The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur’s weighted orbital integrals via counting points on the fundamental domain.

The purpose of the talk is to explain a result in collaboration with B. Pozzetti and A. Wienhard expressing the Hausdorff dimension of certain attractors as a critical exponent. This class of attractors consists of limit sets of Anosov representations in PGLd (hence of non-conformal nature) that verify an extra open condition. If time permits, we will discuss implications of the formula to the geometry of the Hitchin component.

Séminaire Laurent Schwartz — EDP et applications

Mapping class groups associated to surfaces whose fundamental group is not finitely generated are called « big », and their study is linked to classical problems in dynamics. In this talk, we discuss some of the basic properties of big mapping class groups, their simplicial actions, and how these can be used to prove that big mapping class groups « detect » surfaces or (if time allows) that the space of non-trivial quasimorphisms of a big mapping class group is infinite dimensional.

In this talk I will focus on BPS states in supersymmetric field theories with N=2. In this theories one can consider a certain class of supersymmetric line operators. Such operators support a new class of BPS states, called framed BPS states. I will discuss a formalism based on quivers to understand these objects and their properties. Time permitting I will discuss a relation with the theory of cluster algebras.

The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier:

affine spheres as level surfaces
metrization of the interior of K by the Hessian metric F”
primal-dual symmetry
interpretation as a minimal Lagrangian submanifold in a certain para-Kähler space form
complex-analytic structure on 3-dimensional cones.

Kontsevich’s 1997 proof of the formality conjecture provides a universal quantization of every Poisson manifold, by a formal power series whose coefficients are integrals over moduli spaces of marked discs. In joint work with Peter Banks and Brent Pym, we prove that these integrals evaluate to multiple zeta values, which are interesting transcendental numbers known from the Drinfeld associator and as the periods of mixed Tate motives. Our proof is algorithmic and allows for the explicit computation of arbitrary coefficients in the formality morphism, in particular the star product. The essential tools are Francis Brown’s theory of polylogarithms on the moduli space of marked genus zero curves, single-valued integration due to Oliver Schnetz, and an induction over the natural fibrations of moduli spaces.

The statistical measure Evolutionary Entropy characterizes Darwinian Fitness and predicts the outcome of competition for limited resources between related entities at various levels of organization: metabolic, cellular, organismic and social. I will discuss the mathematical basis of the selection principle and describe its application to the evolution of aging and the origin of age-related diseases.

We shall describe a synthesis of results obtained by Chevalley, Kostant, Bourbaki, Demazure and myself about the construction of a group scheme associated to a semisimple complex Lie group. The method is explicit, unlike the one of Demazure (and Grothendieck). It is easy to mimic the same methods to accomodate Kac-Moody algebras and groups.

Metals are an interesting class of gapless quantum many-body systems. Many metals are described by the famous « Fermi liquid theory » at low temperatures, but there are also many metallic materials for which Fermi liquid theory is an inadequate description. In this talk, I will argue that a productive way to think about certain properties of metals, beyond Fermi liquid theory, is in the language of emergent symmetries and anomalies, thus importing ideas originally developed in the context of gapped topological phases of matter and their boundaries. From this point of view, I will show how to derive a vast generalization of Luttinger’s theorem, the result that relates the volume enclosed by the Fermi surface of a Fermi liquid to the microscopic charge density. From this one can derive a number of consequences, including strong constraints on the emergent symmetry group of compressible metals. I also discuss implications for electrical resistivity.

 

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We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.