Le but de l'exposé est de présenter un résultat d'existence et d'unicité globales pour le système magnétohydrodynamique incompressible en 3D pour des données initiales assez proches de l'état d'équilibre e3=(0,0,1).

The geometric Langlands correspondence was introduced by Beilinson and Drinfeld as a tool to solve quantum Hitchin systems such as the Gaudin model. The correspondence can be understood following Kapustin and Witten as arising from S-duality in a topologically twisted 4-dimensional super Yang-Mills theory. In this talk I will describe a multiplicative deformation of the Hitchin system and explain a conjectural statement of the corresponding q-deformed correspondence. I will also give some motivations for the statement from the theory of higher deformation quantization and a deformed setup of Kapustin and Witten.

Les problématiques de formation de singularité en équations aux dérivées partielles non-linéaires sont au coeur d'une dynamique de recherche intense ces vingt dernières années. Je présenterai quelques résultats récents concernant la construction de solutions explosives dans le cadre dit énergie sur critique. Dans le cas parabolique, je présenterai notamment la construction de toutes nouvelles bulles a géométrie fortement anisotrope.

Séminaire Laurent Schwartz — EDP et applications

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

A. Venkatesh a formulé une conjecture ambitieuse sur l'action des espaces de cohomologie motivique sur la cohomologie de la variété localement symétrique (adélique) d'un groupe réductif G.  La cohomologie motivique en question est celle du motif adjoint du motif automorphe attaché à une représentation automorphe cuspidale de G ;  la conjecture explique en quelque sorte la contribution de cette représentation à plusieurs degrés de cohomologie, et relie des questions profondes sur la K-théorie algébrique à des considérations non moins profondes sur les déformations de représentations galoisiennes automorphes.  Dans mon exposé je vais présenter les premiers résultats d'un travail en commun avec Venkatesh sur une version de sa conjecture pour la cohomologie cohérente d'un fibré automorphe L sur une variété de Shimura, dans le cas où L a de la cohomologie (cuspidale) en plusieurs degrés.  Le premier exemple concerne le fibré des formes modulaires de poids 1 sur une courbe modulaire.  On voit des liens inattendus entre la vérification de la conjecture dans ce cas et des questions difficiles sur la version p-adique, dûe à Harris-Tilouine et à Darmon-Rotger, de la fonction L (de Garrett) d'un triplet de formes modulaires.

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

In studying the arithmetic of automorphic Galois representations, an important role is played by global cohomology classes coming from algebraic cycles on Shimura varieties, or more generally from algebraic K-theory; these are the building blocks of Euler systems. Unfortunately, it is surprisingly difficult to prove that these cohomology classes are non-zero! One of the key inputs for the recent progress in the theory of Euler systems was a new approach to solving such problems, developed by Bertolini, Darmon and Prasanna, in which the non-vanishing of Galois cohomology classes can be obtained by relating them to p-adic period integrals via Besser's rigid syntomic cohomology. I will explain some examples of this strategy, for Galois representations arising from products of modular curves and Hilbert modular surfaces, and survey the problems that must be solved to extend this method to more general Shimura varieties.

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

In recent years there has been notable progress on the construction of Euler systems and their connection to special values of classical and p-adic L-functions. In this talk I will describe Euler systems associated to a triple (f,g,h) of classical (cuspidal or Eisenstein) modular forms and their relation with p-adic L-functions constructed by Hida, Harris and Tilouine, following ideas of Kato. As an application, I will explain how these Euler systems can be used to obtain new results on the arithmetic of elliptic curves when the rank of the Mordell-Weil group is 0, 1 or 2.

A review of results concerning the classical XY model in various dimensions is presented.

 

I start by showing that the XY model does not exhibit any phase transitions in a non-vanishing external magnetic field, and that connected spin-correlations have exponential decay. These results can be derived from the Lee-Yang theorem.

