Motivated by the 32-dimensional extension of the spin representation of the compact Lie algebra so(10) to the ‘maximal compact’ subalgebra of the real Kac-Moody Lie algebra of type E10 described by Damour et al. and Henneaux et al. Hainke and myself introduced the concept of a generalized spin representation that allows similar constructions for the ‘maximal compact’ subalgebras of real Kac-Moody Lie algebras of arbitrary simply laced type.

By work of Ghatei, Horn, Weiss and myself, integration of these representations leads to two-fold spin covers of the ‘maximal compact’ subgroups of the corresponding split real Kac-Moody groups. The problem that semisimple elements generally do not have a locally finite action and therefore obstruct integration is circumvented by an amalgamation method using the Iwasawa decomposition and the theory of buildings. The existence of these spin covers has been conjectured by Damour and Hillmann; it contains an extended Weyl group, which in the E10 case is relevant to fermionic billards.

We construct simply laced semisimple Lie algebras L with structure constants built into the Lie bracket. This also gives structure constants for the simply connected Chevalley group associated to the Lie algebra L. For infinite dimensional simply laced Kac-Moody algebras and their associated Kac-Moody groups, we obtain a complete description of the structure constants for root vectors corresponding to real roots whose sum is real. This is joint work with Pierre Cartier.

Research in quantitative evolutionary genomics and systems biology led to the discovery of several universal regularities connecting genomic and molecular phenomic variables. These universals include the log-normal distribution of the evolutionary rates of orthologous genes; the power law-like distributions of paralogous family size and node degree in various biological networks; the negative correlation between a gene’s sequence evolution rate and expression level; and differential scaling of functional classes of genes with genome size. The universals of genome evolution can be accounted for by simple mathematical models similar to those used in statistical physics, such as the birth-death-innovation model. These ‘laws’ of evolutionary genomics, analogously to the laws of statistical physics, appear to emerge from the Maximum Entropy principle which dictates that the probability distribution of any variable in a large ensemble of data or measurements tends to the distribution with the maximum entropy within the applicable constraints. In the case of genome evolution, these constraints are readily interpretable as effects of purifying selection against gene malfunction.

Motivated by the question, « what coordinates choices are `good’ for general relativity? » I will discuss gauge theories from the from the free evolution point of view, in which initial data satisfying constraints of a theory are given, and because the constraints are compatible with the field equations they remain so. I will present a model constrained Hamiltonian theory and identify a particular structure in the equations of motion from which statements can be made about the status of the initial value problem, just by examining a subset of the equations, called the pure gauge subsystem. I will then present the application of these ideas to GR, which results in a five parameter generalization of the `good’ gauge used in Choquet-Bruhat’s original treatment of the initial value problem.

The « very good » property for algebraic stacks was introduced by Beilinson and Drinfeld in their paper « The Quantization of Hitchin’s Integrable System and Hecke Eigensheaves ». They proved that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is « very good » as long as X has genus g > 1. We will introduce the « very good » property in the context of a group action on an algebraic variety, and prove it for a moduli space of parabolic bundles on P1 arising from quiver representations. As a special case, we will consider the « very good » property for the diagonal action of the group PGL(n) on a product of partial flag varieties and its relationship with the space of solutions to the additive Deligne-Simpson problem.

Massive neutrinos make up a fraction of the dark matter, but due to their large thermal velocities, cluster significantly less than cold dark matter (CDM) on small scales. An accurate theoretical modelling of their effect during the non-linear regime of structure formation is required in order to properly analyse current and upcoming high-precision large-scale structure data, and constrain the neutrino mass. Taking advantage of the fact that massive neutrinos remain linearly clustered up to late times, we treat the linear growth of neutrino overdensities in a non-linear CDM background. The evolution of the CDM component is obtained via N-body computations. The smooth neutrino component is evaluated from that background by solving the Boltzmann equation linearised with respect to the neutrino overdensity. CDM and neutrinos are simultaneously evolved in time, consistently accounting for their mutual gravitational influence. This method avoids the issue of shot-noise inherent to particle-based neutrino simulations, and, in contrast with standard Fourier-space methods, properly accounts for the non-linear potential wells in which the neutrinos evolve. Inside the most massive late-time clusters, where the escape velocity is larger than the neutrino thermal velocity, neutrinos can clump non-linearly, causing the method to formally break down. It is shown that this does not affect the total matter power spectrum, which can be very accurately computed on all relevant scales up to the present time.

The principal approximation methods used to compute the inspiral of compact binary systems are the post-Newtonian (pN) expansion, in which an orbital angular velocity MΩ serves as the expansion parameter; and the self-force or extreme-mass-ratio-inspiral approach, in which the small parameter is the mass ratio m/M of the binary’s two components. We work in an overlapping regime where both approximations are valid and find numerical values of pN coefficients at orders beyond the reach of current analytical work. In this talk we present a novel analytic extraction of high-order pN parameters that govern quasi-circular binary systems using ultra-high accuracy numerical computations.

: I will discuss some examples where differential geometry, topology and combinatorics of polyhedra can benefit from each other, and review some tools that make this possible. In particular, I will focus on the following two topics

— the Hirsch conjecture, concerned with bounding the running time of the simplex algorithm, and an application of CAT(1)-geometry to this problem.

— two problems of Legendre–Steinitz and Perles–Shephard concerning realization spaces of polytopes, and their solution based on solving discrete PDEs.

All notions will be introduced in the talk; I intend to make a friendly introduction to the theory, so non-experts are welcome.