Tous les événements à venir

Les surfaces del Pezzo et leurs groupes d'automorphismes jouent un rôle important dans l'étude des sous-groupes algébriques du groupe de Cremona du plan projectif.

Sur un corps algébriquement clos, il est classique qu’une surface del Pezzo est soit isomorphe à $\mathbb{P}^{1} \times \mathbb{P}^{1}$ soit à l’éclatement de $\mathbb{P}^{2}$ en au plus $8$ points en position générale, et dans ce cas, les automorphismes des surfaces del Pezzo (de tout degré) ont été décrits. En particulier, il existe une unique classe d'isomorphismes de surfaces del Pezzo de degré $5$ sur un corps algébriquement clos. Dans cet exposé, nous nous intéresserons aux surfaces del Pezzo de degré $5$ définies sur un corps parfait. Dans ce cas, il y a beaucoup de surfaces supplémentaires (comme on peut déjà le voir si le corps de base est le corps des nombres réels), et la classification ainsi que la description du groupe d’automorphismes de ces surfaces sur un corps parfait $\mathbf{k}$ se ramènent à comprendre les actions du groupe de Galois $\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$ sur le graphe des $(-1)$-courbes.

Consider a control system 𝛛t f + Af = Bu. Assume that 𝛱 is

a projection and that you can control both the systems

 𝛛t f + 𝛱Af = 𝛱Bu,

 𝛛t f + (1-𝛱)Af = (1-𝛱)Bu.

Can you conclude that the first system itself is controllable ? We

cannot expect it in general. But in a joint work with Andreas Hartmann,

we managed to do it for the half-heat equation. It turns out that the

property we need for our case is:

 If 𝛺 satisfies some cone condition, the set {f+g, f∈L²(𝛺), g∈L²(𝛺),

f is holomorphic, g is anti-holomorphic} is closed in L²(𝛺).

The first proof by Friedrichs consists of long computations, and is

very "complex analysis". But a later proof by Shapiro uses quite

general coercivity estimates proved by Smith, whose proof uses some

tools from algebra : Hilbert's nullstellensatz and/or primary ideal

decomposition.

In this first talk, we will introduce the algebraic tools needed and

present Smith's coercivity inequalities. In a second talk, we will

explain how useful these inequalities are to study the control

properties of the half-heat equation.

We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter eps>0. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every eps>0, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free. This is part of joint works with L. Nguyen, M. Rus, V. Slastikov and A. Zarnescu.

Consider a control system 𝛛t f + Af = Bu. Assume that 𝛱 is

a projection and that you can control both the systems

 𝛛t f + 𝛱Af = 𝛱Bu,

 𝛛t f + (1-𝛱)Af = (1-𝛱)Bu.

Can you conclude that the first system itself is controllable ? We

cannot expect it in general. But in a joint work with Andreas Hartmann,

we managed to do it for the half-heat equation. It turns out that the

property we need for our case is:

 If 𝛺 satisfies some cone condition, the set {f+g, f∈L²(𝛺), g∈L²(𝛺),

f is holomorphic, g is anti-holomorphic} is closed in L²(𝛺).

The first proof by Friedrichs consists of long computations, and is

very "complex analysis". But a later proof by Shapiro uses quite

general coercivity estimates proved by Smith, whose proof uses some

tools from algebra : Hilbert's nullstellensatz and/or primary ideal

decomposition.

In this first talk, we will introduce the algebraic tools needed and

present Smith's coercivity inequalities. In a second talk, we will

explain how useful these inequalities are to study the control

properties of the half-heat equation.

We discuss a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with position $x$ and mass $m$. There are three key aspects to the SBGD dynamics: (i) persistent transition of mass from agents at high to lower ground; (ii) a random marching direction, aligned with the steepest gradient descent; and (iii) a time stepping protocol which decreases with $m$.

The interplay between positions and masses leads to dynamic distinction between `heavier leaders’ near local minima, and `lighter explorers’ which explore for improved position with large(r) time steps. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer. 

The Beurling--Selberg extremal approximation problems aim to find optimal unisided bandlimited approximations of a target function of bounded variation. We present an extension of the Beurling--Selberg problems, which we call “of higher-order,” where the approximation residual is constrained to faster decay rates in the asymptotic, ensuring the smoothness of their Fourier transforms. Furthermore, we harness the solution’s properties to bound the extremal singular values of confluent Vandermonde matrices with nodes on the unit circle. As an application to sparse super-resolution, this enables the derivation of a simple minimal resolvable distance, which depends only on the properties of the point-spread function, above which stability of super-resolution can be guaranteed.

.

The regular model of a curve is a key object in the study of the arithmetic of the curve, as information about the special fiber of a regular model provides information about its generic fiber (such as rational points through the Chabauty-Coleman method, index, Tamagawa number of the Jacobian, etc). Every curve has a somewhat canonical regular model obtained from the quotient of a regular semistable model by resolving only singularities of a special type called quotient singularities. We will discuss in this talk what is known about the resolution graphs of $Z/pZ$-quotient singularities in the wild case, when $p$ is also the residue characteristic. The possible singularities that can arise in this process are not yet completely understood, even in the case of elliptic curves in residue characteristic 2.

Annulé

.

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $\s(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

This is a joint work with S. Kupin and A. Vishnyakova.