========================================================== PADOVA SEMINARS IN PROBABILITY AND FINANCE webpage: http://www.math.unipd.it/~bianchi/seminari/ ========================================================== Dear Colleagues, We would like to invite you to the "Padova seminars in Probability and Finance" this Friday, June 30 starting from 14.30 in Room 2BC30 at the department of Mathematics of the University of Padova. The seminar can also be followed via Zoom from a link to be found at https://www.math.unipd.it/~bianchi/seminari/ This week we will have two contributions: 14:30 - Giulia Livieri (LSE London) 15:30 - Federico Sau (University of Trieste) See further details below. The organizers _____________________________________________________ Speaker: Giulia Livieri (LSE London) Title: Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis Abstract: Causal operators (CO), such as various solution operators to stochastic differential equations, play a central role in contemporary stochastic analysis; however, there is still no canonical framework for designing Deep Learning (DL) models capable of approximating COs. This paper proposes a "geometry-aware'" solution to this open problem by introducing a DL model-design framework that takes suitable infinite-dimensional linear metric spaces as inputs and returns a universal sequential DL model adapted to these linear geometries. We call these models Causal Neural Operators (CNOs). Our main result states that the models produced by our framework can uniformly approximate on compact sets and across arbitrarily finite-time horizons Hölder or smooth trace class operators, which causally map sequences between given linear metric spaces. Our analysis uncovers new quantitative relationships on the latent state-space dimension of CNOs which even have new implications for (classical) finite-dimensional Recurrent Neural Networks (RNNs). We find that a linear increase of the CNO's (or RNN's) latent parameter space's dimension and of its width, and a logarithmic increase of its depth imply an exponential increase in the number of time steps for which its approximation remains valid. A direct consequence of our analysis shows that RNNs can approximate causal functions using exponentially fewer parameters than ReLU networks. Speaker: Federico Sau Title: Spectral gap of the symmetric inclusion process Abstract: In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous'spectral gap conjecture --- originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (SNU, South Korea). The Zoom meeting can be joined via the link to be found at https://www.math.unipd.it/~bianchi/seminari/