This is a colloquium-style seminar. It runs on Wednesdays. The current organizers are Melkana Brakalova, Wen Li and Han-Bom Moon.
The seminar is broadcasted via zoom. Check the announcement emails for the link.
Date: October 28, 3:00 pm
Place: JMH 406 (RH)
Speaker: Jussi Behrndt (Graz University of Technology)
Title: Spectral theory for differential operators with singular potentials
Abstract: In this talk, we discuss qualitative spectral properties of self-adjoint Schrödinger and Dirac operators. We first briefly review some of the standard results for regular potentials from the literature and turn to more recent developments afterwards. Our main objective in this lecture is to discuss differential operators with singular potentials supported on curves or hyperplanes, where in the case of Dirac operators it is necessary to distinguish the so-called non-critical and critical cases for the strength of the singular perturbation. In particular, it turns out that Dirac operators with singular potentials in the critical case have some unexpected spectral properties. This talk is based on some recent joint works with B. Benhellal, P. Exner, M. Holzmann, V. Lotoreichik, T. Ourmieres-Bonafos, and K. Pankrashkin.
Date: October 14, 3:00 pm
Place: JMH 406 (RH)
Speaker: Özgür Yilmaz (University of British Columbia)
Title: How much data do we need? Sampling, sparsity, and learned priors
Abstract: Suppose we want to recover a signal from a small number of linear measurements. How many measurements do we need, and how should we choose them? The classical answer, going back to Shannon and Nyquist, is that a bandlimited signal is determined by its samples above the Nyquist rate. Compressed sensing, developed twenty years ago, showed that if the signal is sparse in some basis then random measurements can recover it with far fewer samples than Nyquist would predict. In the last decade, deep generative models have introduced a new kind of prior: the signal is assumed to lie near the range of a learned generator (for example, a GAN generator or a VAE decoder), replacing the sparse prior with a data-driven manifold. In this talk, I will describe a framework that unifies the sparse and generative cases through a single geometric quantity: the local coherence between the signal class and the measurement basis. This quantity determines how many measurements suffice, how they should be distributed, and how well recovery tolerates noise. I will explain what the theory predicts, what it does not yet cover, and what recent numerical experiments say about how the framework fares on diffusion-based generative models. Based on joint work with Aaron Berk, Simone Brugiapaglia, Babhru Joshi, Yaniv Plan, Matthew Scott, and Xia Sheng.
Date: February 25, 3:00 pm
Place: JMH 406 (RH)
Speaker: Alessio Caminata (University of Genova)
Title: F-signature: measuring singularities with Frobenius
Abstract: The F-signature is a numerical invariant defined for local rings in positive characteristic using the Frobenius homomorphism. It encodes subtle information about singularities and has received a lot of attention over the last twenty years. However, in general it is very difficult to compute, and it is explicitly known only for certain special classes of rings. In this talk, I will give an introduction to the F-signature and then survey a series of recent results on the computation of this invariant, as well as the corresponding numerical function. The talk is based on joint works with A. De Stefani, S. Shideler, K. Tucker, and F. Zerman.
Date: March 4, 3:00 pm
Place: JMH 406 (RH)
Speaker: Brian Rider (Temple University)
Title: The general Tracy-Widom laws
Abstract: Many years ago I overheard a colleague referring to the Tracy-Widom law(s) as the Gaussian for the 21st century. As time goes on this seems less and less an exaggeration. First discovered in the context of the largest eigenvalues for certain random matrices, the Tracy-Widom laws are now understood to govern a range of nonlinear phenomena arising in combinatorics, statistics physics, stochastic partial differential equations and more. I will describe a natural one-parameter generalization of these laws available through certain random differential equations and various consequences of this picture.
Date: April 1, 3:00 pm
Place: JMH 406 (RH)
Speaker: Liviana Palmisano (KTH, Royal Institute of Technology)
Title: Order in chaos
Abstract: I will introduce a fundamental object in Dynamical Systems: the attractor of a system, and illustrate it through several examples. In particular, I will present chaotic attractors arising from real biological and physical processes, and discuss the He ́non attractor, one of the simplest models showing complex and seemingly unpredictable behavior. I will describe how the orbits of these systems behave and what makes their dynamics chaotic.
A natural question is what happens when we perturb such a system slightly. Does its behavior change completely, or does some of its properties persist? In other words, how stable are these chaotic attractors?
Date: April 29, 3:00 pm
Place: JMH 406 (RH)
Speaker: Paul Jung (Fordham University)
Title: The Far-reaching Consequences of comparing Sampling with Replacement vs Sampling without Replacement
Abstract: A freshman can calculate that the probability of picking $k$ blue balls after sampling $n$ balls from a bin of $K$ blue balls and $N-K$ red balls is, sampling without replacement,
$$\frac{\dbinom{n}{k} \dbinom{N-n}{K-k}}{\dbinom{N}{K}}.$$
If one samples with replacement it is
$$\dbinom{n}{k} \left(\frac{K}{N}\right)^k\left(\frac{N-K}{N}\right)^{n-k}.$$
We will show that comparing these probabilities leads to proofs of De Finetti's Theorem, the Aldous-Hoover Theorem, and even a weak form of Szemeredi's Regularity Lemma which plays a crucial role in the study of graphons. This last application played a role in the awarding of two Abel Prizes (Szemeredi 2012, Lovasz 2021).