Week 01.12.2024 – 07.12.2024

In this talk, I first give a very general overview of the concept of MBL (many-body localization). Then I introduce the so-called quantum sun model (aka avalanche model/ quantum grain model). Then  I discuss the significance of this model in the MBL community. Afterwards, I present our result which is proof of the localization and Poisson stat. for this model in certain range of parameters. Finally,  I give some rough ideas about the proof: we first prove the result for the "free model" (sum of free disordered spin z), then we show that the interacting model is sufficiently "similar" to the free model by controlling the ratio of "in-resonance" levels. This is joint work with Wojciech De Roeck.

We consider systems of interacting particles governed by the generalized/relativistic Langevin dynamics in the presence of singular repulsive interacting forces. For each system, we establish a rate of convergence toward the unique invariant probability measure, which relies on novel construction of Lyapunov functions. We also study asymptotic limits of these systems when passing to the limit the interested parameters (the small-mass limit and Newtonian limit, respectively).

This talk is based on joint works with H. D. Nguyen (University of Tennessee).

In this seminar, I will explore issues around teaching problem solving in mathematics. I will examine different ways in which the term ‘problem solving’ can be understood and suggest a framework for thinking about it. Problem solving is often felt to be conspicuous by its absence from much undergraduate mathematics. I will suggest an approach to teaching problem solving that focuses on the systematic and explicit teaching of specific, carefully-chosen problem-solving tactics, as opposed to more widely applicable generic strategies. I hope that colleagues will be willing to share their thoughts and experiences.

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

Time crystals are many-body systems that spontaneously break time-translation symmetry, and thus exhibit long-range spatiotemporal order and robust periodic motion. Recent results have demonstrated how to build time-crystal phases in driven diffusive fluids, based on a particular class of symmetry-breaking dynamical phase transitions present in their rare event statistics. A main tool in this idea is the application of an external packing field coupled to density fluctuations, that triggers an instability to a time-crystal phase. In this seminar we will explore the connection between time crystals, rare events and dynamical phase transitions. We will also describe how to exploit this packing-field mechanism to engineer and control on demand programmable continuous time crystals characterized by an arbitrary number of rotating condensates, which can be further enhanced with higher-order modes. We will elucidate the underlying critical point, as well as general properties of the traveling condensates, illustrating our findings in various paradigmatic driven diffusive systems. Overall, these results demonstrate the versatility and broad possibilities of this promising route to time crystals.

The perturbative expansion of quantum field theory expresses physical quantities as series of numbers (or functions) associated to combinatorial graphs, called Feynman integrals. These integrals are hard to compute, and furthermore their sum forms a series that is in fact divergent. To gain insights into the large order behaviour, Feynman integrals can be approximated astonishingly well by easily computable combinatorial invariants of graphs. I will discuss two such approximations: the tropical Feynman integral and the Martin invariants, using phi^4 theory as an example. The Martin invariants are related to the O(-2) symmetric vector model and can be generalized to an integer sequence. I will end explaining how this sequence encodes the exact value of a Feynman integral through a limit used by Apery to prove the irrationality of zeta(3).

I will talk about zeros of Gaussian Analytic Functions of the form $f(z) = \sum c_k z^k$ with independent standard complex normal coefficients and conditioned by the event that $|f(0)|^2=t$. The probability law of the zero set of $f(z)$ can be derived from that of the spectrum of random sub-unitary matrices. I will explain how this link can be used to obtain the full conditional distribution of radial zero counting function in terms of a $q$-series and use asymptotic expansions of the $q$-series to prove asymptotic normality of the counting function, develop precise large deviation estimates and asymptotic expansion of the conditional hole probability. It turns out that to leading order, the conditional hole probability does not depend on parameter $t$ for $t>0$ and coincides with the hole probability for unconditioned GAF of the form $\sum \sqrt{k+1} c_k z^k$. My talk is based on joint work with Yan Fyodorov and Thomas Prellberg.

Confounding is a central concept in causal inference, but it is notoriously difficult to give it a positive definition. In this talk, I will break down this concept in two steps. In the first step, I will use the linear equation model and linear algebra to understand the semantics of an acyclic directed mixed graphs (ADMGs) and introduce a notational system for “words” (special types of walks/paths) in ADGMs. In the second step, I will break down Pearl’s celebrated back-door criterion and introduce an alternative approach to confounder selection by iterative graph expansion.
This talk is based on several working papers: https://arxiv.org/abs/2407.15744, https://arxiv.org/abs/2309.06053 (with F Richard Guo), and to a less degree, https://www.statslab.cam.ac.uk/~qz280/publication/admg-model/paper.pdf.