I will explain a recent work by Hidetoshi Oyama on a reformulation of the mod p analogue of the Simpson correspondence by Ogus and Vologodsky in terms of crystals on certain sites. He follows a similar work of the speaker on the p-adic Simpson correspondence by Faltings.
In the new formulation, the correspondence between D-modules and Higgs modules and the comparison between de Rham cohomology and Higgs cohomology are both given by the direct and inverse image functors of a certain morphism of topos.

We constructed a family of Calabi-Yau varieties over the l-line P1 Z[1/2]
{0, 1, ¥}, which is a projective smooth model of the affine scheme

[ w2
= x1·s xn(1-x1)·s(1-xn)(1 – l x1·s xn), ]

such that the generalized hypergeometric series n+1Fn(1/2, ·s,1/2; 1, ·s, 1; l) appear in the middle cohomology as a period function. In this talk we recall the construction of the family and how to calculate various cohomologies (Betti, de Rham, etale, and crystalline), discuss torsion freeness, up to 2-torsions, of integral cohomologies, and prove the integral version of degeneration of the Hodge to de Rham spectral sequence.

Instantons on R4, namely anti-self-dual Yang-Mills connections, are in bijection with framed locally free sheaves on CP2. Ramified instantons have an imposed singularity along R2 in R4 that translates to a parabolic structure along a CP1 divisor, or equivalently to a cyclic orbifold.  Such a singularity (Gukov-Witten defect) can be obtained in 4d N=2 supersymmetric Yang-Mills theory by adding 2d N=(2,2) degrees of freedom on R2, and gauging a global symmetry of the 2d theory using the R2 restriction of the 4d gauge connection.  The moduli space of ramified instantons should thus be related to a moduli space of instanton-vortex configurations of the 4d-2d pair of gauge theories.  I propose an incomplete definition of the latter moduli space by fibering (over the instanton moduli space) a recent description of the vortex moduli space as based maps to the Higgs branch stack.  As evidence I compare Nekrasov partition functions, namely equivariant integrals over these moduli spaces.  The equality relies on Jeffrey-Kirwan technology, applicable thanks to the ADHM construction of the moduli spaces as Kähler quotients.

Hadamard Lectures 2018

 

Retrouvez toutes les informations sur le site de la Fondation Mathématique Jacques Hadamard :

 

https://www.fondation-hadamard.fr/fr/financements-accueil-206-cours-avances/accueil-lecons-hadamard

« Seminar on homological algebra »

 

A Maurer-Cartan (MC) element in a differential graded (dg) algebra A is an odd element x satisfying the equation dx+x2=0. The group of invertible elements of A acts on MC element by gauge transformations:  g(x):=gxg-1-dgg-1.  MC elements are an abstraction of the notion of a flat connection and are fundamental in many problems of homological algebra, deformation theory, differential geometry etc.

 

There is a notion of a (Sullivan) homotopy of MC elements: two such are homotopic if they could be extended to a family over the de Rham algebra on the interval R[x,dx]. A fundamental result (over 40 years old) due to Schlessinger and Stasheff (SS) states that (under certain assumptions) two MC elements are gauge equivalent if an only if they are homotopic.

 

There is also another notion of homotopy of MC elements, based on the singular cochain complex of the interval, and a corresponding SS type theorem.

 

 

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0le nle 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=1/sqrt{2+sqrt{2-n}}$. For $0<nle 2$, the transition line has been further conjectured to separate a regime with short loops when $x<x_c(n)$ from a regime with macroscopic loops when $xge x_c(n)$.

 

In this talk we will prove that for $nin [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. A main tool in the proof is a new positive association (FKG) property shown to hold when $n ge 1$ and $0<xlefrac{1}{sqrt{n}}$. This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing property). We develop a `domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$.

2D continuum Gaussian free field (GFF) is a canonical model for random surfaces. It has various nice properties like conformal invariance or the Markov property, but also a notable disadvantage when thought of as a surface – it is merely a random generalized function that cannot be defined pointwise. Nevertheless, when one is stubborn enough and insists on studying its geometry, beautiful things start to appear: for example, connections to SLE processes of Schramm or to Brownian loop soups. I would like to give a short overview of some of the results obtained in this direction in collaboration with T. Lupu, E. Powell, A. Sepulveda and W. Werner. In particular, I would like to explain how to decompose the 2D continuum GFF into an independent sum of measures.

ATTENTION : Horaire d'hiver

 

Bloch and Kato formulated conjectures relating sizes of p-adic Selmer groups with special values of L-functions. Iwasawa theory is a useful tool for studying these conjectures and BSD conjecture for elliptic curves. For example the Iwasawa main conjecture for modular forms formulated by Kato implies the Tamagawa number formula for modular forms of analytic rank 0. 
In this talk I'll first briefly review the above theory. Then we will focus on a different Iwasawa theory approach for this problem. The starting point is a recent joint work with Jetchev and Skinner proving the BSD formula for elliptic curves of analytic rank 1. We will discuss how such results are generalized to modular forms. If time allowed we may also explain the possibility to use it to deduce Bloch-Kato conjectures in both analytic rank 0 and 1 cases. In certain aspects such approach should be more powerful than classical Iwasawa theory, and has some potential to attack cases with bad ramification at p.

The dimer model is a model of perfect matching whose popularity stems from the fact that it is exactly solvable. It is believed that the large-scale fluctuations of the height function of the dimer model is universal in a certain sense and should not depend on the microscopic properties of the graph. It turns out that in this level of generality, the well-established methods using Kasteleyn matrices become intractable.

 

 We propose a new method for examining the fluctuation of the height function which enables us to obtain a universality result for general graphs with various boundary conditions and even when the underlying surface is a Riemann surface. This provides a new proof of some old results and solves several open questions. Our methods use exact solvability in a weak sense and use some new results in the continuum instead which enables us to get universal results.

 

Ongoing joint work with Nathanael Berestycki and Benoit Laslier.

What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums. 

Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).

A spherical CR structure on a 3-manifold is a geometric structure modeled on the boundary at infinity of the complex hyperbolic plane, or in other words a (G,X)-structure with G=PU(2,1), X=S3. I will discuss spherical CR uniformizations, which are a special kind of spherical CR structure that arises by taking the manifold at infinity of a quotient of the ball by the action of a discrete group of isometries. I will explain how to construct some explicit uniformizations, including a 1-parameter family of (pairwise non-conjugate) spherical CR uniformizations of the figure eight knot complement.

I will report on joint work with Jenia Tevelev on a question of Orlov about exceptional collections on moduli spaces of pointed stable rational curves.