Weekly Bulletin

A cyclic L-infinity algebra is a shifted symplectic formal derived stack. Using a new geometric approach to homological perturbation theory, we construct a shifted symplectic form on the associated derived stack. (This is a derived analogue of the correspondence between Lie algehroids and Lie groupoids.)

A fundamental problem in quantitative symplectic geometry is to understand in which ways a Hamtilonian flow can "squeeze" phase space. The special case of ellipsoids has been a great source of motivation for the last several decades, in many ways mirroring various important developments in the field (e.g. Gromov-Witten theory, Floer homology, symplectic field theory, embedded contact homology, and more). In this talk, I will survey some new developments in the study of high dimensional symplectic embeddings, and in particular the recent resolution of the so-called stabilized ellipsoid conjecture. Our framework sets up a bridge between quantitative symplectic geometry and the classical study of singular algebraic curves, studying the latter using tools from log Calabi-Yau mirror symmetry. I will not assume familiarity with any of this background.

I discuss the appearance of certain notions and results from contact topology in both equilibrium and non-equilibrium thermodynamics. These include non-smooth Legendrian submanifolds, Reeb chords, and the partial order on the space of Legendrians. Based on joint work with Michael Entov and Lenya Ryzhik.

We will start reviewing the Casten-Holland and Matano theorem for interior reactions. It establishes the nonexistence of nonconstant stable solutions in convex domains. We will then present a forthcoming result stating that the analogue result for boundary reactions is not true. This will require the development of a new Ginzburg-Landau theory for real valued functions, as well as the study of the half-Laplacian on the real line ---for which some open problems will be presented.

We will begin with a quick reminder of algebraic K-theory, and a few classical, vanishing results for negative K-theory. The talk will then focus on a striking 2019 article by Antieau, Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures. The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.

<p>Translation structures on surfaces can be obtained in many different ways: as suspensions of interval exchange transformations, from holomorphic differentials, as cotangent vectors to Teichmüller space, or by unfolding polygonal billiards. We ask how such translation surfaces generically look like for large genus.</p> <p>In the talk, I will introduce translation surfaces through their connections to dynamical systems and give an overview on the question of large-genus asymptotics. The focus will be on a recent joint result with Howard Masur and Kasra Rafi on the distribution of the number of short saddle connections.</p>

The uniformization problem for metric surfaces asks under which condition a metric space <i>X</i>, homeomorphic to a model surface <i>M</i>, admits a parametrization <i>u: M → X</i> with good geometric and analytic properties. In this talk, we focus on the case where <i>X</i> has locally finite Hausdorff 2-measure. After revisiting the breakthrough results of Bonk-Kleiner and Rajala, we will demonstrate that no additional assumptions are necessary for the existence of a "good" parametrization. If <i>X</i> is locally geodesic, such parametrizations can be constructed by exploiting existence and regularity properties of energy-minimizing Sobolev mappings.

We establish a direct correspondence between discrete‐time branching processes and a family of nonlinear evolutionary PDEs generalizing the inviscid Burgers equation. Starting from a simple nonlinear PDE, we show its stochastic analogue is a single‐type branching process with Poisson offspring, closely related to the Erdos-Renyi random graph. By relaxing the PDE’s constraints, this framework naturally extends to multitype branching processes that admit an interpretation as higher‐order coagulation with a multiplicative kernel. Our convergence analysis relies on a new large‐deviation result for the size distribution of finite progenies. Beyond providing an interesting link between PDEs and branching, the representation yields explicit bounds—and in some cases exact expressions—for the blow up time in these nonlinear PDEs.

We consider the problem of testing whether a single coefficient is equal to zero in linear models when the dimension of covariates p can be up to a constant fraction of sample size n. In this regime, an important topic is to propose tests with finite-sample valid size control without requiring the noise to follow strong distributional assumptions. In this paper, we propose a new method, called the residual permutation test (RPT), which is constructed by projecting the regression residuals onto the space orthogonal to the union of the column spaces of the original and permuted design matrices. RPT can be proved to achieve finite-sample size validity under fixed design with just exchangeable noises, whenever p < n/2. Moreover, RPT is shown to be asymptotically powerful for heavy-tailed noises with bounded (1+t) th order moment when the true coefficient is at least of order n−t/(1+t) for t∈ [0,1]. We further proved that this signal size requirement is essentially rate-optimal in the minimax sense. Numerical studies confirm that RPT performs well in a wide range of simulation settings with normal and heavy-tailed noise distributions.

Fano manifolds are complex manifolds with positive first Chern class. In this talk, I will discuss long-time behavior of Ricci flow on those particular complex manifolds. I will report some recent progress and related Laplacian comparison estimates which also hold for Riemannian manifolds with certain integral curvature conditions.

(joint work with T. Untrau) The talk will explain how Wasserstein distances provide a natural and flexible way to quantify equidistribution theorems in number theory. As an illustration, we will present a quantitative version of Deligne's Equidistribution Theorem.