The aim of the mini course is to give a self-consistent introduction into the basic theory about Stochastic Differential Equations (SDE) driven by the Brownian motion. The following topics are planed:
(a) Brownian motion
(b) Stochastic integration (Itô integral) with respect to the Brownian motion
(c) Ito’s formula
(d) Existence and uniqueness of solutions to SDEs under Lipschitz conditions
(e) Feynman-Kac theory for parabolic PDEs
(f) A remark on weak solutions and SDEs under non-Lipschitz conditions
The aim of the mini course is to give a self-consistent introduction into the basic theory about Stochastic Differential Equations (SDE) driven by the Brownian motion. The following topics are planed:
(a) Brownian motion
(b) Stochastic integration (Itô integral) with respect to the Brownian motion
(c) Ito’s formula
(d) Existence and uniqueness of solutions to SDEs under Lipschitz conditions
(e) Feynman-Kac theory for parabolic PDEs
(f) A remark on weak solutions and SDEs under non-Lipschitz conditions
When discretizing partial differential equations, one can choose local (finite differences, volumes, elements) or global (spectral) methods. The most common spectral basis is built on trigonometric polynomials, i.e. Fourier series. It constrains the boundary conditions to be periodic and has been an important tool in physics, used for instance to study theoretical scalings of turbulence. While spectral methods show "exponential convergence" for smooth functions, large DNS simulations also become too expensive for e.g. when reaching very large Reynolds numbers. In practice, it is possible to solve a coarser version of the DNS by removing the largest wavenumbers in spectral space (cut-off) and modeling transfers at the smallest (sub-grid) scales instead. The definition of such a model has been an open problem for a long time and classical ones are either too diffusive or unstable. Machine learning started to be an interesting alternative few years ago and people quickly found that learning a model that performs better on a priori (instantaneous) metrics is possible. We have shown that in order to lead to stable simulations in a posteriori tests, the temporal dimension must be taken into account during the learning process. This problem has now been largely explored with periodic boundary conditions, but when it comes to spectral methods with orthogonal polynomials and fixed boundaries, new challenges appear.
In this work in collaboration with T. Iguchi, we show that the Saint-Venant equations in 2D with a partially submerged obstacle is well-posed. To do so, we show that the problem is equivalent to the usual Saint-Venant equations in an external domain, with additionnal non-standard boundary conditions because they are not local in space and time. These conditions do not fit into any category of dissipativity for which the hyperbolic theory is well posed, but we introduce a new class of well-posed hyperbolic boundary problems: that of weakly dissipative boundary conditions. We then show that our system belongs to this class and is therefore well posed.
J'estimerai la croissance asymptotique de l'espérance mathématique de l'aire des amibes des courbes planes complexes aléatoires. Cela nécessitera, étant donnée une collection de bi-disques de taille inverse à la racine carrée du degré, de minorer la probabilité que l'un de ces bi-disques soit une carte de sous-variété d'une courbe plane. Il s'agit d'un travail en collaboration avec Ali Ulaş Özgür Kişisel.
Implicative algebras, recently discovered by Miquel, are combinatorial structures unifying classical and intuitionistic realizability as well as forcing. In this talk we introduce implicative assemblies as sets valued in the separator of an underlying implicative algebra. Given a fixed implicative algebra A, implicative assemblies over A organise themselves in a category AsmA with tracked set-theoretical functions as morphisms. We show that AsmA is a quasitopos with a natural numbers object (NNO).
Mass movements and delta collapses are significant sources of tsunamis in lacustrine environments, impacting human societies enormously. Paleotsunamis studies play an essential role in understanding historical events and their consequences, along with their return periods. This study investigates a paleotsunami induced by a subaqueous mass movement during the Younger Dryas to Early Holocene transition, ca. 11,700 years ago in Lake Aiguebelette (NW Alps, France). Utilizing high‐resolution seismic and bathymetric surveys associated with sedimentological, geochemical, and magnetic analyses, we uncovered a paleotsunami triggered by a seismically induced mass transport deposit. Numerical simulations of mass movement have been conducted using a visco‐plastic Herschel‐Bulkeley rheological model and corresponding tsunami wave modeled with dispersive and nondispersive models. Our findings reveal for the first time that dispersive effects may be negligible for subaqueous landslides in a relatively small lake. This research reconstructs a previously unreported paleotsunami event and enhances our understanding of tsunami dynamics in lacustrine environments.
Among 3-dimensional sets of given Newtonian capacity, which shape minimizes the q-th moment (q>0) of electrostatic equilibrium measure? One readily shows it is the ball. But what if the set is confined to the plane? A centered disk is then the natural minimizer, yet the proof is quite different and involves a cylindrical variant of Baernstein’s star-function. The approach succeeds when 0 <q <= 2. Higher moments (q>2) remain a tantalizing open problem, as do the analogous questions for Riesz equilibrium measures.
Note: this talk does not assume any previous knowledge about capacities.
(Joint work with Carrie Clark, Univ. of Illinois Urbana–Champaign.)
In 2020, Parusinski and Rond proved that every algebraic set $V \subset \mathbb{R}^n$ is homeomorphic to a $\bar{\mathbb{Q}}^r$-algebraic set $V' \subset \mathbb{R}^n$, where $\bar{\mathbb{Q}}^r$ denotes the field of real algebraic numbers. Latter very general result motivates the following open problem: $\mathbb{Q}$-algebraicity problem: (Parusinski, 2021) Is every algebraic set $V \subset \mathbb{R}^n$ homeomorphic to some $\mathbb{Q}$-algebraic set $V' \subset \mathbb{R}^m$, with $m \ge n$? The aim of the talk is to introduce above open problem and to explain how our new approximation techniques over $\mathbb{Q}$ allowed us to provide some classes of real algebraic sets that positively answer the $\mathbb{Q}$-algebraicity problem.
