Tous les événements à venir

Dilation surfaces are surfaces modeled after the complex plane whose structure group is generated by the group of translations and dilations. Given a dilation surface, for any direction in $S^1$ there exists a corresponding directional foliation on the surface. In this talk, we will study the four possible types of dynamical behaviour that such a foliation may have (i.e completely periodic, Morse-Smale, minimal or Cantor-like) and deduce a dynamical decomposition theorem for the directional foliation on dilation surfaces using results of C.J. Gardiner and G. Levitt from the 1980s.

In a second step, we study the first return map of the directional foliation on a dilation surface, which is a so-called affine interval exchange transformation (AIET). We introduce a powerful tool called Rauzy-Veech induction in order to develop a renormalization scheme which allows to find a decomposition of any given AIET into finite union of intervals which exhibit only one of the four types of dynamical behaviour. This provides an alternative, purely combinatorial approach to the decomposition results of Levitt and Gardiner and is joint work with Corinna Ulcigrai and Charles Fougeron.

Cet exposé concerne un travail en collaboration avec Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang et Xun Yu. Soit X une variété algébrique complexe. Les formes réelles de X sont les variétés réelles W dont la “complexification”, en tant que variété complexe, est isomorphe à X. Bien entendu, certaines variétés complexes n’ont pas de forme réelle. Un fait plus surprenant, mis en évidence par Lesieutre en 2016, est l’existence d’une variété complexe admettant une infinité de formes réelles. Dans cet exposé, on présente une surface de rang de Picard relativement petit possédant une infinité de formes réelles. L’exemple en question est obtenu en adaptant une construction de Dinh-Oguiso-Yu à base de surfaces K3 via une technique due à Mukai. En fin de compte, on fabrique une surface d’Enriques dont l’éclatement en un point très général d'une courbe bien choisie possède une infinité de formes réelles. Si le temps le permet, on expliquera aussi pourquoi le groupe d’automorphismes de cet éclatement n’est pas de type fini.

We take a section P of infinite order on an elliptic surface and consider points where some multiple nP is tangent to the zero section (These are "unlikely intersections" and our consideration of them is motivated by a question in geography of surfaces. It is also analogous to the question of whether elements of an elliptic divisibility sequence are square-free.) In characteristic zero, we show finiteness and give a sharp upper bound, relying heavily on a canonical parallel transport in a family of elliptic curves (the "Betti foliation") and a certain real-analytic one-form. Although the finiteness statement looks completely reasonable in characteristic p, it's not clear what would replace the (non-algebraic) 1-form. Time permitting, I will explain how ongoing work with Felipe Voloch connects tangencies to the p-descent map and allows us to bound them in characteristic p as well.

(w/ G. Urzua and F. Voloch)

Three ball inequalities are a useful tool in the study of unique continuation properties in the continuum. Our goal is to extend these inequalities to certain discrete lattices, known as periodic graphs. Periodic graphs are graphs in $R^d$ that remain invariant under translations of particular vectors. We prove that such inequalities holds for Schrödinger operators on a family of periodic graph, and for Laplace operators on a wider family.

Les inégalités à trois boules sont un outil utile dans l’étude des propriétés de continuation unique dans le continuum. Notre but est d’étendre ces inégalités à certains treillis discrets, connus sous le nom de graphes périodiques. Les graphes périodiques sont des graphes dans $R^d$ qui restent invariants sous les translations de vecteurs particuliers. Nous prouvons que ces inégalités sont valables pour les opérateurs de Schrödinger sur une famille de graphes périodiques, et pour les opérateurs de Laplace sur une famille plus large.

In this talk, we will focus on the well-posedness and an optimal control problem for a phase transition problem related to thermal energy storage. Specifically, we want to improve the melting rate of a phase-change material by incorporating hard inclusions with very high diffusivity. This is represented by a semilinear system of three coupled parabolic equations, with Robin conditions on the inclusions' boundary involving the temperatures outside and inside these, i.e. the coupling of the solutions is also effective in the boundary conditions inherent in the system. 

