swarms, consensus, synchronized states etc. In this talk I will survey recent mathematical developments in collective dynamics driven by alignment. Alignment protocols reflect the tendency of steering towards average headings, and are governed by different classes of pairwise communication kernels. A main question of interest is how different kernels affect the long-time, large-crowd dynamics. In particular, we discuss emergent behavior for a general class of pressure tensors without a closure assumption, proving the flocking of p-alignment hydrodynamics. |
| Date: | March 3, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Alejandro Aceves, SMU |
| Title: | On the Fractional nonlinear Schrödinger Equation |
| Abstract: | The concept of the fractional Lapacian as it relates to Levi flights in comparison to Brownian motion appears in many applications in physics. In this talk we will present our work as it relates to optical physics, in particular in the nonlinear regime where both the discrete and the continuous versions are relevant. |
| Date: | March 3, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Lim Yen Kheng, Xiamen University Malaysia |
| Title: | Solving physics problems from the perspective of (tropical) algebraic geometry |
| Abstract: | In the first part of the talk, it will be shown how the partition function in statistical mechanics can be interpreted as an algebraic variety. In accordance to earlier literature, the zero-temperature limit is equivalent to taking the tropical limit of the algebraic variety. Previous literature have also generalised the temperature parameter to an n-vector. Here, we show that in the case of n=2, the two components of this generalised quantity are the inverse temperature and inverse temperature times chemical potential, respectively. Other values of n can also be similarly interpreted as various intensive thermodynamic parameters. The second part of the talk concerns null geodesics in four dimensional spacetimes. In particular, we observe that the condition for null circular orbits defines an A-discriminantal variety. A theorem by Rojas and Rusek for A-discriminants leads to the interpretation that there are two branches of null circular orbits for certain classes of spacetimes. A physical consequence of this theorem is that light rings around generic black holes with non-degenerate horizons are unstable. [Joint work with Mounir Nisse] |
| Date: | March 31, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Terry Harris, Cornell University |
| Title: | Projections and intersections in the first Heisenberg group. |
| Abstract: | In this talk, I will discuss some recent work on the Hausdorff dimension of projections and intersections in the first Heisenberg group. In Euclidean space, it is known that projections of sets onto k-dimensional subspaces almost surely do not decrease Hausdorff dimension, and that projections of sets of dimension greater than k have projections almost surely of positive k-dimensional area. It has been conjectured that these theorems extend to "vertical projections" in the Heisenberg group. This conjecture is still open, but was recently solved in a significant part of the range by Fassler and Orponen, using a "point-plate incidence" method. I will outline some of my recent work, which also uses the point-plate incidence method, and which proves the "positive area" part of the conjecture. One connection of this talk to harmonic analysis is that it uses the (endpoint) trilinear Kakeya inequality, which grew out of multilinear Fourier analysis inspired by the Fourier restriction and Kakeya conjectures. |
| Date: | April 6, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Andrew Comech |
| Title: | Limiting absorption principle and virtual levels of operator in Banach spaces |
| Abstract: | We review the concept of the limiting absorption principle (LAP) and its connection to virtual levels of operators in Banach spaces. Then we discuss a simple construction which allows us to derive optimal LAP estimates on the resolvent near a point of the spectrum which is not a virtual level. In particular, the construction applies to the Laplacian in dimension 2 (where the estimates were not known). |
| Date: | April 7, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Nestor Guillen, Texas State University |
| Title: | Cancelled!!!Landau’s approximation and related equations |
| Abstract: | The Landau approximation is an important equation in kinetic theory arising as an asymptotic limit for the Boltzmann equation. Despite not being parabolic, this equation is amenable to study by methods from the theory of parabolic equations. Determining whether blow up occurs or not is one of the most important and challenging the questions in the field. I will give a brief introduction to the equation, explain what makes it challenging from a PDE perspective including a comparison with the semi linear heat equation. Then I will mention some recent partial results by several researchers aimed at resolving the blow up question. In particular, I will discuss recent work with Maria Gualdani (UT Austin) on an isotropic version of the equation that was introduced by Krieger and Strain. |
