In this talk, I will speak about character varieties, roughly speaking this the set of conjugacy classes of representations from a finitely generated group to a reductive complex algebraic group G. We will see some examples. We will mainly focus on character varieties into SL(2,C) and PSL(2,C). After this, we will talk about character varieties into more general complex groups, focusing on local properties : tangent spaces and singularities.
The discrete Pompeiu problem stemmed back to an integral-geometric question on the plane. The problem is whether we can reconstruct a function if we know the average values of the function on every congruent copy of a given pattern. Introducing a theorem of spectral analysis on discrete Abelian groups, I show some results of the discrete version of the topic. As an application we get a result for a coloring problem of the plane. I also mention some unsolved questions of this type.
Shimura curves arise as a natural generalisation of elliptic curves. As modular curves, they are constructed as Riemann surfaces, and they turn out to have structure of algebraic curve, i.e. they can be described by some algebraic equations with coefficients in some finite extension of Q. Number theorists are interested in the reductions modulo p of these equations. The problem is that these equations are very difficult to compute. I will describe a method to find these reductions without actually knowing the equation.
Deformation quantization is one of the possible mathematical formalization of the process that associatesto the description of a physical system in the language of classical mechanics a description of this system in the framework of quantum mechanics. In 2001, in his study of deformation quantization of Poisson complex manifolds, M. Kontsevich introduced a new type of object called Deformation Quantization (DQ) modules. These objects are sheaves of modules over a non-commutative deformation of the sheaf of holomorphic functions.
In this talk, we will provide an overview of the theory of DQ-modules and of some of their applications, explains how they are related to complex symplectic geometry and to the theory of microdifferential modules (the analytic counterparts of pseudodifferential operators).
Meaningful geometric informations on spaces with rough structure (such as graphs or degenerate limits of Riemannian manifolds) can be derived from the behaviour of a well-chosen random process on the space. This point of view naturally leads to a study of the so-called Markov generators “à la Bakry-Emery”. In a Riemannian setting controls on the Ricci curvature can be obtained.
After recalling the basics of the theory of curvature, a fundamental equivalence between a dimension-free functional inequality (log-Sobolev inequality), a coarse control of the geometry of the space (infinite-dimensional Böchner inequality) and a dynamical property of the heat operator (gradient-semigroup commutation property) will be derived on Riemannian manifolds and used to generalize the notion of “Ricci curvature bounded below”.
If time permits, the limit of the theory will be discussed with a particular focus on the Heisenberg group.
Classical calculus extends to a geometric setting by the theory of differentiable manifolds. Probabilistic extensions are given using tools from stochastic analysis. Stochastic differential geometry brings these two ideas together. I will introduce this topic by constructing a Brownian motion on a Riemannian manifold. I will then discuss probabilistic formulae for the heat kernel and certain integrals of it (the heat kernel is given by the transition densities of Brownian motion). These formulae relate to some examples of stochastic processes which are constrained to hit a fixed submanifold at fixed positive time, called submanifold bridge processes. The motivation for studying such processes is to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold.
My talk will consist in introducing my research field from an heuristic point of view in the world of science in general. If time permits, I will point out its interactions with differential geometry, harmonic analysis, partial differential equations and special functions theory. Link to the extend abstract.
The first part of the talk will center around the relationship between geometry, second order operators and stochastic processes on manifolds. We will discuss the relations of these things in general, then focus on the special case of Riemannian manifolds that are foliated, that is “cut into layers”. For such manifolds, we consider a geometry focused on the bundle orthogonal to the foliation, known as sub-Riemannian geometry. This geometry is connected to second order operators on manifolds satisfying the so-called Hörmander condition and their corresponding stochastic processes. The second part will focus on a specific geometric question that appears in this theory, related to totally geodesic foliations. To deal with this question, we introduce horizontal holonomy groups, which are groups related to parallel transport along loops tangent to a specific subbundle of the tangent bundle.
Deformation quantization is one formalization of the general idea of quantization of a classical mechanical system/classical field theory to a quantum mechanical system/quantum field theory. It consists in replacing the commutative algebra of observables of a classical system by a non-commutative algebra of quantum observables. We will present a historical overview of the problem, and how it was solved by Kontsevich for Poisson manifolds. Finally, we consider the case of universal enveloping algebras and hypersurfaces.
We review the concept of (weak) convergence of probability measures, introduce several distances which either metrize weak convergence or whose corresponding convergence concept implies weak convergence. These distances give rise to the concept of distributional approximation of a given (complicated) distribution by a better known or well-studied one. Also, we present techniques of effectively assessing these distances, hereby focusing on Fourier methods as well as on Stein’s method. Finally, we wish to illustrate how Stein’s method may be applied to distributional approximation problems in concrete stochastic models.
Nodal patterns have received a great attention. Indeed, they appear in many problems in engineering, physics and the natural sciences: they describe the sets that remain stationary during vibrations, hence their importance in such diverse areas as musical instruments industry, mechanical structures, earthquake study and other areas. They also arise in the study of wave propagation, and in astrophysics; this is a very active and rapidly developing research area. In this talk, we investigate the geometry of nodal sets of random eigenfunctions. In particular, we consider the case of ``arithmetic random waves” (the Gaussian Laplace eigenfunctions on the two-dimensional torus) and we study their nodal lines. We find that the total length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument is mainly based on the Wiener-Itô chaos decomposition for the nodal length and a deep investigation of its components. This talk is based on joint works with Domenico Marinucci (Università di Roma Tor Vergata), Giovanni Peccati (Université du Luxembourg) and Igor Wigman (King’s College London).
Finding a (periodic) tiling of the Euclidean or hyperbolic space is essentially equivalent to finding a discrete group of isometries acting cocompactly on this space. It is known since the work of Poincaré that there are many ways to tile the hyperbolic plane. In the seventies, Thurston showed that the hyperbolic space in dimension 3 also admits lots of tilings. After giving a historical overview of the problem, I will discuss its pseudo-Riemannian analog.