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.

A formal approach (mathematical models) describing the first two hours of development of the embryo of Drosophila (fruit fly) is presented. The first part of the talk is centered on the biological and biochemical events and processes that determine the form and shape (morphogenesis) of the embryo of Drosophila. Then we show the basic equations associated to some of the developmental processes and discuss the calibration and validation of the mathematical models with real experimental data. The models describe all the biochemical morphogenetic parameters measured in the Drosophila embryo with a relative error below 5% and account for the observed phenotypic plasticity and resilience. This approach leads to a precise mathematical definition of resilience, which we associate to the existence of a continuous family of Pareto optimal solutions of the models. Problems associated with conservation laws (symmetries) and optimization for ill posed problems and noisy objective functions will also be discussed.

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).

 

I will explain how General Relativity in four space-time dimensions (with cosmological constant) can be reformulated as an SU(2) gauge theory of a certain type. The action is a (diffeomorphism invariant) functional of an SU(2) connection; no metric is present in the formulation of the theory. This formulation of GR is in many ways analogous to the one proposed many years ago by Eddington (Eddington’s Lagrangian is a function of just the affine connection and is given by the square root of the determinant of the Ricci tensor). The new formulation has some remarkable properties. First, on a 4-manifold, the space of SU(2) connections modulo gauge transformations has just 9 components per space-time point, as compared to 10 components of a metric tensor. Correspondingly, the action of the new formulation can be interpreted as a functional on the space of conformal classes of metrics. Thus, the conformal factor is not free to propagate in this formulation even off-shell. This has the effect that the action functional is (at least locally) convex – there is no conformal mode problem of the usual metric formulation of GR. Another remarkable property of the new formulation is that diffeomorphisms are very easy to deal with. The most natural gauge-fixing of these does not make the corresponding components of the connection propagate. As a result, in (linearized) quantum theory only 8 components of the connection propagate as compared to 10 metric components in the usual formulation. All these 8 components fit into a single irreducible representation of the Lorentz group, which makes the propagator very simple. There are also some remarkable simplifications in the structure of the interaction vertices. The full off-shell 3-vertex (in de Sitter space) contains just 3 terms as compared to a couple of dozen terms in the metric formulation. The 4-vertex is a couple of lines as compared to a couple of pages in the standard description. As an illustration of the formalism I will describe how the graviton scattering amplitudes are computed in this approach.

The new gauge-theoretic reformulation of GR also leads to (an infinite-parameter) family of modified gravitational theories, all propagating just two polarizations of the graviton, as GR. This leads to a rather strong claim that, in spite of the standard GR uniqueness theorems, General Relativity is not the only interacting theory of massless spin two particles. However, GR appears to be the only parity-invariant gravitational theory, as all the « deformations of GR » can be shown to be parity violating. I will also describe how matter is coupled in this approach, and give some speculations as to a possible UV completion of gravity.

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.
 

Let K be a CDVF of mixed characteristic (0,p) and G the absolute Galois group of K. When the residue field of K is perfect, Laurent Berger constructed a p-adic differential equation NdR(V) for any de Rham representation V of G. In this talk, we will generalize his construction when the residue field of K is not perfect. We also explain some ramification properties of our NdR , which are due to Adriano Marmora in the perfect residue field case.

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Séminaire Laurent Schwartz — EDP et applications

Computational anatomy has emerged as a new subject focusing on the quantitative analysis of the variability of biological shapes with interesting challenges both from the mathematical and computational point of view. In this talk I will explain how a geometrical point of view based on the idea of shapes as structured spaces, actions of groups of diffeomorphisms and (sub)-Riemannian geometry provide nice vehicles to build a full processing framework called here diffeomorphometry.

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.
 

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.