MenuEvents for 10/07/2022 from all calendars
Probability and Mathematical Physics
Time: 09:00AM - 10:00AM
Location: Zoom
Speaker: Mostafa Sabri, Cairo University
Title: Quantum ergodicity for periodic graphs
Abstract: Quantum ergodicity for graphs is a delocalization result which says the following. Suppose that a sequence of finite graphs $Gamma_N$ converges to some infinite graph $\Gamma$. Then most eigenfunctions $\psi_j^{(N)}$ of the adjacency matrix $\mathcal{A}_N$ on $\Gamma_N$ become equidistributed on $\Gamma_N$ when $N$ gets large. More precisely, for most $j$, the probability measure $\sum_{v\in \Gamma_N} |\psi_j^{(N)}(v)|^2 \delta_v$ approaches the uniform measure $\frac{1}{|\Gamma_N|}\sum_{v\in \Gamma_N} \delta_v$, in a weak sense. Potentials $Q$ can sometimes be added, so that one now considers the eigenfunctions of $H_N = \mathcal{A}_N+Q_N$. Usually, the proof partially relies on certain nice properties of the infinite graph $\Gamma$. In particular, quantum ergodicity theorems have previously been established when $\Gamma$ is a tree. In this talk, I will present recent results of quantum ergodicity when $\Gamma$ is invariant under translations of some basis of $\mathbb{Z}^d$, and the ``fundamental block'' is endowed a potential Q which is copied across the blocks, so that $H = \mathcal{A}_\Gamma+Q$ is a periodic Schr\"odinger operator. This framework includes $\Gamma=\mathbb{Z}^d$, the honeycomb lattice, strips, cylinders, etc. I will first discuss the Bloch theorem and give some examples of its limitations, presenting along the way some very homogeneous graphs which violate quantum ergodicity. I will then discuss our main results, contrasting them with the tree case, give various examples of applications, and sketch the proof. An open problem concerning Schr\"odinger operators with a periodic potential on $\mathbb{Z}^d$, $d>1$, will also be presented. This talk is based on a joint work with Theo McKenzie (Harvard).
Student/Postdoc Working Geometry Seminar
Time: 1:30PM - 2:30PM
Location: BLOC 628
Speaker: Suhan Zhong, TAMU
Title: Dehomogenization for Completely Positive Tensors
Abstract: A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach.
Mathematical Physics and Harmonic Analysis Seminar
Time: 1:50PM - 2:50PM
Location: BLOC 306
Speaker: Rodrigo Matos , TAMU
Title: Eigenvalue statistics for the disordered Hubbard model within Hartree-Fock theory
Abstract: Localization in the disordered Hubbard model within Hartree-Fock theory was previously established in joint work with J. Schenker, in the regime of large disorder in arbitrary dimension and at any disorder strength in dimension one, provided the interaction strength is sufficiently small. I will present recent progress on the spectral statistics conjecture for this model. Under weak interactions and for energies in the localization regime which are also Lebesgue points of the density of states, it is shown that a suitable local eigenvalue process converges in distribution to a Poisson process with intensity given by the density of states times Lebesgue measure. If time allows, proof ideas and further research directions will be discussed, including a Minami estimate and its applications.
Algebra and Combinatorics Seminar
Time: 3:00PM - 4:00PM
Location: BLOC 302
Speaker: Matthew Faust, TAMU
Title: On the Irreducibility of Bloch and Fermi Varieties
Abstract: Given an infinite ZZ^n periodic graph G, the Schrodinger operator acting on G is a graph Laplacian perturbed by a potential at every vertex. Complexifying and choosing an M-periodic potential for some full rank free module M of ZZ^n fixes a representation of our operator as a finite matrix whose entries are Laurent polynomials. The vanishing set of the characteristic polynomial yields the Bloch variety, the vanishing set for fixed eigenvalues gives the Fermi variety. Questions regarding the algebraic properties of these objects are of significant interest in mathematical physics. We will focus our attention on the irreducibility of these varieties. Understanding the irreducibility of Bloch and Fermi varieties is important in the study of the spectrum of periodic operators, providing insight into the structure of spectral edges, embedded eigenvalues, and other applications. In this talk we will present several new criteria for obtaining irreducibility of Bloch and Fermi varieties for infinite families of discrete periodic operators. This is joint work with Jordy Lopez.
Geometry Seminar
Time: 4:00PM - 5:00PM
Location: BLOC 302
Speaker: Arpan Pal, TAMU
Title: Concise tensors of minimal border rank
Abstract: We know that if a collection of square matrices are simultaneously diagonalizable then they commute, however the converse does not hold. It has been a classical problem in linear algebra to classify the closure of the space of simultaneously diagonalizable matrices. This problem is closely related to a problem regarding tensors. In this talk I shall describe the problem, the relation to classical question, and recent progress towards classifying minimal border rank tensors. This is joint work with JM Landsberg and Joachim Jelisiejew.
Free Probability and Operators
Time: 4:00PM - 5:00PM
Location: BLOC 306
Speaker: Michael Anshelevich, TAMU
Title: Equivalent definitions of finite free convolution
Abstract: I will prove the theorem of Marcus, Spielman, and Srivastava on the equivalence of two definitions of finite free convolution: one in terms of random matrices, the other explicit and combinatorial. The proof uses no machinery beyond (fairly) elementary matrix algebra.
