We revisit the optimal exit problem by adding a moral hazard problem where a firm owner contracts out with an agent to run a project. We analyse the optimal contracting problem between the owner and the agent in a Brownian framework, when the latter modifies the project cash-flows with an hidden action. The analysis leads to the resolution of a constrained optimal stopping problem that we solve explicitly.

We consider the problem of unveiling the implicit network structure of user interactions in a social network, based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $\ell_1$ and trace norms. We provide a first theoretical analysis of the generalization error for this problem, that includes sparsity and low-rank inducing priors. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest. The analysis is based on a new supermartingale property of the trace exponential, based on tools from stochastic calculus. A consequence of our analysis is the construction of sharply tuned $\ell_1$ and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the strong improvements achieved by the use of such data-driven penalizations.

I will introduce the general idea of p-adic Hodge theory from the view point of a beginner. Also, I will give a sketch of the proof of the crystalline comparison theorem in the case of good reduction using 'almost mathematics'.

We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
 

In this talk I will describe two instances of Langlands functoriality concerning the group $\mathrm{Sp}_{2n}$. I will then very briefly explain how this enables one to attach Galois representations to automorphic representations of (inner forms of) $\mathrm{Sp}_{2n}$. 

Abstract: The simplest solutions of integrable systems are special functions that have been known since the time of Newton, Gauss and Euler. These functions satisfy not only differential equations as functions of their independent variable but also difference equations as functions of their parameter(s).  We show how the inverse scattering transform method, which was invented to solve the Korteweg-de Vries equation, can be extended to its discrete version.

S.Butler and N.Joshi, An inverse scattering transform for the lattice potential KdV equation, Inverse Problems 26 (2010) 115012 (28pp)

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Logic.