This week

We define stochastic volatility for infinite dimensional processes, using an infinite dimensional Ornstein-Uhlenbeck process. Particular attention is paid to a fractional volatility model. Moreover, we also discuss realized covariation in this context, presenting a law of large numbers and a central limit theorem. The main application we refer to is modelling commodity forward term structures. The presentation is based on joint work with Fabian Harang (BI), Dennis Schroers (Bonn) and Almut Veraart (Imperial).

Although time is continuous, many physical processes and models are either constructed with a discrete time variable, or require time discretization to be tractable. While Markovian processes in discrete time have been studied in great details over many decades, memory effects induced by hidden degrees of freedom remain greatly unexplored in discrete-time dynamics. In this talk, I will focus on the general case where the evolution of the system state after n time steps depends on all its previous states in a linear way. In particular, I will identify a well-delineated regime where the dynamics can be faithfully approximated by a Markovian-like, first-order recurrence relation. This regime is defined through strict inequalities rather than comparison of orders of magnitude, as is often the case when justifying memoryless approximations. I will show how this formalism can be applied in concrete examples -- both quantum and classical -- and how it paves the way for the derivation of approximations in the strong-memory regime and for systems with periodic driving.

Many local conformal field theories can be analytically continued to a family of nonlocal, long-range CFTs by varying the exponent of the kinetic term $(p^2)^{\zeta}$ in the action away from the usual $\zeta=1$. It is natural to then ask what singles out the original local CFT within each such family. The answer is neat -- it has the most degrees of freedom, as counted by the (universal part of the) sphere free energy, also sometimes called the central charge. This provides a simple organising principle for models such as the critical Ising and O(N) vector CFTs in any d, and a compact way of organising perturbative data for their scaling dimensions. For unitary CFTs, this extremum is a maximum, which can be proven in conformal perturbation theory. In this talk, I will first give an introduction to nonlocal CFTs, then discuss counting degrees of freedom in CFTs, and finally prove the main result.

A major goal of the Langlands program is to describe the multiplicity of an irreducible discrete automorphic representation of a connected reductive group G in its discrete L2-spectrum. The first goal of this talk is to explain work from the last few years which gives the first conjectural formula for this multiplicity for general G over a global field, as envisioned by Kottwitz in 1984. We then discuss recent work which we hope lays the foundations for proving cases of these formulas using the geometric framework of Fargues and Scholze.