Weekly Bulletin

In this talk we focus our attention on compactly supported Hamiltonian diffeomorphisms of the plane. To each pair of distinct fixed points we can associate an integer, the linking number of the pair. Moreover, we can describe the dynamics of the diffeomorphism itself via the Morse complex of a generating function. This Morse complex is classically known to be filtered by action. In this talk we are going to show how we can equip its tensor power with a secondary filtration which keeps track of the linking numbers of pairs of orbits. The proof relies on foundational work on twist maps carried out by Patrice Le Calvez in the '90s, and on uniqueness properties of generating functions. Moreover, we are going to explain how using this language one may provide a finite-dimensional proof of Hofer-lower semicontinuity of the  topological entropy, first proved by Alves and Meiwes. One can define a similar filtration on Floer complexes using intersection products of pseudo-holomorphic cylinders as defined by Siefring: if time permits, we are going to provide a sketch of this construction.

This talk is about joint work with Yann Guggisberg. We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $R^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $R^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption.

We consider the inhomgeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is one of the most classical instances of the Rayleigh-Taylor instability. In the talk we will see how weak locally dissipative solutions to the Euler equations reflecting a turbulent mixing of the two fluids in a quadratically growing zone can be constructed. If time allows, we will discuss an arising selection problem for the averaged motion of solutions. The core of the talk is based on a joint work with József Kolumbán.

In the divergence case of Khintchine's theorem, Schmidt established an asymptotic formula for the number of rational approximations of bounded height to almost every real number. Using tools from homogeneous dynamics and the geometry of numbers, we prove a version of this theorem for intrinsic Diophantine approximation on projective quadrics, Grassmannians, and other examples of flag varieties.

We outline a strategy for showing that the Chow ring of A_4 is tautological. The geometric setup involves a parametrization of the complement of the Jacobian locus by a moduli space of 6-nodal quartics in P^3. An ongoing collaboration with Sam Canning and Aitor Iribar-López.

A one-weight code, or constant-weight code, is an error-correcting code in which all codewords share the same weight. In 1984, Bonisoli provided a classification of one-weight linear codes by leveraging the connection between codes equipped with the Hamming metric and projective systems. More recently, similar geometric techniques have been applied to the study of linear codes in the rank metric. In this talk, we discuss one-weight linear rank-metric codes, by exploiting a new geometric framework. This is a joint work with Martino Borello and Alessandro Neri.

In his seminal 1988 Bulletin article, Thurston showed that the stretch factor of a pseudo-Anosov map of a closed orientable surface is an algebraic integer of degree bounded from above by the dimension of the Teichmüller space of the surface. Thurston further claimed, without proof, that a construction of examples presented later in the article shows that this bound is sharp—a construction which nowadays is known as <i>Thurston’s construction</i> or <i>Thurston—Veech construction</i>. Margalit remarked in 2011 what Strenner wrote down in his 2017 article on algebraic degrees of stretch factors, namely that no proof of Thurston's claim has ever been published. In this talk, we present almost explicit examples of pseudo-Anosov maps obtained via the Thurston—Veech construction with stretch factors of maximal algebraic degree, finally substantiating Thurston's claim. This is joint work with Erwan Lanneau.

Ferrers diagram rank-metric codes were first studied in 2009 by Etzion and Silberstein, motivated by their application in network coding. Concretely, they arise from subspace codes entirely contained in a unique Schubert cell. In their work, the authors proposed a conjecture on the largest dimension of a linear space of matrices over a finite field whose nonzero elements are supported on a given Ferrers diagram and have all rank lower bounded by a fixed positive integer r. Since then, their conjecture has been proved only in some few cases, and as of today it still remains widely open. In this talk, I will give an overview of the main combinatorial and algebraic properties of Ferrers diagram rank-metric codes and on the state of art on the Etzion-Silberstein conjecture, starting from the first findings, until some very recent results.

This project is an ongoing work with Armand Riera. The Brownian half-plane is a random surface that arises as the universal scaling limit in the Gromov-Hausdorff sense of large graphs embedded in the closed half-plane. We will present two exploration processes of this surface; the horohulls and the hulls, which correspond roughly to exploring it from a point at infinity, and from its root. We characterize the law of the horohulls, which in turn provides information on the local topology of the half-plane. We describe the joint law of the horohulls and the hulls, and support the idea that they are dual processes. Then, we exhibit a simple martingale which is a function of the hulls, which allows to define another surface whose law is absolutely continuous to whose of the Brownian half-plane. We show that this surface exhibits hyperbolic features, with some links to 3D non random geometry. We end with a few conjectures/open questions.

The weight decomposition and Fourier transform are among the most powerful tools in the study of intersection theory on an abelian scheme. In proper families abelian schemes tend to degenerate to certain singular objects -- semiabelic schemes or compactified abelian schemes for simplicity -- on which those tools are essentially lost. In this talk, I will discuss how to partially restore them in the case of certain families of compactified abelian schemes of central interest in the study of curves,  the compactified Jacobians, and how this leads to the determination of a large part of their tautological intersection theory. I will end by discussing some conjectures about weight decompositions on compactified Jacobians. This is joint work with Bae-Pixton.