One of the key properties of convex problems is that every stationary point is a global optimum, and nonlinear programming algorithms that converge to local optima are thus guaranteed to find the global optimum. However, some nonconvex problems possess the same property. This observation has motivated research into generalizations of convexity. This talk proposes a new generalization which we refer to as optima-invexity: the property that only one connected set of optimal solutions exists. We state conditions for optima-invexity of unconstrained problems and discuss structures that are promising for practical use, and outline algorithmic applications of these structures.

We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice L in an n-dimensional Euclidean space V and a positive constant a, the goal is to find the points z in V that minimize the sum of the potential exp(-a ||x - z||^2) over all the points x in L.

By a result of Bétermin and Petrache from 2017 it is known that for steep potential energy functions (when a tends to infinity) the minimum in the limit goes to a deep hole of the lattice.

The goal of this talk is to strengthen this result for lattices with a lot of symmetries: We prove that the deep holes of root lattices are already the exact minimizers for all a>a0 for some finite a0. Moreover, we prove that such a stability result can only occur for lattices with strong algebraic structure.

After introducing the problem, we will discuss how to design and solve exactly an LP bound for spherical designs, which allows to prove that the deep holes are local minimizers.

The end of the argument follows from a covering argument involving a precise control of the parameters around the lattice points.

Joint work with C. Bachoc, F. Vallentin and M. Zimmermann

In this talk, we are interested in the asymptotic dynamics of a fast rotating incompressible fluid in the regime of vanishing Rossby number. We assume that the fluid moves in a three-dimensional domain with topography (including the possible presence of a land area) and we impose no-slip conditions at the boundary. By proving a "weak implies strong" convergence principle and constructing Ekman layers adapted to the geometry of the domain, we characterise the limit velocity profile and show that it evolves following a linear dynamics.

The talk is based on a joint work with J.-Y. Chemin (Université Claude Bernard Lyon 1) and I. Gallagher (École Normale Supérieure - Paris).

The behavior of the random feature model in a high-dimensional framework has recently become a popular topic of interest in the machine learning literature. This model is generally considered for feature vectors composed of independent and identically distributed (iid) entries. We move beyond this specific assumption, which may be restrictive in various applications. To this end, we propose studying the performance of the random feature model with non-iid data by introducing a variance profile to the feature matrix. The performance of this model is linked to the spectrum of the random feature matrix, which turns out to be a nonlinear mixture of random variance profiled matrices. We have computed the limiting traffic distribution of such matrices using an extension of the method of moments. Knowledge of this distribution allowed us to introduce a new random matrix, which we call the « linear plus chaos » matrix, and which shares the same limiting spectrum as the random feature matrix. This linear plus chaos model proves to be simpler to study and has enabled us to derive deterministic equivalents that describe the asymptotic behavior of the performance of the random feature model.

In this talk, we consider one-parameter families of pseudodifferential

operators whose Weyl symbols are obtained by dilation and a smooth

deformation of a symbol in a weighted Sjöstrand class. We show that

their spectral edges are Lipschitz/Hölder continuous functions of the

dilation or deformation parameter. Suitably local estimates hold also

for the edges of every spectral gap.

These statements extend Bellissard’s seminal results on the Lipschitz

continuity of spectral edges for families of operators with periodic

symbols to a large class of symbols with only mild regularity

assumptions.

If time permits, we will also discuss how these results can be used to

study the behavior of the frame bounds of Gabor systems generated from

a dilated non-uniform set of time-frequency shifts.

Les travaux de Mañé-Sad-Sullivan et Lyubich (années 80) caractérisent le lieu de bifurcation d'une famille de fractions rationnelles ou de polynômes d'une variable complexe, vus comme des systèmes dynamiques. Par la suite (années 2000) DeMarco, Bassanelli, Berteloot et d'autres ont, à l'aide de méthodes issues de la théorie du pluripotentiel, introduit une mesure naturelle appelée la mesure de bifurcation, dont le support est strictement inclus dans le lieu de bifurcation, et qui détecte les bifurcations "maximales". On présentera un résultat récent sur l'existence de disques holomorphes contenus dans le support de cette mesure, dans le cas où la famille est celle des polynômes cubiques.

