Weekly Bulletin

In this talk, the idea is to first motivate the appearance of half-densities along the graph of a symplectic groupoid multiplication as a semiclassical approximation to a star product. Then, we introduce the corresponding associativity axiom and present an existence and classification result for such structures. Finally, we apply this theory to understand structurally the "1-loop factor" in Kontsevich's star product for a coordinate Poisson manifold. We also discuss in detail the case of a linear Poisson structure in which "Duflo factors" appear naturally as an underlying canonical half-density. This is based on joint work with G. Ledesma.

In this talk, based on joint work with Gonzalo Contreras, I will sketch a proof of the following theorem: on any closed manifold of dimension at least two with non-zero first Betti number, a C^∞ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. I will derive this result as a consequence of the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable C^∞-close Riemannian metric.

<p>Kashiwara's conjectures are deep statements about the cohomology of semisimple perverse sheaves, including the decomposition theorem and the Hard Lefschetz theorem. I will describe a new specialization method for perverse sheaves, which allows one to extend geometrically irreducible perverse sheaves from the generic fiber to a relatively perverse sheaf over an arbitrary quasi-excellent base scheme. This specialization method implies Kashiwara's conjectures for arithmetic complexes and implies stronger semisimplicity statements than were available before. The goal of this talk is to describe the extension theorem and its applications. This is work in progress.</p>

Duhamel and the fixed point theorem are often your best allies when solving non-linear PDEs like the cubic non-linear Schrödinger equation (NLS). This PDE has its origins in quantum mechanics, and it plays a prominent role in the modeling of dispersive wave phenomena, as for instance Bose-Einstein condensation. Dispersion is affected by the nature of the underlying geometry of the space it evolves, and in this talk we will address the basic question: Can one solve the cubic NLS modeling dispersion on the Sierpinski gasket? The latter is a common prototype of a compact fractal set. To approach the problem, we will call our allies Duhamel and the fixed point theorem. With a mixture of surprise (?) and perplexity (?) we will realize they are of no help to prove existence of solutions when the initial data has regularity below the threshold for the Sobolev embedding. What a different picture than in the torus or the sphere! Afterwards, if time permits, we will connect this result to (the absence of) Strichartz estimates. In any case, excitement will fill the stage :) Results are part of joint work with Gigliola Staffilani (MIT).

This lecture requires no previous knowledge of algebraic number theory, as even the most elementary notions of that theory are computationally too complicated to be accessible by algorithms that run in polynomial time. Thus, the field described by the title is quite restricted in scope. The best results were obtained by means of a relatively recently developed technique that is closely related to the method of "blowing up" known from algebraic geometry. It is hoped that the latter field will also profit from the insights gained by the new algorithmic approach. The lecture represents joint work with Daan van Gent and Alexander Spieksma.

<p>The winding statistics of closed geodesics on the modular surface depend strongly on how one samples the geodesics: allowing deep excursions into the cusp yields heavy-tailed Cauchy laws, whereas suppressing these excursions produces Gaussian limits. I will explain this transition using continued fraction expansions and the associated digit sums, and discuss results for low-lying geodesics, where geometric length and symbolic length become comparable.</p>

In the talk I will disuss the existence of the geodesic flow and the invariance of the Liouville measure in non-smooth settings. A central role will play the so-called metric-measure boundary, an object from the geometric measure theory, which detects the boundary of a Riemannian manifold in the smooth setting and controls the average non-flatness of the space in more general situations. The talk will be based on a joint work with Vitaly Kapovitch and Anton Petrunin and an ongoing project with Daniele Semola and Stephan Stadler.

Dynkin's duality between Markov fields and Markov paths relate Gaussian free fields with Brownian paths. Symanzik's program for the Phi^4 field consists in using this relation to write the 2k-points functions of the Phi^4 fields as the expectation of the intersection local time of k Brownian bridges. In this talk, I will present this program in a different framework, that of the Abelian Higgs-Yang-Mills field in a planar domain, for which a complex scalar field interacts with a vector (gauge) field. In this framework, we will see that the role played by the intersection local time is played instead by the Amperean area of the paths, a random variable associated with the path which I will present during the talk. Due to the fact we consider two fields interacting with each other, it is necessary to take some partition functions into account. I will indeed explain how the partition function of a Gaussian field, in presence of an external magnetic field, can be written as an expectation of magnetic flux over a Brownian loop soup. The talk is freely based on two papers, one of which is joint with P. Perruchaud (ArXiv ids: 2402.00767 and 2412.16781). I will try to focus on presenting the big picture and I will stay rather informal at some stages.