This week

Coarse correlated equilibria are generalizations of Nash equilibria which have first been introduced in Moulin et Vial (1978). They include a correlation device which can be interpreted as a mediator recommending strategies to the players, which makes it particularly relevant in a context of market failure. After establishing an existence and approximation results result in a fairly general setting, we develop a methodology to compute mean-field coarse correlated equilibria (CCEs) in a linear-quadratic framework. We identify cases in which CCEs outperform Nash equilibria in terms of both social utility and control levels. Finally, we apply such a methodology to a CO2 abatement game between countries (a slightly modified version of Barrett (1994)). We show that in that model CCEs allow to reach higher abatement levels than the NE, with higher global utility. The talk is based on joint works with F. Cannerozzi (Milan University), F. Cartellier (ENSAE) and M. Fischer (Padua University).

Consider the following mean-field equation on R^d:
d X_t = V(X_t, mu_t) dt + d B_t,
where mu_t is the law of X_t, the drift V(x, mu) is smooth and confining, and (B_t) is a standard Brownian motion.
This McKean-Vlasov equation may admit multiple invariant probability measures.
I will discuss the (local) stability of one of these equilibria.
Using Lions derivatives, a stability criterion is derived, analogous to the Jacobian stability criterion for ODEs.
Under this spectral condition, the equilibrium is shown to be attractive for the Wasserstein metric W1.
In addition, I will discuss a metastable behavior of the
associated particle system, around a stable equilibrium of the mean-field equation.

In this talk, I will employ an ADM deparametrization strategy to discuss the radial canonical formalism of asymptotically AdS_3 gravity. It leads to the identification of a radial 'time' before quantization, namely the volume time, which is canonically conjugate to York time. Holographically, this allows to interpret the semi-classical partition function of TTbar theory as a Schrodinger wavefunctional satisfying a Schrodinger evolution equation in volume time. The canonical perspective can be used to construct from the Hamilton-Jacobi equation the BTZ solution, and corresponding semi-classical Wheeler-DeWitt states. Based on upcoming work with Matthew J. Blacker, Blanca Hergueta and Sirui Ning.

This talk will discuss the Bombieri--Vojta proof of the Mordell conjecture, using gap principles for points of large height.

More information about the London (algebraic) number theory study group can be found here: https://sites.google.com/site/netandogra/seminars/uniform-mordell

Bianchi modular forms (i.e. automorphic forms over imaginary quadratic fields) share many similarities with their classical cousins. One such similarity is the period polynomial, studied for classical modular forms by Manin, Kohnen and Zagier, as well as many others. In this talk we define period polynomials of Bianchi modular forms, show how to compute them in practice, and use them to (conjecturally) extract information about congruences between Bianchi forms of various types (base-change and genuine forms\DSEMIC cusp forms and Eisenstein series). All of this is done through an example space of Bianchi forms, from which we find new congruences modulo 43 and 173. Time permitting, we will also describe some open problems relating to these methods, and how these relate to the classical picture. No prior knowledge of Bianchi modular forms is assumed.

The description of subspaces invariant under the Volterra operator goes back to a problem of Gelfand from 1938. Invariant subspaces for differentiation on $C^{\infty}$ were studied much later by Aleman and Korenblum and continued by Aleman, Baranov and Belov. Both problems contain a wealth of interesting ideas and have several interesting connections to exponential systems, among other things. I intend to give a review of some of these results and then continue with a more abstract setting consisting of an unbounded operator D with a compact quasi-nilpotent right inverse V. It turns out that under certain general conditions one can prove similar results for a large class of examples (for D) containing Schrödinger operators, Dirac operators and other Canonical systems of differential equations. This is a report about recent joint work with Alexandru Aleman.

One of the most important problems of modern science is that of emergence. How do laws of motion emerge at large scales of space and time, from much different laws at small scales? A foremost example is the theory of hydrodynamics. Take molecules in air, which simply follow Newton’s equations. When there are very many of them, these equations becomes untractable\DSEMIC seeking the knowledge of each molecule’s individual trajectory is completely impractical. Happily it is also unnecessary. At our human scale, new, different equations emerge for aggregate quantities: those of hydrodynamics. And these are apparently all we need to know in order to understand the weather! Despite its conceptual significance, the passage from microscopic dynamics to hydrodynamics remains a notorious open problem of mathematical physics. This goes much beyond molecules in air: similar principles hold very generally, such as in quantum gases and spin lattices, where the resulting equations themselves can be very different. In particular, integrable models, where an extensive mathematical structure allows us to make progress, admit an entirely new universality class of hydrodynamic equations. In this talk, I will discuss in a pedagogical and mathematically precise fashion the general problem and principles of hydrodynamics as an emergent theory, and some recent advances in our understanding, including those obtained in integrable models