Weekly Bulletin

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.

In this talk we will introduce a new PDE way to construct hypersurfaces which are critical for anisotropic integrands. Namely, we study energy concentration for rescalings of an anisotropic version of Allen-Cahn. Besides a Gamma-convergence result, we will sketch a proof of the fact that energy of stable critical points (of the rescaled Allen-Cahn) concentrates along an integer rectifiable varifold, a weak notion of hypersurface, using stability (or finite Morse index) to compensate for the lack of monotonicity formulas. Among the new technical ingredients, we will see a generalization of Modica's bound and a diffuse version of the stability inequality for hypersurfaces. This is joint work with Antonio De Rosa (Bocconi University)

<article class="text-token-text-primary w-full focus:outline-none scroll-mt-(--header-height)" dir="auto" tabindex="-1" data-turn-id="2713db4b-07eb-4f73-8bf4-27192e14efc2" data-testid="conversation-turn-1" data-scroll-anchor="false" data-turn="user"> <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> </article>

In this talk, I will introduce a general operation of producing shifted symplectic stacks from given ones. Basic examples like cotangent bundles, critical loci, and Hamiltonian reduction can be understood as special cases of this operation. Moreover, this unification enables us to provide an etale local structure theorem for shifted symplectic Artin stacks. As an application, I will explain how to construct cohomological Hall algebras for 3-Calabi-Yau categories, which is joint work with Tasuki Kinjo and Pavel Safronov.

The signature is one of the most versatile invariants in knot theory. For example, it can be used to detect the chirality of a knot, and yields lower bounds on the unknotting number, the Seifert genus, as well as the four-genus of a knot. This invariant also extends nicely to links, offering new insights such as bounds on the splitting number. The aim of this talk is to give a gentle introduction to this powerful invariant, tracing its development from Trotter’s original definition in 1962 to the most recent advances.

Are two functions identical if they take the same values on discrete samples? According to classical Shannon sampling, the answer to this type of uniqueness question is yes when the functions are band-limited. In fact, the uniqueness question is determined by the density of the discrete samples. We study non-linear versions of this problem, where it is assumed that the functions agree only up to multiplicative symmetries. We develop a non-linear Shannon-type theory, obtaining sharp results that characterize when uniqueness holds and when it fails. In special cases, this framework recovers results from phase retrieval. Finally, we investigate the problem in finite dimensions, where it is related to information-completeness in quantum theory. In this setting, we demonstrate that the problem exhibits behavior markedly different from the infinite-dimensional case of band-limited functions. The talk is based on joint work with Tomasz Szczepanski (University of Alberta).

Bernstein-gamma (BG) functions, introduced by Patie and Savov (and also considered in earlier works by Berg, Bertoin, Hirsch, Yor, and others), solve a gamma-type recurrence with a Bernstein function in place of the identity. Their relevance for studying exponential functionals of Lévy processes stems from the fact that the Mellin transform of an EF factors through BG functions. This representation lets us extract asymptotics via Mellin inversion, Tauberian arguments, and links to Wiener-Hopf factors, and, in some cases, it yields weak limits for suitably scaled EF laws. We will sketch some concrete arguments and discuss how these ideas could extend to Markov additive processes through a matrix- or operator-valued analogue of BG functions, noting new obstacles. Joint work with Mladen Savov.

Many applications, ranging from reinforcement learning to the analysis of time series, involve high-dimensional random matrices that are generated by a stochastic process. Such settings can be challenging to analyze due to the dependence involved in the process. In this talk, I present a new universality principle for sums of matrices generated by a Markov chain that enables sharp concentration estimates when combined with recent advances in the Gaussian literature. A key challenge in the proof is that techniques based only on classical cumulants, which have been used by Brailovskaya and Van Handel in a setting with independent summands, fail to produce efficient estimates in our dependent setting. We hence developed a new approach based on Boolean cumulants and a change--of--measure argument. Based on joint work with Jaron Sanders, available at arXiv:2307.11632.

New deep neural network architectures based on high-order weak approximation algorithms for stochastic differential equations (SDEs) are proposed. The core of these architectures is formed by high-order weak approximation algorithms of the explicit Runge--Kutta type, in which the approximation is realised solely through iterative compositions and linear combinations of the vector fields of the target SDEs.

We propose a linear algebraic framework for performing density estimation. It consists of three simple steps: convolving the empirical distribution with certain smoothing kernels to remove the exponentially large variance; compressing the empirical distribution after convolution as a tensor train, with efficient tensor decomposition algorithms; and finally, applying a deconvolution step to recover the estimated density from such tensor-train representation. Numerical results demonstrate the high accuracy and efficiency of the proposed methods.

We define and prove properties of a GL(2, Z)-invariant function, a Lyapunov exponent associated to the modular function j, generalizing a function defined by Spalding and Veselov in the case of the constant function 1. Our results were motivated by conjectures of Kaneko about the "values" of j at real quadratic irrationalities. This is joint work with Paloma Bengoechea and Özlem Imamoglu.

I will discuss recent progress on conjectural formulas for quintuple Hodge integrals. A proof of these formulas will be presented in two distinct limits, and I will discuss the obstructions to extending the arguments beyond the two regimes. I will also mention their implications for Gromov–Witten theory of toric Calabi–Yau fivefolds and string theory, with particular attention to constant map contributions.