Preliminary Schedule
Abstracts
Aren Martinian (YRS)
Title: The empirical spectral distribution of the Hessian of the loss function of neural networks
The empirical spectral distribution (ESD) of the loss function of a neural network contains important qualitative information about the learning process and can be used to estimate the rate at which the loss approaches its minima. However, the ESD is typically difficult to compute. Generalizing the work of Liao and Mahoney, our work uses free probability, random matrix theory, and a block matrix framework to develop a recursive algorithm to estimate the ESD of a single-layer network with many-dimensional output. We find that this spectrum depends on the chosen loss and the limiting ratio of data points to width. In this talk, I will introduce the concepts underlying our research and discuss avenues for future study. This talk is based on joint work with Benjamin Eisley, under the guidance of Professor Federico Pasqualotto. The research was partially funded by the SURF summer fellowship for Berkeley students.
Karl Zieber (YRS)
Title: Localization in the nonstationary unbounded 1D Anderson model
Spectral localization of the 1D Anderson model has been studied extensively, with many known results for iid potentials. Recently, spectral localization in the nonstationary, bounded case was established. In this talk, we will present a new approach to the nonstationary case that removes the boundedness assumption by leveraging recent advances in the theory of random non-stationary matrix products.
Alexandro Luna (YRS)
Title: Regularity of Non-stationary Stable Foliations and Applications to Spectral Properties of Sturmian Hamiltonians
We discuss regularity results concerning non-stationary stable foliations of hyperbolic maps satisfying a common cone condition. We then describe how to use these results to obtain properties concerning the dimension of the spectra of Sturmian Hamiltonians. This work is joint with Seung uk Jang.
Liyang Shao (YRS)
Title: Weighted inhomogeneous bad is winning and null
We will introduce the notion of inhomogeneous weighted badly approximable vectors. We discuss that this set can be very large (winning) in a sense and in some other sense it is very small (Lebesgue measure zero). In particular, we talk about such largeness and smallness via studying weighted inhomogeneous bad intersected with manifolds and support of certain measures. This is joint work with Shreyasi Datta.
Michael Aizenman
Title: Entropy of (quantum) entanglement in pure states of rapid decorrelation
I. The entropy of the restriction of a pure quantum state to a subsystem is a measure of the entanglement between the system's two components.
II. After explaining the concepts, the talk will focus on conditions implying an area-type bound on the entanglement in pure states of quantum lattice models.
Alexander Elgart
Title: Localization properties of the random XXZ spin chain
I will discuss the spectral and dynamical properties of the random Heisenberg XXZ spin chain in arbitrary (but system-size independent) energy intervals. The results include the almost sure exponential decay of Green functions and eigen-correlators associated with the underlying Hamiltonian, as well as spectral localization for the infinite spin-chain. Based on joint works with Abel Klein.
Stanislav Molchanov
Title: Localization on the Exner’s lattice
Exner’s lattice is a periodic quantum graph with unit edges connecting neighboring vertices. The Laplacian is defined on smooth functions on edges with Kirchhoff-type gluing conditions at the vertices. P. Exner studied the spectrum of such Laplacians and discovered that the spectra of a wide class of these operators (with appropriate gluing conditions) contain infinitely many gaps — in contrast to the classical periodic Schrödinger operator. This talk (in collaboration with O. Safronov) presents recent results on localization inside the gaps of the Exner’s Laplacian perturbed by a random potential concentrated on the vertices. The work was inspired by studies from the Optical Center (UNC Charlotte) related to optical computing. The talk will also include a review of recent works on localization on quantum graphs.
Grigorii Monakov
Title: Central limit theorem for non-stationary random products of SL(2, ℝ) matrices
Consider a sequence of independent and identically distributed SL(2, ℝ) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. I will talk about a recent joint work with Anton Gorodetski and Victor Kleptsyn, where we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, ℝ) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.
Omar Hurtado
Title: Localization for the non-stationary Anderson model in 2 dimensions
We discuss a localization result for variants of the Anderson model with non-stationary random potentials. Under some assumptions on the non-stationary potential, we are able to obtain localization at the bottom of the spectrum. The localization argument is based on that introduced by Bourgain and Kenig and adapted to the discrete setting by Ding and Smart; it combines probabilistic unique continuation bounds with combinatorial bounds to obtain a Wegner type estimate, for use in a multiscale analysis argument.
