Traditionally, theories of “Sobolev” spaces on metric spaces have used local Lipschitz constants as a substitute for the gradient of functions. However, a recent study by Kajino and Murugan revealed that such an idea does not work for a class of self-similar sets including the planar Sierpinski carpet. The notion of conductive homogeneity was proposed to construct a counterpart of Sobolev spaces and Sobolev p-energy even for such cases. In this talk, I will review the method of construction of Sobolev spaces under the conductive homogeneity and give a class of regular polygon-based self-similar sets having the conductive homogeneity. Our condition is the local symmetry of the space with some (or no) global symmetry. In particular, we show that any locally symmetric triangle-based self-similar sets possess the conductive homogeneity. This is joint work with Y. Ota.[-]

It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional limit theorem for the sine-process, which turns out to be very different from the Donsker Invariance Principle. We show that the anti-derivative of our process can be approximated by the sum of a linear Gaussian process and small independent Gaussian fluctuations whose covariance matrix we compute explicitly.[-]

We consider the monogenic representation for self-similar random fields. This approach is based on the monogenic representation of a greyscale image, using Riesz transform, and is particularly well-adapted to detect directionality of self-similar Gaussian fields. In particular, we focus on distributions of monogenic parameters defined as amplitude, orientation and phase of the spherical coordinates of the wavelet monogenic representation. This allows us to define estimators for some anisotropic fractional fields. We then consider the elliptical monogenic model to define vector-valued random fields according to natural colors, using the RGB color model. Joint work with Philippe Carre (XLIM, Poitiers), Céline Lacaux (LMA, Avignon) and Claire Launay (IDP, Tours).[-]

It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. We start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.
This is joint work with Mengchu Li and Yi Yu.[-]

We will consider the discretization of the stochastic differential equation$$X_t=X_0+W_t+\int_0^t b\left(s, X_s\right) d s, t \in[0, T]$$where the drift coefficient $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is measurable and satisfies the integrability condition : $\|b\|_{L^q\left([0, T], L^\rho\left(\mathbb{R}^d\right)\right)}<\infty$ for some $\rho, q \in(0,+\infty]$ such that$$\rho \geq 2 \text { and } \frac{d}{\rho}+\frac{2}{q}<1 .$$Krylov and Röckner [3] established strong existence and uniqueness under this condition.Let $n \in \mathbb{N}^*, h=\frac{T}{n}$ and $t_k=k h$ for $k \in \left [ \left [0,n \right ] \right ]$. Since there is no smoothing effect in the time variable, we introduce a sequence $\left(U_k\right)_{k \in \left [ \left [0,n-1 \right ] \right ]}$ independent from $\left(X_0,\left(W_t\right)_{t \geq 0}\right)$ of independent random variables which are respectively distributed according to the uniform law on $[k h,(k+1) h]$. The resulting scheme Euler is initialized by $X_0^h=X_0$ and evolves inductively on the regular time-grid $\left(t_k=k h\right)_{k \in \left [ \left [0,n \right ] \right ]}$ by:$$X_{t_{k+1}}^h=X_{t_k}^h+W_{t_{k+1}}-W_{t_k}+b_h\left(U_k, X_{t_k}^h\right) h$$where $b_h$ is some truncation of the drift function $b$. When $b$ is bounded, one of course chooses $b_h=b$. Then the order of weak convergence in total variation distance is $1 / 2$, as proved in [1]. It improves to 1 up to some logarithmic correction under some additional uniform in time bound on the spatial divergence of the drift coefficient. In the general case (1), we will see that for suitable truncations $b_h$, the difference between the transition densities of the stochastic differential equation and its Euler scheme is bounded from above by $C h^{\frac{1}{2}\left(1-\left(\frac{d}{\rho}+\frac{2}{q}\right)\right)}$ multiplied by some centered Gaussian density, as proved in [2].[-]

Microparasites (virus, bactéries, protozoaires ...) et macroparasites (métazoaires : helminthes, arthropodes...) sont omniprésents dans les écosystèmes terrestres et marins. Le nombre total d'espèces parasites sur la planète est supérieur à celui des espèces libres qu'ils colonisent, temporairement ou non, au point que ces organismes interfèrent à toutes les échelles d'organisation du vivant. Les pathologies qu'ils peuvent parfois engendrer sont dépendantes de conditions particulières, soit liées à leur propre virulence, soit à un ensemble de facteurs environnementaux. Dans ce contexte, les modèles mathématiques constituent des outils précieux en épidémiologie, permettant de mieux comprendre les modalités de leur propagation dans les populations d'hôtes. Aborder les stratégies démographiques des micro ou des macroparasites implique des approches mathématiques différentes. Le développement de ces modèles ouvre des perspectives intéressantes pour décrire, analyser et même prévoir les comportements démographiques de ces systèmes couplés. En milieu marin, les macroparasites peuvent aussi poser des problèmes de santé à leurs hôtes quand les équilibres de différentes natures sont déplacés, avec ou sans l'intervention de l'homme (espace protégé, pêche, aquaculture...). En prenant l'exemple de parasites de Poissons téléostéens, l'accent sera mis sur la complexité des processus biologiques en cause, et son intégration dans des modèles mathématiques.

[-]

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.
braid groups - Garside groups - Nielsen-Thurston classification - pseudo-Anosov - conjugacy problem[-]

The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs measure using appropriate stochastic dynamics of Langevin type. Rigorous ergodicity results as well as elements on the estimation of numerical errors are provided. The last part is devoted to the computation of transport coefficients such as the mobility or autodiffusion in fluids, relying either on integrated equilibrium correlations à la Green-Kubo, or on the linear response of nonequilibrium dynamics in their steady-states.[-]

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint with with Houman Owhadi and Clint Scovel (Caltech).[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]