We briefly review Hida's white noise space theory and some  of its applications to linear stochastic systems and stochastic processes. To define derivatives and stochastic integrals we introduce a family of algebras of stochastic distributions, which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra. Non-commutative counterparts and applications to free stochastic processes and their derivatives will be also discussed. 

The talk is based on various collaborations, with in particular Haim Attia, Palle Jorgensen,  Alon Kipnis, David  Levanony, Guy Salomon and Tryphon Georgiou 

We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC'24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS'23).

https://sites.uci.edu/inverse/

In this talk, we study arithmetic properties of the pentagram map, a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of tori. We show that the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties of algebraic curves. In work in progress, we explore the question of which pentagram-like maps are integrable vs. chaotic.