Travail en collaboration avec Davoud Cheraghi et Arnaud Chéritat.

There are three underlying classes of algebraic varieties: General Type varieties, Calabi-Yau varieties, and Fano varieties. One of the key goals in complex geometry is to study when the above classes of varieties admit canonical metrics. An important example of such metrics is the Kähler-Einstein metrics. General type and Calabi-Yau varieties always admit a unique Kähler-Einstein metric. Fano varieties, however, are more complicated since it is known that some Fano varieties do not admit a Kähler-Einstein metric. To prove that varieties are K-stable it is possible to compute stability thresholds (δ-invariants). Generalizing this concept, one can also consider log Fano pairs so that stability conditions will tell us if there is a Kähler-Einstein metric on the regular locus of a variety with volume equal to the algebraic volume. In my talk, I will discuss log Fano planes, meaning that I will consider curves on a projective plane for degrees 1, 2, 3, and 4. I will describe a classification of such curves and present a computation of δ-invariant for one of the curves as an example.  I will also summarize my results and show possible applications.

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À préciser

It is well known since the pionnering work of Bardos-Lebeau-Rauch 92’ that the observability and control of linear waves on bounded domains of the euclidian space $\mathbb{R}^n$ are governed by the so-called geometric control condition (GCC). It is a microlocal condition, i.e. a property in the cotangent bundle, linking the set $\omega$ on which the control is acting and the generalized bicharacteristics of the wave operator. The aim of this work is to investigate geometries where GCC is not not satisfied. More precisely, given an open set $\omega$, and without assuming GCC, we aim to determine which subsets of data in the energy space can be observed from $\omega$. 

Regional and microlocal results are obtained, as well as corresponding control results, under microlocal conditions much weaker than GCC. 

In doing this, we have to manage in crucial way the central question of unique continuation property.

This talk comes from a joint work with Sylvain Ervedoza ( Univ. Bordeaux ) and Enrique Zuazua ( Univ. Erlangen ) .

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TBA

A définir

On the mesoscopic level, motion of individual particles can be modeled by a kinetic transport equation for the population density f(t,x,v) as a function of time t, space x and velocity v \in V. A relaxation term on the right hand side accounts for scattering due to self-induced velocity changes and typically involves a parameter K(x,v,v') encoding the probability of changing from velocity v' to v at location x:

\partial_t f(t,x,v) + v \cdot abla f(t,x,v) = \int K(x,v,v') f(t,x,v') - K(x,v',v)f(t,x,v) dv'

This hyperbolic model is widely used to model bacterial motion, called chemotaxis.

We study the inverse parameter reconstruction problem whose aim is to recover the scattering parameter $K$ and that has to be solved when fitting the model to a real situation. We restrict ourselves to macroscopic, i.e. velocity averaged data $\rho = \int f dv$ as a basis of our reconstruction. This introduces additional difficulties, which can be overcome by the use of short time interior domain data. In this way, we can establish theoretical existence and uniqueness of the reconstruction, study its macroscopic limiting behavior and numerically conduct the inversion under suitable data generating experimental designs.

This work based on a collaboration with Kathrin Hellmuth (Würzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).

Dans cet exposé, j'expliquerai que les multiplicateurs aux cycles de périodes $1$ et $2$ fournissent une bonne description de l'espace $\mathcal{P}_{d}$ des polynômes de degré $d$ modulo conjugaison par une transformation affine. Plus précisément, les fonctions symétriques élémentaires des multiplicateurs aux cycles de périodes $1$ et $2$ induisent un morphisme birationnel fini de $\mathcal{P}_{d}$ sur son image. Ce résultat est une conséquence directe des deux énoncés suivants : (1) Pour tout $p \geq 2$, une suite de polynômes complexes de degré $d$ avec multiplicateurs bornés en ses cycles de période $p$ est nécessairement bornée dans $\mathcal{P}_{d}(\mathbb{C})$. (2) Une classe de conjugaison générique de polynômes complexes de degré $d$ est déterminée de façon unique par ses multiplicateurs en ses cycles de périodes $1$ et $2$. Je présenterai une version quantitative de l'énoncé (1). L'énoncé (2) démontre une conjecture de Hutz et Tepper et précise un résultat récent de Ji et Xie dans le cas polynomial.

