Abstracts

We will study the high energy asymptotic behavior of the nodal volume associated with random linear combinations of Laplace eigenfunctions in various geometric contexts. In particular, we will show the almost sure and expected asymptotics are in some way universal, i.e. they do not depend on the base manifold, nor on the particular choice of random coefficients. The talk will be based on joined works with G. Poly and L. Gass.

Motivated by biological application, such as cell-biology, partial differential equations on curved (moving) domains have become a flourishing mathematical field. Moreover, including uncertainty into these models is natural due to the lack of precise initial data or randomness of the processes itself. One of the basic questions in these models is how to represent random field on a curved domain?

In this presentation we will first give a brief insight into different possibilities of representing isotropic Gaussian random fields defined on a flat domain and their importance. In particular, we will recall the standard Karhunen-Loeve expansions. Next, we will consider Gaussian random fields on a sphere. The main goal of the talk will be to present the construction of a multilevel expansions of isotropic Gaussian random fields on a sphere with independent Gaussian coefficients and localized basis functions (modified spherical needlets). In the last part we show numerical illustrations and an application to random elliptic

PDEs on a sphere. This is a joint work with Markus Bachmayr.

We present probabilistic and geometric extensions to Babbit's hexachordal theorem. This result coming from the mathematical theory of Music is initially a combinatorical observation that concerns the groups of six notes in Z/12Z. Joint work with Moreno Andreata and Corentin Guichaoua.

Random fields on manifolds can be used as building blocks for solutions to stochastic partial differential equations or they can be described by stochastic partial differential equations. In this talk I present recent developments in numerical approximations of random fields and solutions to stochastic partial differential equations on manifolds and connect the two. More specifically, we look at the stochastic wave equation on the sphere and approximations of Gaussian random fields on manifolds using suitable finite element methods. Throughout the talk, theory and convergence analysis are combined with numerical examples and simulations.

This talk is based on joint work with David Cohen, Erik Jansson, Mihály Kovács, and Mike Pereira

The incompressible Euler’s equations are a mathematical model of an incompressible inviscid fluid. We will discuss some aspects of a perturbation of the Euler system by a rough-in-time, divergence-free, Lie-advecting vector field. We are inspired by the problem of parametrizing unmodelled phenomena and representing sources of uncertainty in mathematical fluid dynamics. We will begin by presenting a geometric fluid dynamics inspired variational principle for the equations and the corresponding Kelvin balance law. Then we will give sufficient conditions on the data to obtain i) local well-posedness of the system in any dimension in $L^2$-Sobolev spaces and ii) a Beale-Kato-Majda (BKM) blow-up criterion in terms of the $L_t^1L^\infty_x$-norm of the vorticity. The $L^p$-norms of the vorticity are conserved in two dimensions, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation in two dimensions.

This talk is based on joint work with Dan Crisan

The theory of rough paths is now a vivid field of research at the intersection of many domains such as analysis (stochastic and classical), algebra, geometry, data science and so on. Its first objective was to construct integrals and differential equations driven by irregular signal, before expanding in many directions.

The various interpretations of this theory all rely on variants of the so-called sewing lemma. In this talk, we consider how to construct directly flows from numerical schemes using a "non-linear sewing lemma”.

and present some of the main properties that can be reached. We put them in parallel with some results in the theory of ordinary differential equations and show how they are expanded.

A second part will be devoted to the relationship between such flows and other objects already existing in the theory of rough paths.

From a joint work with A. Brault.

Slides for the talk: https://drive.google.com/file/d/1HGcDabFKWHfLTSBaKTE_yKlD9lmi734O/view?usp=sharing

I shall discuss the intrinsic geometry of a family of vector fields with constant rank, and its application in understanding sub-elliptic diffusions. I intend to follow closely the book of `On the geometry of diffusion operators and stochastic flows'. See

http://www.xuemei.org/On_the_geometry_of_diffusion_operators_and_stochastic_flows-book-1%20copy.pdf

Second lecture: . I will discuss some recent progress and problems with diffusion models.

Link to the lecture notes:

Lecture 1

https://drive.google.com/file/d/18hicncsKuuXJEev1YgFHrWXRBVKekzKC/view?usp=sharing

Lecture 2

https://drive.google.com/file/d/1fNBYaf1V36NLSV9eFRUeLx6ONPAigd65/view?usp=sharing

In this talk I will discuss some recent results that allow to approximate the zero set of a real polynomial of degree $d$ with the zero set of a polynomial of degree $O(d^{1/2} \log d)$, without changing its topology, and with "high probability". (The statement involves some special gaussian fields on the sphere.) The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes to infinity).

This is based on a combination of joint works with P. Breiding, D. N. Diatta and H. Keneshlou.

Distribution dependent SDEs (or McKean—Vlasov equations) are important from both the point of view of mathematical analysis and applications; in the case of Brownian noise they are closely related to nonlinear parabolic PDEs. In this talk I will present some recent joint work with L. Galeati & F. Harang, in which we prove a variety of well-posedness results for McKean—Vlasov equations driven by either additive continuous or fractional Brownian noise. In the former case we extend some of the recent results by Coghi, Deuschel, Friz & Maurelli to non-Lipschitz drifts, establishing separate criteria for existence and uniqueness and providing a small extension of known propagation of chaos results. However, since our results in this case also apply for zero noise they do cannot make use of any regularisation effects; in contrast, for McKean—Vlasov equations driven by fBm we extend the results of Catellier & Gubinelli for SDEs driven by fBm to the distribution dependent setting. We are able to treat McKean—Vlasov equations with singular drifts provided the dynamics are driven by an additive fBm of suitably low Hurst parameter.

Symmetric spaces are fundamental in differential geometry and harmonic analysis. Examples n-spheres and Grassmann manifolds, the space of positive definite symmetric matrices, Lie groups with a symmetric product, and elliptic and hyperbolic spaces with constant sectional curvatures.

Symmetric spaces are characterised by having an isometric symmetry in each point, giving rise to a symmetric product structure on the manifold.

We give an introduction to symmetric products and Lie triple systems, which describe their tangent spaces.

A new geometric numerical integration algorithm for differential equations evolving on symmetric spaces is discussed. The integrator is constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the exponential from the LTS to the symmetric space.

Slides for the talk: https://drive.google.com/file/d/1tu9EV9Etb6o32z4vQvvKhF5PG3VoiWBi/view?usp=sharing

The aim of the lectures is to explain Arnold’s discovery from 1966 that solutions to Euler’s equations for the motion of an incompressible fluid correspond to geodesics on the infinite-dimensional Riemannian manifold of volume preserving diffeomorphisms. In many ways, this discovery is the foundation for the field of geometric hydrodynamics, which today encompasses much more than just Euler’s equations, with deep connections to many other fields such as optimal transport, shape analysis, and information theory.

Slides for the first lecture: https://drive.google.com/file/d/17jae_svnl1h4ZgHNQjQylrLS-FZxzhvz/view?usp=sharing

Geometric statistics, the statistical analysis of manifold and Lie group valued data, can be approached from a probabilistic viewpoint where families of parametric probability distributions are fitted to data.

This likelihood-based approach gives one way to generalize Euclidean statistical procedures to the non-linear manifold context. Stochastic processes here play an important role in providing geometrically natural ways of defining probability distributions. In the talk, I will discuss such constructions and how they lead to new geometric evolution equations for the most probable paths to observed data. In particular, we will see how such paths for an anisotropically scaled Brownian motion arise as geodesics of a sub-Riemannian metric on the frame bundle of the manifold.