ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

Précédemment pour étudier la $mathbb{Q}_l$-cohomologie des variétés de Shimura de type Kottwitz-Harris-Taylor (KHT), on a utilisé la filtration par les poids du faisceaux pervers des cycles évanescents. La suite spectrale de cohomologie associée $E_1^{p,q} Rightarrow E_infty^{p+q}$, dégénère alors en $E_2$ mais pas en $E_1$, ce qui la rend inutilisable sur tout $mathbb{Z}_l$-analogue.

Dans cet exposé, j’expliquerai comment construire une nouvelle suite spectrale de nature géométrique dégénérant en $E_1$ et permettant, outre la simplification des arguments combinatoires sur $mathbb{Q}_l$, de fournir un procédé assez général de construction de classes de torsion.

In nearly all of the contexts in biology in which groups of cilia or flagella are found they exhibit some form of synchronized behaviour. Since the experimental observations of Lord Rothschild in the late 1940s and G.I. Taylor’s celebrated waving-sheet model, it has been a working hypothesis that synchrony is due in large part to hydrodynamic interactions between beating filaments. But it is only in the last few years that suitable methods have been developed to test this hypothesis. In this talk I will summarize our recent experimental and theoretical work addressing this important issue.

Deformations of cell sheets are ubiquitous in early animal development, often arising from a complex and poorly understood interplay of cell shape changes, division, and migration. In this talk I will describe an approach to understanding such problems based on perhaps the simplest example of cell sheet folding: the “inversion” process of the algal genus Volvox, during which spherical embryos literally turn themselves inside out through a process hypothesized to arise from cell shape changes alone.  Through a combination of light sheet microscopy and elasticity theory a quantitative understanding of this process is now emerging.

The field of "active matter" focuses on the collective behaviour of large numbers of individual units (molecular motors, cells, organisms) which inject energy into a fluid at the small scales, creating large-scale nonequilibrium patterns.  In this talk I will link together two historically important examples of active matter – concentrated suspensions of bacteria and cytoplasmic streaming in plant and animal cells – to illustrate recent experimental and theoretical developments in the area of self-organization.

From Leonardo da Vinci to the Brothers Grimm our fascination with hair has endured in art and science. We love it for its “body” or “volume”, the fluffiness and elasticity that comes from its random waves and curls. But apart from a purely tactile response, can we take a more quantitative approach to hair, to explain these macroscopic properties in terms of the behaviour of individual hairs? We know that the important physics governing hair involves the interplay of its elasticity, weight, and curliness, but it is only recently that these have been synthesized into a mathematical theory. This talk will cover those recent advances in the description of physical fiber bundles, including the "Ponytail shape equation" and aspects of "Hairodynamics".

We study the role of the Ding-Iohara-Miki (DIM) algebra, which is the simplest example of quantum toroidal algebra, in gauge theories, matrix models, q-deformed CFT and refined topological strings. We use DIM algebra to write down the Ward identities for the matrix models and show how it is connected to quiver W-algebras of the A-series. We describe the integrable structure of refined topological strings arising from DIM algebra: the R-matrix, T-operators and RTT relations. Finally, we write down the q-KZ equation for the DIM algebra intertwiners and interpret its solutions as refined topological string amplitudes.

In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa(ℚ_p). These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec(ℤ). The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

In the first part of the talk, I will discuss a joint project with Chris Elliott on realizing the geometric Langlands correspondence as an instance of S-duality by careful analysis of Kapustin and Witten’s work using derived algebraic geometry. In the second part of the talk, I will report on work in progress to produce new instances of Langlands duality in geometric representation theory through the lens of quantum field theory.

Séminaire Laurent Schwartz — EDP et applications

Une variété affine (au sens de la géométrie différentielle) est une variété admettant un atlas de cartes à valeur dans un espace affine V et à changements de cartes localement constants dans le groupe affine Aff(V). A la fin des années 50, Chern a conjecturé que la caractéristique d’Euler de toute variété affine compacte s’annule. Je discuterai cette conjecture, et sa preuve dans le cas où X est spéciale affine (i.e. X est affine et admet une forme volume parallèle).

The sheaf of chiral differential operators is a sheaf of vertex algebras defined by Gorbounov, Malikov, and Schechtman in the early nineties that exists on any manifold with vanishing second component of its Chern character. Later on it was proposed by Witten to be related to the chiral operators of the (0,2)-supersymmetric sigma-model. Recently, we have proved this using an approach to QFT developed by Costello: the BV-quantization of the holomorphic twist of the (0,2) theory is isomorphic to the sheaf of chiral differential operators. Along with Gorbounov and Gwilliam, we prove this using the language of holomorphic factorization algebras in one complex dimension. In this talk I will sketch the proof of this result while also motivating a family of BV theories that produce sheaves of higher dimensional holomorphic factorization algebras that deserve to be called “higher” CDOs. We discuss the meaning of the OPE for these theories as encoded by the higher dimensional factorization structure.

The space of projective measured foliations is (one of) the boundaries of Teichmüller space. One can consider a special subclass of this set that define Teichmüller geodesics whose projection to moduli space is contained in a compact set. These can be thought of as analogous to badly approximable rotations. The main result of the talk is that this set is path connected in high enough genus. This is joint work with Sebastian Hensel.