Mini-cours

In a 2-page note of 1969, Victor Kac described automorphisms of
finite order of simple Lie algebras over the field of complex numbers
C. He used certain diagrams that were later called Kac diagrams. In
this talk I will explain the method of Kac diagrams for calculating
the Galois cohomology set H^1(R,G) for a connected semisimple
algebraic group G (not necessarily simply connected or adjoint) over
the field of real numbers R. I will use real forms of the half-spinor
group of type D_{2n} as examples.

This is a joint work with Dmitry A. Timashev.

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.

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.

In 50 hours, the Xenopus embryo turns from a single egg cell into a complex organism with highly differentiated tissues: beating heart, flowing blood, contracting muscles, functional sensory organs. This process has been examined by scientists for over 150 years yet we can not claim a thorough systems-level understanding of how it works. Today, embryology is enjoying a technological revolution. We are able to observe the molecular (DNA, RNA, metabolites and protein) makeup of life at unprecedented resolution: the complete genome sequence, expression level of all proteins at genomic scale, messenger RNA expression at a single cell level as it changes over time. This enormous amount of data already constitutes a treasure trove for embryologists working on particular molecular circuits, but we are only just beginning to sense paradigm shift towards mathematical description and modeling of embryonic development.

 

In this talk I will review these new data acquisition tools, explain what are the challenges and achievements, and discuss what we can expect in the near future and what we have learned already. I expect to combine slide presentation with chalk-talk and open discussion.

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.

Séminaire Laurent Schwartz — EDP et applications

I'll discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I'll show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of (0,4) SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular. I'll show how to construct its modular completion and prove the modular invariance of the twistorial construction of D-instanton corrected hypermultiplet moduli space.

Séminaire Laurent Schwartz — EDP et applications