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.

: 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.

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.

Dans un travail commun avec D. Hulin nous montrons que toute quasiisométrie entre variétés de Hadamard pincées est à distance bornée d'une unique application harmonique.

Au début des années 90, R.E. Schwartz a montré que le théorème de Pappus permet de définir une famille d'actions du groupe modulaire sur le plan projectif avec des propriétés géométriques et dynamiques remarquables. Ces propriétés sont semblables à celles satisfaites par les représentations d'Anosov. Dans sa thèse (sous ma direction), V. Pardini Valerio a élucidé ce fait en montrant que les représentations de Schwartz, restreintes à un sous-groupe d'indice deux, sont limites de représentations d'Anosov.

 

Je présenterai ce travail de thèse, et les progrès récents obtenus conjointement avec Gye-Seon Lee.

We prove that when 3g − 3 + p > 3, the Teichmüller space of the closed surface of genus g with p punctures contains balls which are not convex in the Teichmüller metric. We analyze the quadratic differential associated to a Teichmüller geodesic and, as a key step, show that the extremal length of a curve (as a function of time) can have a local maximum. This is a joint work with Maxime Fortier Bourque.

A convex projective manifold C = Ω/Γ is the quotient of convex subset of projective space, Ω, by a discrete group of projective transformations Γ ⊂ PGL(n+1,R). A generalized cusp in dimension 3 is a convex projective manifold that is the product of a ray and a torus. The holonomy centralizes a 1-parameter subgroup of PGL(n,R). I have shown: A generalized cusp on a properly convex projective 3-dimensional manifold is projectively equivalent to one of 4 possible cusps.

 

For a generalized cusp C = Ω/Γ in dimension n, we require that ∂C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0,∞) × ∂C → C. Together with Sam Ballas and Daryl Cooper we have classified generalized cusps in dimension n, and explored new geometries arising from such cusps. We show the holonomy of a generalized cusp is a lattice in one of a family of Lie groups G(λ) parameterized by a point λ = (λ1, …, λn) ∈ Rn. More generally a maximal-rank cusp in a hyperbolic n-orbifold is determined by the similarity class of lattice in Isom(En-1).

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

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.

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.