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 

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications 

The presence of nearby conformal field theories (CFTs) hidden in the complex plane of the tuning parameter was recently proposed as an elegant explanation for the ubiquity of « weakly first-order » transitions in condensed matter and high-energy systems. In this work, we perform an exact microscopic study of such a complex CFT (CCFT) in the two-dimensional O(n) loop model. The well-known absence of symmetry-breaking of the O(n>2) model is understood as arising from the displacement of the non-trivial fixed points into the complex temperature plane. Thanks to a numerical finite-size study of the transfer matrix, we confirm the presence of a CCFT in the complex plane and extract the real and imaginary parts of the central charge and scaling dimensions. By comparing those with the analytic continuation of predictions from Coulomb gas techniques, we determine the range of validity of the analytic continuation to extend up to ng≈12.34, beyond which the CCFT gives way to a gapped state. Finally, we propose a beta function which reproduces the main features of the phase diagram and which suggests an interpretation of the CCFT as a liquid-gas critical point at the end of a first-order transition line. Participer à la réunion Zoomhttps://us02web.zoom.us/j/85766610072?pwd=VFl2UUpPOURpMUk5SGJpMmsxMFpmQT09ID de réunion : 857 6661 0072Code secret : 966618==================================================================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 quantum_encounters_seminar PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarRandom walks on groups are nice examples of Markov chains which arise quite naturally in many situations. Their key feature is that one can use the algebraic properties of the group to gain a fine understanding of the asymptotic behaviour. For instance, it has been observed that some random walks exhibit a very sharp phase transition called the cut-off phenomenon. I will first explain this phenomenon on a concrete example, introducing all the necessary material. Then, I will give some ideas on how the setting may be enlarged to encompass more general stochastic processes (called non-commutative Markov chains) by using a strange algebraic structure called a quantum group.  ========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.