Let p be a rational prime, $q>1$ a p-power and P a non-constant irreducible polynomial in $F_q[t]$. The notion of Drinfeld modular form is an analogue over $F_q(t)$ of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite $nu$-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi’s theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

I will explain how the conformal bootstrap can be adapted to place rigorous bounds on the spectra of automorphic forms on locally symmetric spaces. A locally symmetric space is of the form HG/K, where G is a non-compact semisimple Lie group, K the maximal compact subgroup of G, and H a discrete subgroup of G. If we take G = SL(2,R), then spaces of this form are precisely hyperbolic surfaces and hyperbolic 2-orbifolds. Automorphic forms then come in two types: modular forms, and eigenfunctions of the hyperbolic Laplacian, also known as Maass forms. The bootstrap constraints arise from the associativity of function multiplication on the space HG, and are very similar to the usual correlator bootstrap equations, with G playing the role of the conformal group. For G=SL(2,R), I will use this method to prove upper bounds on the lowest positive eigenvalue of the Laplacian on all closed hyperbolic surfaces of a fixed genus. The bounds at genus 2 and genus 3 are very nearly saturated by the Bolza surface and the Klein quartic. This is based on upcoming work with P. Kravchuk and S. Pal.

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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.

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.

In joint work with François Maucourant, we study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologically transitive, and that the natural invariant measure, the so-called « Burger-Roblin measure », is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codimension of the unipotent flow inside the maximal unipotent flow. The latter result generalises a theorem of Mohammadi and Oh.

In the talk, I will present the main ideas of this work.

 

We introduce Feynman categories and show that they naturally define bi-algebras. In good circumstances these bi-algebras have Hopf quotients. Corresponding to several levels of sophistication and decoration (both terms have technical definitions), we recover the Hopf algebras of Goncharov and Brown from number theory, a Hopf algebra of Baues used in the analysis of double loop spaces and the various Hopf algebras of Connes-Kreimer used in QFT as examples of the general theory. Co-actions also appear naturally in this context as we will explain.

We will first define the moduli space of algebraic-group-valued representations of finitely presented groups. Then we will briefly describe how non-commutative rings influence the structure of the coordinate ring of these moduli spaces. Lastly, we will illustrate this general relationship by constructing minimal generating sets of the coordinate rings of these moduli spaces in some specific examples.

Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that there exists an isometric embedding of the hyperbolic plane in Minkowski space, which is the analogue of the embedding of the round sphere in Euclidean space. However, differently from the Euclidean case, the embedding of the hyperbolic plane is not unique up to global isometries. In this talk I will discuss several results on the classification of these embeddings, and explain how this problem is related to Monge-Ampère equations, harmonic maps, and Teichmüller theory. This is joint work with Francesco Bonsante and Peter Smillie.

Hitchin representations are an important class of representations of fundamental groups of closed hyperbolic surfaces into PSL(n,R), at the heart of higher Teichmüller theory. Given such a representation j, there is a unique j-equivariant harmonic map from the universal cover of the hyperbolic surface to the symmetric space of PSL(n,R). We show that its energy density is at least 1 and that rigidity holds. In particular, we show that given a Hitchin representation, every equivariant minimal immersion from the hyperbolic plane into the symmetric space of PSL(n,R) is distance-increasing. Moreover, equality holds at one point if and only if it holds everywhere and the given Hitchin representation j is an n-Fuchsian representation.