 

Subsequently, I study the XY model in zero magnetic field: The McBryan-Spencer upper bound on spin-spin correlations in two dimensions is derived. The XY model is then reformulated as a gas of vortices of integer vorticity (Kramers-Wannier duality). This representation is used to explain some essential ideas underlying the proof of existence of the Kosterlitz-Thouless transition in the two-dimensional XY model. Remarks on the existence of phase transitions accompanied by continuous symmetry breaking and the appearance of Goldstone modes in dimension three or higher come next.

 

Finally, I sketch the random-walk representation of the XY model and explain some consequences thereof – such as convergence to a Gaussian fixed point in the scaling limit, provided the dimension is > 4; and the behaviour of the inverse correlation length as a function of the external magnetic field.

 

: In my talk I consider a q-deformation of the so-called Yangian Y(gl(m)). The standard Yangian Y(gl(m)) (associated with the Yang R-matrix) was introduced by V.Drinfeld and is rather well known. It possesses a lot of interesting properties and has applications in integrable models of mathematical physics (for example, in the non-linear Schroedinger model), W-algebras and so on. Its q-analog, called the q-Yangian, is usually defined as a « half » of a quantum affine group. D. Gurevich and me suggest a new construction for such a q-analog of the Yangian Y(gl(m)). We call it « braided Yangian ». We associate the braided Yangians with rational and trigonometric quantum R-matrices, depending on a formal parameter. These R-matrices arise from constant involutive or Hecke R-matrices by means of the Baxterization procedure. Our braided Yangians admit the evaluation morphism onto quantum matrix algebras and due to this one can construct a rich representation theory for them. In my talk I also plan to define analogs of symmetric polynomials (full, elementary and powers sums) which form a commutative subalgebra in the braided Yangian and exibit some noncommutative matrix identities similar to the Newton-Cayley-Hamilton identities of the classical matrix anlysis.

2-D Gaussian free field (GFF) is an intriguing mathematical object emerging in a wide range of contexts in probability theory and statistical physics. Several important properties of GFF have been explored. Among them are its various metric properties which have attracted a substantial amount of research in recent years. In this talk, we will discuss three of them, namely the Liouville FPP, the Liouville graph distance and an effective resistance metric. We will discuss the contexts in which they arise, state the current results, try to give rough sketches of the proofs and mention some open problems for future research. The content of this talk is based on joint works with Jian Ding and Marek Biskup. 

The talk summarizes my work on the classification of knots and closed plane curves, most of which was done jointly with O.Karpenkov and S.Avvakumov. The main idea is to supply the knot (or the closed plane curve) with an energy functional and to classify these objects via their {it normal forms}, which we define as the shapes of these objects that minimize the functional.

 

For any closed $mathcal C^2$ curve $gamma$, the functional that we choose is the {it Euler functional} $E(gamma)$ equal to the integral along $gamma$ of the squared curvature of $gamma$. We prove (using some fairly sophisticated methods of the calculus of variations) that:

 

(1) the critical points of $E$ are given by circles passed once or several times and by $infty$-shaped curves passed once or several times.

 

(2) the minima of $E$ are given by $infty$-shaped curves passed once and by circles passed once or several times. }

 

This solves a (long forgotten) problem set by Euler in 1774. The same result was obtained at about the same time (2012) by Yu.Sachkov by completely different methods. The result also gives a new proof of the famous Whitney–Graustein theorem on the classification of plane curves. It will be illustrated in the talk by an animation that shows in real time how a plane curve is homotoped to its normal form (by gradient descent along the functional).

 

For knots $k:S^1 to R^3$, we use a functional $F$ equal to the sum of the Euler functional $E(k)$ and a simple {it repulsive functional} $R(k)$ (the latter prevents self-intersections of $k$). We construct an algorithm (implemented in an animation that will be shown in the talk) which yields the isotopy of a given knot to its normal form (corresponding to the minimum of $F$) via a discretized version of gradient descent along $F$.

 

We then discuss to what extent this algorithm gives a practical solution of the knot classification problem and compare our theoretical normal forms with those obtained by physical experiments (which will be shown during the talk) with models of knots made out of flexible wire.

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