Local and global geometric structures of perfectoid Shimura varieties
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.
Orthogonal and unitary tensor decomposition from an algebraic perspective
« Return of the IHES Postdoc Seminar »
Abstract: While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. In this talk I will present an intrinsic characterization of those tensors that do, by means of polynomial equations of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. This is a joint project with J. Draisma, E. Horobet, and E. Robeva.
Correspondance de Langlands géométrique en caractéristique p, d’après Bezrukavnikov, Braverman, Chen, Travkin et Zhu.
Soit X une courbe projective lisse et G un groupe réductif au dessus d'un corps de base k.
La correspondance de Langlands géométrique vise à comparer la catégorie des D-modules sur le champ Bun_G des G-fibrés sur X avec la catégorie des faisceaux quasi-cohérents sur le champ LocSys_{G^L}, où G^L est le groupe dual de G au sens de Langlands. On peut imaginer une telle équivalence comme une transformation de Fourier non-abélienne. Pour cette raison, elle est extrêmement difficile à démontrer.
Cependant, si k est de caractéristique positive, les deux côtés deviennent assez proches de leurs limites quasi-classiques (à savoir, les espaces de Hitchin correspondants) que l'on peut comparer grace à la transformation de Fourier-Mukaï habituelle (c'est-à-dire, abélienne). Cette idée a été suggérée dans l'article de Bezrukavnikov-Braverman et puis développée par Bezrukavnikov, Chen, Travkin et Zhu.
Dans cet exposé j'expliquerai les idées principales de cette théorie, ainsi qu'une application à la théorie en caractéristique 0, à savoir la construction de faisceaux automorphes à partir des *opers*, selon un article récent de Bezrukavnikov-Travkin.
Prismatic Dieudonné Theory
I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.
Measure of stability violation as probable mathematical marker for pre-pathological unsustainability of human metabolism at health-to-disease transitions
Living organism, considered as a complex biological system, sustains its functionality through interconnected molecular transformations denoted as metabolism. Modern biomedical technologies allow simultaneous measurement of near all (small) organic molecules in a microsample of a biofluid such as blood. For the analysis of system viability, a complete set of metabolites from blood – blood metabolome – is of a prime resolution power, because i) metabolites are the final entities of the body’s life-sustaining functionality in the line gene → proteine → metabolite, and as such totalize in their concentrations the organismal processes at all organisational levels, and ii) circulating blood, serving as a transmitter for all metabolites, is a whole-body metabolic information integrator. We collect blood metabolome data in a format of a comprehensive set of concentrational curves on a time trajectory health-to-disease, and explore the applicability of mathematical approaches for stability measurement to these big biodata sets in search for markers of pre-pathological metabolism sustainability violations.
I shall give a brief introduction to Mathieu Moonshine, an observation made in 2010 in the context of string theory compactified on a K3 surface and whose significance in string theory remains elusive. Attempts to understand the mathematical structure behind this observation have included techniques from Number Theory, Group Theory and Geometry. I will discuss how geometry provides an interesting angle when attempting to explain the presence of the huge Mathieu 24 discrete symmetry in string theories compactified on a K3 surface.
