In 1960, Higman asked whether the number f(n,q) of conjugacy classes of n x n unitriangular matrices U(n,q) over the finite field with q elements, is polynomial in q for every n.  I will survey what is known about this problem, explain the connections to realization spaces of matroids, enumerative combinatorics and group representation theory.  I will then describe our recent efforts to prove the conjecture for small n and disprove it for large n.

Joint work with Andrew Soffer.

I will show a simple example of fast-slow partially hyperbolic systems in which the fast variable acts similarly to a Gaussian  noise and will obtain, for such a deterministic setting, results similar to the one obtained in Freidlin–Wentzell theory. In particular, I will show conditions under which there exists a unique SRB measure with exponential decay of correlations.
 
(Work in collaboration with Jacopo de Simoi).

 

Information plays a critically important role in ecology and evolution but is very often subjective or analog or both. This is a problem because most information theory has been developed for objective and discrete information. Can information theory be extended this theory to incorporate multiple forms of information, each with its own (physical) carriers and dynamics? Here I will not review all the possible roles information can play, but rather what conditions an appropriate theory should satisfy.
 

Séminaire Laurent Schwartz — EDP et applications

According to a widely held opinion, Life corresponds to a physical state far from equilibrium. Thus, whereas such fundamental notion of equilibrium physics as ‘ground state’ is widely used to describe the properties of biological macromolecules or even macromolecular complexes, it is considered of no use for the description of a whole living cell. I would like to challenge this preconception, by discussing how the idea of a cell in a ground state is possible, and what could be the nature of the forces responsible for its stability. Strikingly, this line of inquiry leads to a novel justification of the self-organization principle, as the action of the restoring forces responsible for the stability of the ground state amounts to “optimization without natural selection of replicators”. Unlike the statistical-mechanical approaches to self-organization, our approach does not encounter the problem of ‘tradeoff between stability and complexity’ at the level of individual cell.
 

The infrared structure of quantum gravity is explored by solving a lattice version of the Wheeler-DeWitt equations. For this talk, first, the case of 2+1 dimensions is presented. The wavefunction solutions only depend on the geometric quantities indicating preservation of diffeomorphism. Properties of the lattice vacuum are consistent with the existence of an ultraviolet fixed point in G located at the origin, thus precluding the existence of a weak coupling perturbative phase. The correlation length exponent is determined exactly and found to be nu=6/11. The results obtained lend support to the claim that the Lorentzian and Euclidean formulations belong to the same field-theoretic universality class. I then discuss some results in 3+1 dimensions. Investigations of the vacuum wave functional further indicate that for weak enough coupling, G < Gc, a pathological ground state with no continuum limit appears, where configurations with small curvature have vanishingly small probability.

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.

In the AdS/CFT correspondence, gauge theory calculations beyond the planar approximation correspond to quantum corrections in gravity or in string theory. Recently, the partition function on the three-sphere of Chern-Simons-matter theories has been computed at all orders in the 1/N expansion, and this leads to predictions for quantum corrections in M-theory/string theory. Using the ideas of effective field theory, we show that some of these corrections can be calculated reliably by doing one-loop calculations in supergravity. A similar reasoning has been used recently to calculate logarithmic corrections to black hole entropy, and we use it here to perform a successful test of AdS4 /CFT3 beyond the leading, planar approximation.

I review the construction of maximal supergravities in two dimensions. These are parametrized by an embedding tensor transforming in the basic representation of the affine algebra E9. Among the examples is the SO(9) theory describing the low-energy effective action upon reduction of the type IIA theory on the D0-brane near-horizon geometry.

Recently, Weinstein finds some affinoids in the Lubin-Tate perfectoid space and computes their reduction in equal characteristic case. The cohomology of the reduction realizes the local Langlands correspondence for some representations of GLh, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectoid space in equal characteristic case, whose reduction realizes "ramified" representations of conductor exponent h+1. We call them ramified affinoids. We study the cohomology of the reduction and its relation with the local Langlands correspondence. This is a joint work with Takahiro Tsushima.

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In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will construct a motivic structure on the (nilpotent completion of) n-th homotopy group of Pn minus n+2 hyperplanes in general position.

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