Given a smooth proper formal scheme over a p-adic ring of integers, the étale cohomology of the generic fiber can be related to the crystalline cohomology of the special fiber. In my talk, I will present a new perspective on this relation involving a suitable notion of F-crystals on the prismatic site of the formal scheme. Using it, we can compare the two cohomology theories with general coefficients, in the relative setting, and allowing ramified base rings. Joint work with Haoyang Guo.  Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

Probability and analysis informal seminar​(joint work with V. Delecroix, E. Goujard and P. Zograf)I will remind how Maxim Kontsevich and Paul Norbury have counted metric ribbon graphs and how Maryam Mirzakhani has counted simple closed geodesic multicurves on hyperbolic surfaces. Both counts use Witten-Kontsevich correlators (they will be defined in the lecture with no appeals to quantum gravity).I will present a formula for the asymptotic count of square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large genus asymptotic analysis of this formula, performed by Amol Aggarwal, and the uniform large genus asymptotics of intersection numbers of Witten-Kontsevich correlators, proved by Aggarwal, combined with the results of Kontsevich, Norbury and Mirzakhani, allowed us to describe the structure of a random multi-geodesic on a hyperbolic surface of large genus and of a random square-tiled surface of large genus.​ As an application I will count oriented meanders on surfaces of any genus and an asymptotic probability to get a meander by a random identification of endpoints of a random braid on a two-component surface of any genus. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The ultrarelativistic scattering of two « elementary » objects interacting gravitationally is a problem that has inspired several interesting developments over the last four decades. I will review some of these developments in (super)gravity and string theory, and finally discuss how they are helping us to better understand gravitational binaries in general.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

Probability and analysis informal seminarThe talk will present recent progress towards a rigorous understanding of the critical regime associated to continuous phase transitions in the presence of long-range correlations, in dimensions larger than 2. Our findings deal with a 3-dimensional percolation model built from the harmonic crystal, which benefits from a rich interplay with potential theory for the associated diffusion. The results rigorously exhibit the scaling behavior of various observables of interest and unveil the values of the associated critical exponents, which are consistent with scaling theory below the upper-critical dimension (expectedly equal to 6). This confirms various predictions by physicists based on non-rigorous renormalization group techniques, notably that of Weinrib-Halperin concerning the value of the associated correlation length exponent. It also yields a proof of Fisher’s scaling relation for this model. Based on joint works with Alex Drewitz and Alexis Prevost.  ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminar This is a joint work with Reza Gheissari (Northwestern) and Aukosh Jagannath (Waterloo), Outstanding paper award at NeurIPS 2022. We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. Interestingly, we find a critical scaling regime for the step-size below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

 Though proteins are basic blocks of life, mathematicians are only starting to formalize the fundamental concepts of structural biology. The key missing piece was the definition of a practical equivalence on (tertiary structures of) proteins embedded in 3-dimensional space. Since protein structures are determined in a rigid form, the strongest equivalence in practice is rigid motion or isometry also including reflections. We can consider a protein an ordered sequence of ordered alpha-carbons (protein backbone) or a cloud of unlabeled atomic centers, which allows us to compare any molecules under isometry.The pairwise comparisons of all protein chains in the Protein Data Bank (PDB) by complete isometry invariants unexpectedly detected thousands of pairs that have identical coordinates of all alpha-carbon atoms (often all atoms as well). More than 325 billion pairwise comparisons were completed in less than two days on a modest desktop, implemented by Alexey Gorelov in our joint work.Some pairs of chains differ in primary sequences of their amino acids, which seems physically impossible. We discussed the findings with the PDB validation team and several authors confirmed that corrections in the PDB are needed. Using more flexible isometry invariants for rigid clouds of unlabeled atomic centers, we produced a continuous map revealing hot spots in the whole PDB.

The quantization of the mirror curve to a toric Calabi-Yau threefold gives rise to quantum-mechanical operators. Their fermionic spectral traces produce factorially divergent power series in the Planck constant, which are conjecturally captured by the refined topological string in the Nekrasov-Shatashvili limit via the Topological Strings/Spectral Theory correspondence. In this talk, I will discuss how the machinery of resurgence can be applied to study the non-perturbative sectors associated with these asymptotic expansions, producing infinite towers of periodic singularities in the Borel plane and infinitely-many rational Stokes constants, which are encoded in generating functions given in closed form by q-series. I will then present an exact solution to the resurgent structure of the semiclassical limit of the first fermionic spectral trace of the local P2 geometry, which unveils a remarkable arithmetic construction. The same analytic approach is applied to the dual weakly-coupled limit of the conventional topological string on the same background. The Stokes constants are explicit divisor sum functions, the perturbative coefficients are particular values of known L-functions, and the duality between the two scaling regimes appears in number-theoretic form. This talk is based on arXiv:2212.10606.  Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

Dirichlet’s theorem in Diophantine approximation implies that for any real x, there exists a rational p/q arbitrarily close to x such that |x-p/q| < 1/q2. In addition, the exponent 2 that appears in this inequality is optimal, as seen for example by taking $x=sqrt2$. In 1967, Wolfgang Schmidt suggested a similar problem, where x is a real subspace of Rd of dimension ℓ, which one seeks to approximate by a rational subspace v. Our goal will be to obtain the optimal value of the exponent in the analogue of Dirichlet’s theorem within this framework. The proof is based on a study of diagonal orbits in the space of lattices in Rd.

Property (T) is a fundamental notion introduced by D. Kazhdan in the mid 1960’s, that found numerous applications since then, notably in the context of rigidity of group actions. For a group G generated by a finite set S, property (T) means that there is a constant K>0 such that given any unitary representation of G on a Hilbert space without non-zero invariant vectors, every unit vector is displaced by some element of S to a point that is at least K apart. Finite groups have that property. Kazhdan proved that lattices in simple Lie groups of rank at least 2 all do as well.I will introduce a new class of infinite groups enjoying Kazhdan’s property (T) and admitting alternating group quotients of arbitrarily large degree. Those groups are constructed as automorphism groups of the ring of polynomials in n indeterminates with coefficients in the finite field of order p, generated by a suitable finite set of polynomial transvections. As an application, we obtain explicit presentations of hyperbolic Kazhdan groups with infinitely many alternating group quotients, and explicit generating pairs of alternating groups of unbounded degree giving rise to expander Cayley graphs. The talk is based on joint work with Martin Kassabov.

A motive over a scheme S is a bit of linear algebra which is supposed to « universally » capture the cohomology of smooth proper S-schemes.  Motives can be studied via various « realizations », which are objects of more concrete linear algebraic categories attached to S.  It is known that over certain S, these different realizations are related to one another via comparison isomorphisms, as in Hodge theory.  In this talk, I will try to explain that for completely general S, there is a much more subtle kind of relationship between these realizations, which takes a similar form to classical reciprocity laws in number theory. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

Séminaire Laurent Schwartz — EDP et applications