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.

Abstract:
The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. Bruno Duchesne and I examined the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identified them with some kaleidoscopic group acting on dendrites or universal groups acting on biregular trees, realizing them as Polish permutation groups. From these identifications, we deduced algebraic, topological and geometric properties of these groups.This talk will present the identification of the groups and the results of this work.

<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>