Events

In 1922, Birkhoff proved that an essential embedded closed curve that is invariant under a symplectic twist map of the cylinder is a Lipschitz graph over the zero section. There has been several works extending this theorem in higher dimension (Herman, Katznelson–Ornstein, Siburg, Bialy–Polterovich...). In 2010, Arnaud proved that a closed exact Lagrangian in the cotangent bundle of a closed manifold that is invariant under the flow of a Tonelli Hamiltonian is a graph over the base. Some improvements on this theorem have been made on one hand by Bernard-dos Santos and Amorim-Oh-dos Santos, who showed a version of this theorem for Lipschitz Lagrangian submanifolds, and on the other hand by Charfi who replaced the invariance assumption with a recurrence assumption (for a topology he defined and called "reduced complexity convergence"). In this talk, I will explain how to define a new class of "$C^0$ Lagrangians" for which the multidimensional Birkhoff theorem holds, using a new variant of the reduced complexity convergence topology. In particular, this class contains all images of continuous graphs by $C^1$ exact symplectomorphisms, which can be less regular than Lipschitz. This is based on a work in progress with Skander Charfi.

Hill's spherical vortex is a classical explicit solution to the three-dimensional Euler equations. It is an axisymmetric and swirl-free traveling wave with compactly supported yet discontinuous vorticity. In this talk we will explore the space of nearby traveling-wave Euler solutions and, in particular, prove that Hill's vortex is the limit of traveling vortex rings that are smooth to infinite order. Along the way, we shall unveil certain pathological aspects of this space of steady solutions that emerge once swirl is admitted. This is joint work with Răzvan Radu.

Constant Mean Curvature (CMC) c-immersions of a closed orientable surface S (with genus g ≥ 2) into hyperbolic 3-manifolds emerged by the work of K. Uhlenbeck in connection with irreducible representations of the fundamental group of S into the Mobious group. For |c| < 1, such (CMC) c-immersions are always available and labelled by elements of the tangent bundle of the Teichmueller space of S. On the other hand, the (critical) value |c|=1 of the mean curvature plays a crucial role, since (CMC) 1-immersions into the hyperbolic space H^3 (known as Bryant surfaces) furnish the hyperbolic analogue (via bi-holomorphisms) of the classical minimal immersions into the Euclidean 3-space E^3. Naturally, one could try to attain (CMC) 1-immersions as limits for |c| → 1. But possible “blow-up” phenomena could yield (after scaling) to (CMC) 1-immersions with conical singularities, consistently with the presence of smooth ends in Bryant surfaces. I shall discuss how to encompass the blow-up situation in terms of suitable “orthogonality conditions” involving the image Z of the Kodaira map (for genus g = 2) and more generally the (g −1)-secant variety of Z, for larger genus. Subsequently, using Hitchin self-duality theory, I shall indicate how the given “orthogonality conditions” are actually sharp towards “compactness”. In this way, under a generic condition, we can ensure existence and uniqueness of (CMC) 1-immersions of S into (germs) of hyperbolic 3-manifolds, and obtain a parametrization of the corresponding moduli space in terms of the tangent bundle of the Teichmueller space of S.

<p>The wind-tree model is a billiard model of gas diffusion introduced by Ehrenfest and Ehrenfest in the early 20th century. A small particle (the "wind") bounces between periodically placed rectangles (the "trees") on the plane. A lot of progress has been made in the past 15 years by studying the wind-tree model from the point of view of translation surfaces. <br>In particular, Fraczek and Ulcigrai proved in 2014 that in typical directions the billiard flow has no dense trajectories. This opens the natural question of trying to understand what the closures of the trajectories look like. The new result that I will present in this talk is that in a special self-similar case, all closures of trajectories have non-maximal Hausdorff dimension.<br>This is achieved by constructing explicit invariant functions for the flow, and studying their levels sets using a symbolic coding for interval exchange transformations.</p>

In this talk we describe recent progress on the Zimmer Programme, the study of actions of lattices in Lie groups on manifolds by diffeomorphisms. We begin by recalling motivations and some important examples, explain the general strategy introduced by Brown-Fisher-Hurtado when the manifold is of low enough dimension, and discuss modifications that allow us to prove results in more general semisimple and S-arithmetic groups. Time willing, we will also describe some recent progress on actions of dense subgroups of Lie groups on manifolds.

This talk discusses the theory and practice of operator learning, motivated by inverse problems and PDE solution maps. We start with function graph transformers which represent functions by measures supported on their graphs and operators by graph-preserving pushforwards. This viewpoint connects discrete data with continuum operators and yields constructive approximation results. Turning from approximation to sample complexity, we show how compressibility in phase space makes a class of double fibration transforms, including Radon and geodesic ray transforms, learnable from few input–output examples, even when their geometry is unknown. Finally, we use the method of characteristics to motivate Flowers, neural PDE surrogates built from state-dependent pullbacks. Despite their simplicity, Flowers outperform Fourier-, convolution-, and attention-based models of comparable size across a broad range of two- and three-dimensional PDE benchmarks.

I will discuss probabilistic and analytic aspects of two-dimensional Coulomb gases confined to Jordan domains, curves, and arcs. In particular, I will describe the asymptotic behavior of particle distributions as the number of particles tends to infinity. I will also explain how the Loewner energy and related quantities arise in the asymptotic expansion of the free energy. A key analytical tool throughout is the Grunsky operator. The talk is based on joint work with Kurt Johansson and Klara Courteaut.

Stix and Vdovina used quaternion algebras over function fields to construct explicit families of torsion-free arithmetic lattices acting simply transitively on products of Bruhat-Tits trees of equal regularity. In particular, they obtained a lattice of Euler characteristic one over 𝔽₃(t) acting on a product of two trees. In this talk, I will discuss analogous constructions over number fields. I will describe classification results for lattices of small Euler characteristic and present new infinite families of torsion-free lattices acting simply transitively on products of trees of equal regularity. This talk contains joint work with Jakob Stix, Alina Vdovina, and Jiahui Yu.

<p><span style="caret-color: #000000; color: #000000; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">We prove that, for any given sufficiently regular one-particle density, there exists a unique external potential for which the solution to the many-body Schrödinger equation has this prescribed density. This implies in particular that one can exactly reproduce the time-dependent density of an interacting system using non-interacting electrons, which is at the heart of time-dependent density functional theory. Our result covers the Coulomb interaction and densities arising from extended nuclei. Joint work with Asbjørn Bækgaard Lauritsen and Mathieu Lewin.</span></p>

I will discuss the problem of expressing the cobordism class of Mbar_{0,n} as a polynomial in generators of the cobordism ring. With rational coefficients, a method was given by Coates and Givental. With integer coefficients, we will get a recursion by lifting the string equation, introduced by Witten, to algebraic cobordism in genus 0. This recovers known closed formulas for invariants of Mbar_{0,n} such as the Poincaré polynomials, and gives new closed formulas for invariants such as certain twists of the chi_y-genus. I will also explain how equivariant localization in cobordism can alternatively be used, and the outlook in higher genus.