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 product torus in S^2 x S^2 is a Lagrangian torus obtained as the product of circles in the factors. The goal of this talk is to give a classification up to symplectomorphisms of such tori and illustrate that interesting things happen in case the symplectic form is non-monotone. Among other applications, we will answer a question about Lagrangian packings posed by Polterovich--Shelukhin. This is partially based on joint work with Joontae Kim.

Many (partial) CohFTs in the moduli spaces of curves can be canonically lifted to the moduli space of spin structures. This leads to a refined study of these cohomology classes according to the sign (or Arf invariant) of the spin structure. We will review open problems and recent advances on the refined study of several partial CohFTs: Witten’s class, Double Ramification cycles, and strata of differentials.

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.

Can two minimal surfaces touch each other to infinite order at a point without coinciding in a neighborhood of the point? Intuition from the theory of unique continuation for elliptic PDEs suggests this should not happen. Of course, part of the game here is to specify the notion of minimal surface. In joint work with Camillo Brena we give an answer to an instance of the question above: if an m-dimensional area minimizing integral current has infinite order of contact at a point with an m-dimensional surface with zero mean curvature then the current coincides with the surface in a neighborhood of the point.

In part, the goal of homotopy theory is to compute algebraic invariants of topological objects. Of particular interest is the computation of homotopy groups of spheres. Any given result can typically be obtained in several completely distinct ways, using methods that have no visible consistency with each other. It is remarkable that there exists a solution to this highly overdetermined problem. Like many areas of pure mathematics, machine computation can be applied to great effect in this endeavor. I will discuss the history, current state of the art, and future prospects of machine computation in homotopy theory.

Following Birkhoff's proof of the Pointwise Ergodic Theorem, it is natural to consider whether convergence still holds along various subsequences of the integers. In 2020, Bergelson and Richter showed that in uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$, with multiplicities. In this talk, we will see that by removing this assumption, a pointwise ergodic theorem does not hold along $\Omega(n)$. In fact, $\Omega(n)$ satisfies a notion of non-convergence called the strong sweeping out property. We then further classify the strength of this non-convergence behavior by considering weaker notions of averaging. Time permitting, we will introduce a more general criterion for identifying slow growing sequences with the strong sweeping out property (based on joint work with S. Mondal).

Degenerating abelian varieties

I will introduce the notion of a flat extension of a connection on a principal bundle. Roughly speaking, a connection admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented 3-manifolds, I will relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated to the connection. Joint work with Andreas Cap & Keegan Flood.

In this talk we will use the periodic cubic nonlinear Schrödinger equation to present some estimates of the long time dynamics of the energy spectrum, a fundamental object in the study of wave turbulence theory. Going back to Bourgain, one possible way to conduct the analysis is to look at the growth of high Sobolev norms. It turns out that this growth is sensitive to the nature of the space periodicity of the system. I will present a combination of old and very recent results in this direction.

We begin by introducing the notion of Brownian motion indexed by the Brownian tree. We will then present the main aspects of a theory, developed in two recent works with Armand Riera, that describes the evolution of this tree-indexed process between visits to 0. The theory applies to fairly general continuous Markov processes indexed by Lévy trees. Despite the radically different setting, we will see that our results share strong similarities with the celebrated Itô excursion theory for linear Brownian motion. If time permits, we will also discuss some applications to Brownian geometry.

We show that any two same-genus, oriented, boundary parallel surfaces bounded by a non-split alternating link into the 4-ball are smoothly isotopic fixing boundary. In other words, a non-split alternating link bounds a unique Seifert surface up to isotopy in the 4-ball (and up to genus). This is joint with Seungwon Kim and Jaehoon Yoo.

I will discuss some recent joint work on formalising the regular case of Fermat’s Last Theorem in Lean. The regular case of FLT, where the exponent is a prime p that does not divide the class number of the p-th cyclotomic field, has long been known to be much simpler than the full proof, making it a good target for formalisation. In my talk, I will explain what Lean is, why one would want to formalise mathematics, and some of the challenges encountered in this process. No prior knowledge of Lean or formalisation is required.

Quantum spectrum and Gamma structures play key roles in Katzarkov-Kontsevich-Pantev-Yu’s proposal to extract birational invariants from quantum cohomology and Iritani's decomposition theorem for quantum D-modules of blowups. In this talk, we investigate these structures for standard flips in a much simpler setup, by restricting the quantum multiplication to a fiber curve direction. In this setup, we can show that both quantum spectrum and asymptotic behavior can be reduced from the local models, where the small I/J-functions are known explicitly. Using the asymptotic behavior of cohomology-valued Meijer G-functions, we obtain a decomposition of the cohomology of standard flips into asymptotic Gamma classes. This decomposition is compatible with the semi-orthogonal decomposition for standard flips constructed by Bondal-Orlov and Belmans-Fu-Raedschelders. The talk is based on work in progress joint with Mark Shoemaker.