Christoph Fischbacher
Title: Non-selfadjoint operators with non-local point interactions
In this talk, I will discuss non-selfadjoint differential operators of the form
i(d/dx) + V + k⟨δ,·⟩, where V is a bounded complex potential. The additional term, formally given by k⟨δ,·⟩, is referred to as a “non-local point interaction” and has been studied in the selfadjoint context by Albeverio, Cojuhari, Debowska, I.L. Nizhnik, and L.P. Nizhnik.
Konstantin Khanin
Title: First passage percolation in a product-type random environment
We consider a first passage percolation model in dimension 1+1 with potential given by the product of a spatial i.i.d. potential with symmetric bounded distribution and an independent i.i.d. in time sequence of signs. We assume that the density of the spatial potential near the edge of its support behaves as a power, with exponent κ > −1. We investigate the linear growth rate of the actions of optimal point-to-point lazy random walk paths as a function of the path slope and describe the structure of the resulting shape function. It has a corner at 0 and, although its restriction to positive slopes cannot be linear, we prove that it has a flat edge near 0 if κ > 0. For optimal point-to-line paths, we study their actions and locations of favorable edges that the paths tend to reach and stay at. Under an additional assumption on the time it takes for the optimal path to reach the favorable location, we prove that appropriately normalized actions converge to a limiting distribution that can be viewed as a counterpart of the Tracy–Widom law. Since the scaling exponent and the limiting distribution depend only on the parameter κ, our results provide a description of a new universality class.
The talk is based on joint work with Yuri Bakhtin, András Mészáros, and Jeremy Voltz.
Hermann Schulz-Baldes
Title: Local perturbations of Toeplitz matrices
This talk is about the asymptotic spectral theory of tridiagonal Toeplitz matrices with matrix entries, with periodicity broken on a finite number of entries. Varying the ranks of these perturbations allows interpolation between open boundary and circulant Toeplitz matrices. While the continuous parts of the limit spectrum depend only on these ranks and no other aspect of the perturbation, the outliers of the spectrum depend continuously on the local perturbation. The proof is essentially based on a generalized Widom formula for the characteristic polynomial. The mathematical results are illustrated by numerics. Joint work with Lars Koekenbier.
Peter Müller
Title: On Szegő-type asymptotics for quasifree Fermi gases
In the first part of the talk we present a generalization of the Widom-Sobolev formula to matrix-valued symbols. It can be applied to deduce a logarithmically enhanced area law for the entanglement entropy of the ground state of a relativistic free Fermi gas with the Dirac operator as single-particle Hamiltonian.
In the second part of the talk we consider a non-relativistic quasifree Fermi gas in a spatially homogeneous magnetic field, i.e., with the Landau operator as its single-particle Hamiltonian. The density of the gas is assumed to be such that its ground state corresponds to a half-filled lowest Landau level. We show that for suitably chosen ground states of that type the entanglement entropy can exhibit any type of growth between an area and a volume law.
The talk is based on joint work with Leon Bollmann and Leo Wetzel.
Jeff Schenker
Title: New perspectives on disordered quantum systems: open systems and matrix product states
I will give an overview of recent results and open questions on disordered matrix product states and disordered generalized measurement processes. Both areas of study rely on the long-time asymptotics of Ergodic Quantum Processes (EQP), which were introduced by the speaker and Ramis Movassagh in 2020. An EQP is a discrete time evolution described by the composition of a sequence of random quantum channels with a stationary distribution.
(Joint work with Owen Ekblad, Eloy Moreno Nadales, Lubashan Pathirana, Renaud Raquepas, and Eric Roon)
Ilya Kachkovskiy
Title: Quasiperiodic operators with monotone potentials
We address the recent joint result of the author with S. Jitomirskaya, L. Parnovski, and R. Shterenberg. These results include: Non-perturbative arithmetic phase transitions for one-dimensional quasiperiodic operators with monotone potentials. Perturbative localization for multi-dimensional operators of the above class. Existence of gaps in the spectra and, in some cases, Cantor spectra for these operators.
