Prossimi seminari del Dipartimento di Matematica

We will introduce a potential theory in the complex unit ball which respects the complex structure of the unit sphere in C^n. It turns out that this capacity is able to describe the fine properties of functions in the Drury - Arveson space. Some new interesting problems emerge since the kernel which generates this new potential theory is positive but is not the reciprocal of a quasi-distance.

The classical Gauss curvature flow was introduced by Firey in 1974 as an idealized model for the abrasion of convex stones under random collisions on a beach. In 1985, Kai-Seng Chou proved that convex hypersurfaces contract to a point in finite time under this flow. Later, in 1999, Ben Andrews showed that in two dimensions the rescaled flow converges to a round sphere. Subsequent works by Ben Andrews, Bennett Chow, Pengfei Guan, Lei Ni, and Brendle-Choi-Daskalopoulos established analogous convergence results in higher dimensions, as well as for the power Gauss curvature flow. In this talk, based on joint work with Xinqun Mei and Guofang Wang, we present convergence results for capillary variants of the Gauss curvature flow and power Gauss curvature flow under hydrophilic case.

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In this talk we discuss continuous-time and state-space optimal stopping problems from a reinforcement learning perspective. We begin by formulating the stopping problem using randomized stopping times, where the decision maker's control is represented by the probability of stopping within a given time--specifically, a bounded, non-decreasing, càdlàg control process. To encourage exploration and facilitate learning, we introduce a regularized version of the problem by penalizing it with the cumulative residual entropy of the randomized stopping time. The regularized problem takes the form of an (n+1)-dimensional degenerate singular stochastic control with finite-fuel. We address this through the dynamic programming principle, which enables us to identify the unique optimal exploratory strategy. For the specific case of a real option problem, we derive a semi-explicit solution to the regularized problem, allowing us to assess the impact of entropy regularization and analyze the vanishing entropy limit. Finally, we propose a reinforcement learning algorithm based on policy iteration. We show policy improvement results for our proposed algorithm. This talk is based on a joint project together with Giorgio Ferrari and Renyuan Xu.

This lecture deals with evolution problem for the 1-Laplacian with mixed boundary conditions on bounded open sets. We prove existence and uniqueness of strong solutions for square-integrable data means of the theory of maximal monotone operator. We also see that if the flux on the boundary is 1 (that is, the maximum possible) then these strong solutions can be seen as the large solutions introduced in [Journal Functional Analysis 262 (2012), 1566–1602 ]. We give explicit examples of solutions. Joint work with N. Igbida and J. Toledo