Weekly Bulletin

Reduction of Poisson manifolds by coisotropic submanifolds formalizes symmetry reduction of classical mechanical systems, and therefore plays an important role in Poisson geometry. Constraint algebras encode the additional structure on the algebra of functions needed for reduction and their deformations yield (formal) star products compatible with reduction. I will discuss the deformation theory of these constraint algebras and present some results about an adapted Hochschild-Kostant-Rosenberg Theorem computing the cohomology controlling this deformation theory.

In this talk, based on joint work with Marco Mazzucchelli, I will present some new results on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result is a forced existence theorem for orientable closed Riemannian surfaces of positive genus, asserting that the existence of a contractible simple closed geodesic γ forces the existence of infinitely many closed geodesics in every primitive free homotopy class of loops and intersecting γ. I will then explain how this type of result can be used to show the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface.

Gröbner bases are one of the main tools in computer algebra to solve many theoretical problems. For instance, they provide algorithms to solve the ideal membership problem, to perform elimination of variables and to solve zero-dimensional polynomial systems. In this talk, we will recall some basics notions about Gröbner bases and introduce the analogous but less known Khovanskii (or Sagbi) bases. These are particularly well-behaved sets of algebra generators that allow similar algorithms on subalgebras of the polynomial ring. Unlike Gröbner bases, Khovanskii bases are not always finite. We will discuss examples and introduce applications of Khovanskii bases to solve structured equations on projective varieties (based on joint work with M.Panizzut and S. Telen).

We propose a general framework of transverse dynamical system which allows us to simultaneously study point processes related to lattices in locally compact groups; aperiodic tilings in locally compact groups; strata of translation surfaces. In this general context we introduce a notion of intersection space and construct an associated Siegel-Radon transform, which generalizes the classical Siegel and Siegel-Veech transforms. We discuss conditions which guarantee (square-)integrability of the image and applications of the transform to the spectral theory of tiling spaces and various counting problems. This talk is based on joint work with Michael Björklund (Chalmers).

The conformal dimension of a metric space $(X,d)$ measures its optimal shape from the perspective of quasiconformal geometry. It is defined by infimizing dimension over metrics in the quasisymmetric equivalence class of $d$. Introduced by Pierre Pansu in 1989, conformal Hausdorff dimension played an important early role in the development of analysis on metric spaces. Subsequently a variant notion, conformal Assouad dimension, gained prominence. Assouad dimension—which bounds Hausdorff dimension from above—is a scale-invariant, quantitative measurement of optimal coverings of a space. Dimension interpolation is an emerging program of research in fractal geometry which identifies geometrically natural one-parameter dimension functions interpolating between existing concepts. Two exemplars are the Assouad spectrum (Fraser-Yu, 2015), which interpolates between box-counting and Assouad dimension, and the intermediate dimensions (Falconer-Fraser-Kempton, 2020), which interpolate between Hausdorff and box-counting dimension. In this talk, I’ll discuss the mapping-theoretic properties of intermediate dimensions and the Assouad spectrum, with applications to the quasiconformal classification of sets and to the range of conformal Assouad spectrum. The latter results are based on a recent joint project with Efstathios Chrontsios Garitsis (Univ of Tennessee) and an ongoing collaboration with Jonathan Fraser (Univ of St. Andrews).

Let G be a finite subgroup of SL_2. The McKay correspondence relate the representation theory of G to the geometry of the quotient X of C^2 by the action of G. An avatar of this correspondence has been studied by Batyrev and later Denef-Loeser. They show that the p-adoc or motivic volume of X can be explicitly computed as a sum over conjugacy classes of G. Yasuda reinterprets this sum in terms of the inertia of the Deligne-Mumford stack associated to this quotient. I will present a generalization of this formula to the case of a quotient of a smooth variety by a general linear group. This is joint work with François Loeser and Dimitri Wyss.

In 2009, in connection with the smooth 4-dimensional Poincaré conjecture, D. Calegari constructed an infinite family of smooth homotopy 4-spheres arising from monodromies of fibered knots in the 3-sphere. We review this construction and prove that all of these are diffeomorphic to the standard 4-sphere. We also discuss some related applications.

The Feynman-Kac formula bridges PDEs and probability theory by providing a probabilistic representation of PDE solutions, connecting the convergence of numerical schemes for PDEs to limit theorems in stochastic processes. In this talk, we extend this paradigm beyond linear frameworks to sublinear expectations—known as Peng’s G-expectations. We establish convergence rates for the Universal Robust Limit Theorem, including central limit theorems, laws of large numbers, and alpha-stable limit theorems, all under the sublinear expectation framework. Our analysis leverages the monotone scheme analysis of viscosity solutions to fully nonlinear second-order PIDEs associated with nonlinear Lévy processes. Based on joint work with Mingshang Hu and Lianzi Jiang.