Abstract:
In this talk, I first give a general overview of the concept of MBL (many-body localization), and 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, including why it was introduced and its implications for the stability of MBL. Afterward, I present our result, proof of the localization, and Poisson statistics for this model in a 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 a joint work with Wojciech De Roeck.

Abstract:
In this talk I will explore the correspondence, established by Etnyre and Ghrist, between Reeb vector fields and Beltrami vector fields, which are stationary solutions of the Euler equations. This bridge allows us to apply techniques from contact geometry to fluid dynamics. As an example, I will show how universality of Beltrami fields can be deduced via an h-principle in contact geometry, and how Reeb embeddings can be used to construct Turing-complete solutions of the Euler equations, i.e. stationary flows capable of emulating the computation of a universal Turing machine. Extending these ideas, I will discuss how Reeb vector fields on cosymplectic manifolds provide a framework to build Turing-complete stationary solutions of the Navier–Stokes equations, thus giving a positive answer to a question raised by Terence Tao. These results illustrate a striking connection between contact and symplectic geometry, fluid dynamics, and theoretical computer science — all in the flow.

Abstract:
Diffusion phenomena are ubiquitous, from the flow of heat and the dispersion of milk in coffee to the spread of ideas, the invasion of biological species, and the progression of a disease into a pandemic. The immense power of mathematics lies in its intrinsic capacity to unify these seemingly disparate aspects and provide valuable insights for understanding them.This talk will present classical and modern approaches to modeling diffusion. We will highlight how diverse phenomena, even those involving living beings from microorganisms to humans, often exhibit long-tailed distributions of steps and go beyond the classical assumptions of simple random walks.We will then demonstrate how mathematics can effectively deal with these complex cases, offering a richer picture of how things spread in the real world. No preliminary knowledge on these topics is required.

<div class="text-base my-auto mx-auto pt-3 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)"> <div class="[--thread-content-max-width:40rem] thread-sm:[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col" tabindex="-1"> <div class="flex max-w-full flex-col grow"> <div class="min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&]:mt-5" dir="auto" data-message-author-role="user" data-message-id="2713db4b-07eb-4f73-8bf4-27192e14efc2"> <div class="flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start"> <div class="user-message-bubble-color relative rounded-[18px] px-4 py-1.5 data-[multiline]:py-3 max-w-[var(--user-chat-width,70%)]" data-multiline=""> <div class="whitespace-pre-wrap">In this talk we show that under general resonance the classical piecewise linear Fermi-Ulam accelerator behaves substantially different from its quantization in the sense that the classical accelerator exhibits typical recurrence and non-escaping while the quantum version enjoys quadratic energy growth in general. We also describe a procedure to locate the escaping orbits, though exceptionally rare in the infinite-volume phase space, for the classical accelerators, which in particular include Ulam's very original proposal and the linearly escaping orbits therein in the existing literature, and hence provide a complete (modulo a null set) answer to Ulam's original question. For the quantum accelerators, we reveal under resonance the direct and explicit connection between the energy growth and the shape of the quasi-energy spectra.</div> </div> </div> </div> </div> </div> </div>