Les strates de différentielles méromorphes à ordres de singularités prescrits sur la sphère de Riemann forment des espaces de modules appelés strates. L'intégration de la differentielle le long de certaines classes d'homologie relatives fournit à ces strates ce que l'on appelle les coordonnées périodes. Fixer les résidus aux pôles (qui sont des périodes particulières) définit la fibration isorésiduelle au-dessus de l'espace vectoriel des configurations de résidus. Il apparaît que le lieu singulier de cette fibration est un arrangement d'hyperplans complexes: l'arrangement de résonance.

Dans le cas particulier des 1-formes avec un seul zéro, la fibration devient un revêtement ramifié. Nous fournissons une formule pour calculer le degré de ce revêtement et analysons sa monodromie. Nos résultats exploitent la correspondance entre l'analyse complexe et la géométrie plate des surfaces de translation.

La géométrie qualitative de ces surfaces de translation est classifiée à l’aide d’arbres décorés, ce qui ramène le calcul du degré du revêtement à un problème combinatoire. Pour les strates avec deux zéros, les fibres isorésiduelles sont des courbes complexes dotées d’une structure de translation canonique. Les singularités de ces fibres codent, à travers leurs invariants locaux, les dégénérescences correspondantes des objets paramétrés. La monodromie est décrite en termes de connexion de Gauss-Manin, qui possède de riches propriétés géométriques et combinatoires.

Ce travail est une collaboration avec Dawei Chen, Quentin Gendron et Miguel Prado.

Ces dernières années, de nombreux progrès ont été réalisés dans l'étude des métriques de Kähler-Einstein sur les variétés singulières. Cependant, il existe très peu de résultats concernant l'existence des métriques kählériennes à courbure scalaire constante sur les variétés singulières. Dans cet exposé, je discuterai de cette question et présenterai nos résultats sur l'existence de telles métriques lorsque la fonctionnelle de Mabuchi est coercitive. Ce sont des travaux en collaboration avec C-M. Pan et A. Trusiani

Following a suggestion by Mathew, we will explain how to identify some arithmetic invariants of logarithmic schemes. A key ingredient is a form of flat descent for logarithmic invariants that we call "saturated descent”: using this, we compute logarithmic invariants in terms of classical (non-logarithmic) invariants and generalize classical results to logarithmic invariants. As a sample application, we can identify the homotopy-theoretic version of logarithmic prismatic and syntomic cohomology with the site-theoretic version introduced by Koshikawa and Koshikawa-Yao. We will explain how to get from this a log variant of the (graded version of the) Beilinson fiber square of Antieau-Mathew-Morrow-Nikolaus.This is a joint work with F. Binda, T. Lundemo and D. Park.

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Pas de séminaire

We derive an entropy stable extension of the Navier-Stokes-Fourier equations into the transition regime of rarefied gases. We do this through a variational multiscale reformulation of the closure of conservation equations derived from the Boltzmann equation. Our reformulation subsumes existing methods such as the Chapman-Enskog expansion. We apply the linearized version of this extension to the stationary heat problem and the Poiseuille channel and compare our analytical solutions to asymptotic and numerical solutions of the linearized Boltzmann equation. In both model problems, our solutions compare remarkably well in the transition regime. For some macroscopic variables, this agreement even extends far beyond the transition regime.

TBA

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 A définir

Pas de séminaire cette semaine puisqu'il y a la conférence pour les 60 ans de Yuri Bilu : https://yubi60.pages.math.cnrs.fr/

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