Weekly Bulletin

Floer homology and the related invariants of 4-manifolds has given us deep insight in smooth differential topology in dimensions 3 and particularly 4. The theory has yielded insights like existence of exotic differentiable structures on 4 dimensional euclidean space, complex curves minimize genus in complex projective space, killing the Hauptvermuntung, there even appear to be connection to the 4 color map theorem. This course will build up Floer homology of three manifolds from scratch. The focus will be on Instanton Floer homology but we will mention other versions and develop applications as the course goes on.

A generalized configuration space on $X$ consists of a collection of points on $X$ with specific rules governing which points cannot coincide. In this work, I will introduce a new algebraic structure, called a "contractad," on the union of these spaces for $X = \mathbb{R}^n$, which extends the concept of the little discs operad. I will demonstrate how this algebraic framework can be used to extract information regarding the Hilbert series of cohomology rings. Surprisingly, the same approach can be applied to generate series for various combinatorial data associated with graphs, such as the number of Hamiltonian paths, Hamiltonian cycles, acyclic orientations, and chromatic polynomials. Additionally, natural compactifications of these configuration spaces for $X = \mathbb{C}$ generalize the Deligne-Mumford compactification of moduli spaces of rational curves with marked points. If time allows, we will also discuss the generating series for their cohomology. The talk is based on the joint work with D.Lyskov: https://arxiv.org/abs/2406.05909

The central point of this talk is to present a strategy for proving that Lagrangians which are displaceable by a Hamiltonian diffeomorphism admit a "Weinstein neighborhood of non-displacement", i.e. a neighborhood W of the given Lagrangian L such that if the image of L by a Hamiltonian diffeomorphism is included in W, it must intersect L. When the inclusion of L into M induces the 0-map at the level of first homology groups with real coefficients, this non-displacement property also holds for any Lagrangian included in W which is the image of L by a (non necessarily Hamiltonian) symplectomorphism. In both cases, non-displacement follows directly from "local exactness" of nearby Lagrangians, i.e. the fact that any Lagrangian in the Hamiltonian or symplectic orbit of L, included in W, is exact in W seen as a subset of T*L. I will give several natural examples for which such a neighborhood exists. I will then discuss applications of this line of ideas in terms of the topology of the Hamiltonian orbit of L, and in terms of C^0 symplectic geometry. This is joint work with Jean-Philippe Chassé.

In this course we will investigate questions of weak turbulence theory by using as explicit example of wave interactions the solutions to periodic and nonlinear Schrödinger equations. We will start with Strichartz estimates on periodic setting, then we will move to well-posedness. We will then present two different ways of introducing the evolution of the energy spectrum. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. Then, we will give some ideas of how to derive rigorously the effective dynamics of the energy spectrum itself (wave kinetic equation), when one considers weakly nonlinear dispersive equations.

I will discuss a joint work with Keefer Rowan (Courant Institute, NYU) in which we show exponential mixing of passive scalars advected by a solution to the stochastic Navier–Stokes equations with finitely many (e.g. four) forced modes satisfying a hypoellipticity condition. Our proof combines the asymptotic strong Feller framework of Hairer and Mattingly with the mixing theory of Bedrossian, Blumenthal, and Punshon-Smith.

Fix a hyperbolic surface $X$, and for $L > 0$, let $\mathcal{S}_L(X)$ denote the set of simple closed geodesics of length at most $L$ on $X$. Fixing a norm on $H_1(X, \mathbb{R})$, we may ask what is the statistics of the norm of homology class of $\alpha$, denoted by $[\alpha]$, when $\alpha$ is chosen randomly uniformly from $\mathcal{S}_L(X)$, as $L \rightarrow \infty$? For example, does one expect the norm of $[\alpha]$ to be of order $L$ or smaller? We answer this question by proving a CLT-type result for the norm of homology of a randomly chosen curve in $\mathcal{S}_L(X)$. We discuss the main steps to reduce the desired CLT to a CLT for the Kontsevich-Zorich cocycle obtained by Forni-Saqban. (Joint work with F. Arana-Herrera)

Tropical abelian varieties and their moduli

Gromov's PSC conjecture for a discrete group G states that the universal cover of a closed PSC n-manifold with the fundamental group G has the macroscopic dimension less than or equal to n-2. We prove Gromov's conjecture for right-angled Artin groups (RAAGs).

In this talk, we develop a novel approach to causal modeling and inference. Rather than structural equations over a causal graph, we show how to learn stochastic differential equations (SDEs) whose stationary densities model a system's behavior under interventions. These stationary diffusion models do not require the formalism of causal graphs, let alone the common assumption of acyclicity, and often generalize to unseen interventions on their variables. Our inference method is based on a new theoretical result that expresses a stationarity condition on the diffusion's generator in a reproducing kernel Hilbert space. The resulting kernel deviation from stationarity (KDS) is an objective function of independent interest.

This talk will present a regularising diffusion in the space of square integrable probability measures over the reals. One begins by representing each probability measure via a uniquely chosen symmetric non-increasing random variable (over the circle) having that measure as its law under (unit-normalised) Lebesgue measure; this is akin to considering its quantile function representation. A diffusion on this space of functions is constructed via a discrete-in-time splitting scheme. Over one interval, one acts on the representative random variables (symmetrised quantiles) by an increment of stochastic heat driven by coloured noise, then at the end of the time interval, one transforms the output function to its symmetric non-increasing rearrangement. This operation ensures the Markovianity of the resulting evolution of the induced measures. The fine time-mesh limit of the schemes can be shown to admit a well-posed characterisation that we call the rearranged stochastic heat equation. This diffusion's regularisation effect is illustrated via its associated semigroup, which maps bounded functions to Lipschitz ones with an integrable small-time singularity for the Lipschitz constants. A simple-to-write minimisation problem of a non-convex functional of probability measure is used as a prototypical and motivational application for which the stochastic gradient descent driven by the rearranged stochastic heat is studied. Exponential convergence to a unique equilibrium is demonstrated under modest assumptions along with metastability properties exhibiting the same order as the finite dimensional setting. Time permitting, I will discuss open directions of inquiry.

An Enriques surface is the quotient of a K3 surface by a fixed point free involution. There are many intriguing questions about curve counting on the Enriques surface. After giving an overview, we will zoom in on refined curve counting on the local Enriques surface. We discuss both K-theoretic refinement (following Nekrasov-Okounkov) and motivic refinements (following Kontsevich-Soibelman). In particular, we will see a refinement of the Klemm-Marino formula. This investigation leads to some concrete predictions about the moduli space of stable sheaves on Enriques surfaces.