Netanel Levi
Title: Spectral Multiplicity of Jacobi Operators on Star-Like Graphs
I will discuss the spectral multiplicity of Jacobi operators on star-like graphs, formed by attaching a finite number of half-lines to a compact component. It was recently shown that in the singular continuous spectrum, the multiplicity is bounded above by m, the number of half-lines. Our main result improves this bound to m − 1, and we show that this is optimal. The proof uses ideas from generalized eigenfunction expansion and subordinacy theory, along with a detailed analysis of certain invariant subspaces. Time permitting, I will also present examples illustrating the sharpness of the bound and discuss possible generalizations of our results. This is joint work with Tal Malinovitch.
Wencai Liu
Title: Algebraic geometry, analysis and combinatorics in the study of ℤᵈ -periodic graph operators
In this talk, we will discuss how techniques from algebraic geometry, combinatorics, and mathematical physics can be applied to the spectral theory of ℤᵈ-periodic graph operators. Specifically, we will present our recent results on proving the irreducibility of discrete periodic Schrödinger operators and demonstrating rare flat bands in any connected ℤᵈ -periodic graph.This work is partially joint with Matt Faust.
Barry Simon
Title: Gap labelling for periodic Jacobi matrices on trees
I will present a new and simple proof (published with Banks, Breuer, Garza Vargas, and Seelig) of Sunada’s gap labelling for periodic Jacobi matrices on trees. I will discuss what periodic Jacobi matrices on trees are and the historical background of the result as well as present the proof.
Bruno Nachtergaele
Title: The Charge Gap, the Neutral Gap, and Fractional Quantum Hall Systems
We introduce an elementary inequality relating the charge and neutral gaps of a family of quantum many-body systems and discuss an application to the so-called pseudo-potentials modeling fractional quantum Hall systems. These are quantum many-body Hamiltonians that are frustration free and have two symmetries, one related to the conservation of charge (particle number) and another to the conservation of dipole moment (angular momentum), in addition to translation invariance. We show that for such systems the general inequality can be refined, opening the possibility of new applications to the study of the spectral gap of such systems.
(Joint work with Marius Lemm, Simone Warzel, and Amanda Young, arXiv:2410.11645.)
Peter Hislop
Title: Dynamical localization and delocalization for random Schrödinger operators with delta-function potentials in three dimensions
I will discuss joint work with M. Krishna and W. Kirsch in which we prove that random Schrödinger operators on ℝ³ with independent, identically distributed random variables and single-site potentials given by δ-functions on ℤ³ exhibit both dynamical localization and dynamical delocalization with probability one. That is, there are regions in the deterministic spectrum that exhibit dynamical localization, and regions that exhibit nontrivial quantum transport, almost surely. The nontrivial transport is due to the existence of delocalized, generalized eigenfunctions at positive energies E > π². Another new result of independent interest is a proof of the Combes–Thomas estimate on exponential decay of the Green’s function for Schrödinger operators with δ-potentials.
Anton Gorodetski
Title: Anderson localization beyond iid
We will present a series of results on 1D random Schrödinger operators that generalize the classical work by Carmona–Klein–Martinelli on Anderson localization in the iid Anderson–Bernoulli model. First, we will deal with non-stationary Anderson localization, when the potential is defined by independent but not necessarily identically distributed random variables. Using a non-stationary version of the Furstenberg Theorem, one can show that both spectral and dynamical localization hold in this case.
Then we will consider the case of a random potential defined by a block code, in which case the values of the potential are not even independent. In this case, spectral localization holds, but eigenfunctions in general do not have to have a uniform rate of exponential decay, and dynamical localization holds outside of a finite set of exceptional energies.
Finally, we will discuss the topological structure of the spectrum in the case of an ergodic potential with random iid noise and show that in many cases the spectrum is a finite union of intervals, similarly to the classical iid Anderson model. In all these cases, no assumptions on the regularity of the potential are needed.
The talk is based on joint results with Artur Avila, David Damanik, and Victor Kleptsyn.