I have been co-organizing the student probability seminar for three years and gave the following talks. Click on title to see the abstract.
One important area of research and debate in ecology is the so-called complexity vs. stability issue. It was started by a now celebrated article by Robert May (1972), which suggested that a large complex system driven by generic non-linear ordinary differential equations would be unlikely to exhibit a point of stable equilibrium if its variables are 'too well connected' and 'too random' at the same time; this surprising theoretical fact can be directly applied to population dynamics, where the system is typically driven by a system of Lotka-Volterra equations, and many other settings.
Following a review article by Allesina and Tang, we will see how May's result can be recovered and precisely quantified using eigenvalue statistics from non-Hermitian random matrix theory. This analysis has been refined very recently by Jacek Grela, who also studied the existence of stable non-transient trajectories. In this context, transient means that the trajectory goes away from the equilibrium before converging to it; understanding transient behavior implies not only eigenvalues, but eigenvectors of non-Hermitian random matrices.
Several concepts and results of Random Matrix Theory also have a representation theoretic interpretation. The purpose of this talk is to show that this connection is not as esoteric as it sounds. We will introduce the basic concepts of representation theory from scratch, and see a few ways in which characters and representations of the unitary and symmetric groups appear in a random matrix context. In particular, our goal will be to understand an elegant proof of a Central Limit Theorem by Diaconis and Shahshahani that relies on such ideas.
Abstract : While in general there is no exact formula that counts
self-avoiding walks on a given infinite graph, the number of self-avoiding
walks of length k on lattices is known to be logarithmically
equivalent to ck , where c is called the connective
constant of the lattice, and usually can only be approximated. In the
specific case of the honeycomb lattice (i.e. the hexagonal structure one
finds in beehives), physical heuristics led to the conjecture that the
connective constant is
Smirnov proved this conjecture a few years ago in a sensational
thirteen-page-long paper. We will sketch their proof, discuss why it cannot
be easily extended to other settings, and what else can be asked or
expected from a probabilistic point of view.
Abstract : The name of Jean Ginibre was given to a very natural ensemble of random matrices, those with iid complex gaussian coefficents. The eigenvalues of such a matrix form a determinantal point process, and thus exhibit a well-studied repulsion. But this highly correlated system of points happens to decorrelate completely (I say : completely - not asymptotically) when put to a high enough power - that is, having them spin around the origin. This stunning property is known to hold for a wider class of processes; but the proof is particularly straightforward in the Ginibre case, as we shall see.
Abstract : We shall consider the uniform distribution over the
permutation groups and try to answer simple questions such as : how likely
is it for two elements to belong to the same cycle ? How are fixed points
distributed ? What is the typical size of a cycle ?... Some of our answers
will be just as simple as the questions; whereas others will require to
introduce usual tools and methods of random permutation theory, such as the
celebrated Feller coupling, thanks to which we might even end up answering
questions we didn't ask.
Abstract : We will introduce the Circular Unitary Ensemble (CUE) of random unitary
matrices distributed according to the Haar measure on Un(C), and compute the
distribution of its characteristic polynomial thanks to an explicit decomposition of
the law. In particular, the moments of this characteristic polynomial can be
computed ; they are strongly believed to have a link with the moments of the zeta
function along its critical line, and we will give some reasons why this might be
Abstract : The story of the meeting at tea time between Freeman Dyson and Hugh Montgomery, and what happened that the
latter would later recall as 'realserendipity', is now famous; but a good story never bores. We shall tell it again,sketch its mathematical background, and present a few other number-theoretic objects that exhibit amazing similarities with well-known results in Random Matrix theory :not only the zeros of the Zeta function, but also its moments on the critical line, and the number of points of some elliptic curves over finite fields.
Abstract : This talk will recall and sketch famous results about the longest increasing subsequence of a random (uniformly chosen) permutation. For this purpose, we shall learn and play a solitaire game, give a probabilistic proof of two elegant hook-formulas, and explain the Robinson-Schensted algorithm, among other things. These tools will give us two different approaches of the same problem, and even allow us to draw some conclusions.
✍ My Mémoire de Maîtrise,"Corps
de u-invariant pair" was co-written
with Margaret Bilu. The u-invariant of a field k
is the maximal dimension of an anisotropic
quadratic form on k. It was conjectured at
some point that it should be a power of 2 for
every field; which is not the case. In this paper,
following an argument given by A. Merkurjev, we
build a field of u-invariant 2n for any
positive integer n.