Comment un résultat de catégories a inspiré une solution aux problèmes de performance de Darcs. Darcs est un système de contrôle de versions sorti en 2005, basé sur des patchs, ce qui le rendait extrêmement simple à utiliser et particulièrement rigoureux, en particulier dans sa gestion des conflits. Seul problème, il prenait parfois un temps exponentiel en la taille de l'histoire pour son opération la plus courante (appliquer des patchs). Je vous expliquerai comment nous avons résolu le problème, en utilisant des catégories et des structures de données purement fonctionnelles, pour concevoir un algo en temps logarithmique pour tous les cas (sauf un cas dégénéré en temps linéaire).
Le résultat est un système déterministe (ce qui le distingue de tous les autres outils de contrôle de versions), facile à apprendre et à utiliser, tout en passant à des échelles qu'aucun autre système de contrôle de versions distribué ne peut atteindre.
The topic of the talk is existence of weak solutions to the pressureless Navier-Stokes system with nonlocal attraction--repulsion forces. We construct the solutions on the whole three-dimensional space, assuming that the viscosity coefficients are density-dependent. For the nonlocal term it is further assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. The main point of the construction is the derivation of the analogs of the Bresch--Desjardins and Mellet--Vasseur estimates in the nonlocal setting.
Toute courbe complexe plane est munie d’une métrique riemannienne induite par la métrique ambiante de Fubini- Study du plan projectif complexe. Nous donnons des bornes inférieures probabilistes sur certaines quantités métriques et spectrales (telles que la systole ou le trou spectral) des courbes planes lorsque celles-ci sont choisies aléatoirement. Il s’agit d’un travail commun avec Damien Gayet.
I will be the speaker this time and I'll talk about Shape Optimization. I will explain what are the problems that we try to solve and give a few examples then I will present some of the main tools that we use (\gamma-convergence, Buttazzo- Dal Maso existence theorem and Lions Concentration Compactness principle) and finaly if we still have some time, I will try to introduce the work that I'm doing at the moment. All of this with maybe some proofs to give you an idea of how the field works.
Nous montrons comment la théorie de la classification locale des systèmes dynamiques analytiques discrets en une variable peut s'étendre au cadre formel des transséries et de certains germes transsériels. Ces résultats s'étendent également à certains corps de "transséries généralisées" contenus dans le corps des nombres surréels, en s'appuyant sur des considérations inspirées des travaux de Rosenlicht sur les corps de Hardy. Travail joint avec V. Mantova, D. Peran et T. Servi.
In a first part I'll will broadly cover the topic of what the Laplace- Beltrami is, its different characterization and its many uses in computer graphics.
Then I'll cover our works on the topic of building this operator (along with a wider digital calculus framework) on digital surfaces (boundary of voxels). These surfaces cannot benefit directly from the classical mesh calculus frameworks. In our recent work, we propose two new calculus frameworks dedicated to digital surfaces, which exploit a corrected normal field. First we build a corrected interpolated calculus by defining inner products with position and normal interpolation in the Grassmannian. Second we present a corrected finite element method which adapts the standard Finite Element Method with a corrected metric per element. Experiments show that these digital calculus frameworks seem to converge toward the continuous calculus, offer a valid alternative to classical mesh calculus, and induce effective tools for digital surface processing tasks.
Normal form bisimilarities are a natural form of program equivalence resting on open terms, first introduced by Sangiorgi in call-by-name. The literature contains a normal form bisimilarity for Plotkin’s call-by-value 𝜆-calculus, Lassen’s enf bisimilarity, which validates all of Moggi’s monadic laws and can be extended to validate 𝜂. It does not validate, however, other relevant principles, such as the identification of meaningless terms—validated instead by Sangiorgi’s bisimilarity—or the commutation of lets. These shortcomings are due to issues with open terms of Plotkin’s calculus. We introduce a new call-by-value normal form bisimilarity, deemed net bisimilarity, closer in spirit to Sangiorgi’s and satisfying the additional principles. We develop it on top of an existing formalism designed for dealing with open terms in call-by-value. It turns out that enf and net bisimilarities are incomparable, as net bisimilarity does not validate Moggi’s laws nor 𝜂. Moreover, there is no easy way to merge them. To better understand the situation, we provide an analysis of the rich range of possible call-by-value normal form bisimilarities, relating them to Ehrhard’s relational model.
The motion of a compressible viscous barotropic fluid is described by the Navier-Stokes system. It is a system of hyperbolic-parabolic mixed-type PDEs. In this talk, we will study the so-called density patch problem: If we are given a density that is initially discontinuous across a C^(1+\alpha) curve alpha and alpha- Hölder continuous on the two disjoint components delimited by gamma, is this structure preserved in time?
An important quantity in the mathematical analysis of this system is the so-called effective flux, which was discovered by Hoff and Smoller in 1985. More precisely, the mathematical properties of this quantity play a crucial role in the study of the propagation of oscillations in compressible fluids (Serre, 1991), in the construction of weak solutions (P-L Lions 1996) or the propagation of discontinuity surfaces (Hoff 2002), to cite just a few examples. In the case of density-dependent viscosities, the behavior of the effective flux degenerates, which renders the analysis more subtle.
We give a characterization of star-free languages (a well-known subclass of regular languages) in a λ-calculus with support for non-commutative affine types (in the sense of linear logic), via the algebraic characterization of the former using aperiodic monoids. This was the first result in a research program that Cécilia Pradic (Swansea University) and I have carried out since my PhD, on which I will say a few words.