I will review the recent mathematical approach to Conformal Field Theory proposed by my colleagues Nam-Gyu Kang and Nikolai Makarov. Their construction defines a certain class of algebraic operations on correlation functions of the Gaussian Free Field, and these operations can be used to give meaning to "vertex observables" and other well known objects in CFT. Using conformal transformation rules for the GFF these objects can be defined on any simply connected domain, and using Lie derivatives they can be analyzed when the domain evolves according to an infinitesimal flow. Using the flow of Loewner's differential equation produces a connection with the random curves of the Schramm-Loewner evolution, which I will describe along with some recent work in the case of multiple SLE curves.
In this talk we discuss closed self adjoint extensions of the Laplacian and fractional Laplacian on L2 of Euclidean space minus the origin. In some cases there is a one parameter family of these operators that behave like the original operator plus a potential at the origin. Using these operators, we can construct polymer measures which exhibit interesting phase transitions from an extended state to a bound state where the pinning at the origin due to the potential takes over. The talk is based on joint works with Koralov, Molchanov, Squartini and Vainberg.
We recall some classical results on self-similar Markov processes and in particular explain how those relate to a very particular class of stochastic differential equations. Those SDEs have an infinite dimensional counter part which arise naturally in interacting diffusions and can be interpreted as generalized voter process.
It is a classical result, by Dyson, that the behavior of the eigenvalues of a random unitary matrix following uniform measure tend, when the dimension goes to infinity, after a suitable scaling, to a random set of points, called adeterminantal sine-kernel process. By defining the model in all dimensions on a single probability space, we are able to show that the convergence stated above can occur almost surely. Moreover, in an article with K. Maples and A. Nikeghbali, we interpret the limiting point process as the spectrum of a random operator.
We consider the contact process on a random graph chosen with a fixed degree, power law distribution, according to a model proposed by Newman, Strogatz and Watts (2001). We follow the work of Chatterjee and Durrett (2009) who showed that for arbitrarily small infection parameter λ
In this talk we aim at emphasizing the role of information in financial markets (public information versus insider information). In particular, if the information about a particular event (as for instance the default event of a company) is incorporated into a pricing model, then by a change of the underlying filtration, one can compute risk premiums attached to particular events. We also show that modeling of the information leads eventually to modeling of dependencies.