| Date: | April 13, 2023 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Wilhelm Schlag, Yale |
| Title: | Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory |
| Abstract: | In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson |
| Date: | April 14, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Kirill Cherednichenko, University of Bath |
| Title: | Operator-norm homogenisation for Maxwell equations on periodic singular structures |
| Abstract: | I will discuss a new approach to obtaining uniform operator asymptotic estimates in periodic homogenisation. Based on a novel uniform Poincar ́e-type inequality, it bears similarities to the techniques I developed with Cooper (ARMA, 2016) and Velcic (JLMS, 2022). In the context of the Maxwell system, the analytic framework I will present leads to a new representation for the asymptotics obtained by Birman and Suslina in 2007 for the full system and by Suslina in 2004 for the electric field in the presence of currents. As part of the new asymptotic construction, I will link the leading-order approximation to a family of “homogenised” problems, which was not possible using the earlier method. The analysis presented applies to a class of inhomogeneous structures modelled by arbitrary periodic Borel measures. However, the results are new even for the particular case of the Lebesgue measure. This is joint work with Serena D’Onofrio. |
| Date: | April 21, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Ruoyu Wang, Northwestern University |
| Title: | Damped waves with singular damping on manifolds |
| Abstract: | We will discuss a new damped wave semigroup for damping exhibiting H\”{o}lder-type blowup near a hypersurface of a compact manifold. We will use this semigroup to prove a sharp energy decay result for singular damping on the torus, where the optimal rate of energy decay explicitly depends on the singularity of the damping. We also show that no finite time extinction could happen under this setting. This is a joint work with Perry Kleinhenz. |
| Date: | April 28, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 302 |
| Speaker: | Lior Alon, MIT |
| Title: | Quasicrystals and Lee-Yang Polynomials |
| Abstract: | The concept of quasi-periodic sets, functions, and measures is prevalent in diverse mathematical fields such as Mathematical Physics, Fourier Analysis, and Number Theory. In natural science, Shechtman was awarded the 2011 Nobel Prize for the discovery of materials with quasi-periodic atomic structures, which are now known as Quasicrystals. This talk will focus on Fourier quasicrystals (FQ): discrete measures with Fourier transform which is also discrete, and with some growth bound. In particular, we care about sets with a counting measure which is an FQ. By the Poisson summation formula, the counting measure of any discrete periodic set is an FQ. Recently, Kurasov and Sarnak provided a general construction (motivated by quantum graphs) of counting measures that are FQ. Their method is based on restricting the zero set of a multivariate Lee-Yang polynomial to an irrational line in the torus. In particular, they answered a long-standing question of Meyer, providing explicit FQ which is the counting measure of an a-periodic uniformly discrete set. In this talk, we will show that the Kurasov-Sarnak construction generates all FQ counting measures and that generically these sets are a-periodic and uniformly discrete. If time permits, we will see that these measures have well-defined gaps distribution whose properties are deduced from the polynomial's structure. The talk is aimed at a broad audience, no prior knowledge in the field is assumed. Based on joint works with Alex Cohen and Cynthia Vinzant. |
| Date: | May 19, 2023 |
| Time: | 1:50pm |
| Location: | BLOC 306 |
| Speaker: | Ilya Kachkovskiy, Michigan State University |
| Title: | Anderson localization for quasiperiodic operators with monotone potentials: perturbative and non-perturbative methods. |
| Abstract: | The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on $d$-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson localization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types. We will consider quasiperiodic operators $$ (H(x)\psi)_n=\epsilon(\Delta\psi)_n+f(x+n\cdot\omega)\psi_n,\quad n\in \mathbb Z^d, $$ where $\Delta$ is the discrete Laplacian, $\omega$ is a vector with rationally independent components, and $f$ is a $1$-periodic function on $\mathbb R$, monotone on $(0,1)$ with a positive lower bound on the derivative and some additional regularity properties. We will focus on two methods of proving Anderson localization for such operators: a perturbative method based on direct analysis of cancellations in the Rayleigh—Schr\”odinger perturbation series for arbitrary $d$, and a non—perturbative method based on the analysis of Green’s functions for $d=1$, originally developed by S. Jitomirskaya for the almost Mathieu operator. The talk is based on joint works with S. Krymskii, L. Parnovski, and R. Shterenberg (perturbative methods) and S. Jitomirskaya (non-perturbative methods). |
