פעילויות השבוע

Let $T$ be a bounded linear operator on a Banach space $X$ satisfying $\lVert T^n\rVert/n\rightarrow 0$. We prove that $T$ is uniformly ergodic if and only if the one-sided ergodic Hilbert transform $H(T)x:=\lim_{n\rightarrow \infty}\sum_{k=1}^nk^{-1}T^kx$ converges for every $x\in \overline{(I-T)X}$. When $T$ is a power-bounded (or more generally $(C,\alpha)$ bounded for some $0<\alpha<1$), then $T$ us uniformly ergodic if and only if the domain of $H$ equals $(I-T)X$.

An effective approach to the Diophantine problem of enumerating all points on curves with non-abelian fundamental groups, such as those of genus greater than 1, is provided (conjecturally always) by the Chabauty-Kim method. The central object in this method is a Selmer scheme associated to the initial curve of interest and generalizing the association of Selmer groups to elliptic curves. In this talk, we‘ll show that arithmetic dualities produce (derived) symplectic and Lagrangian structures on associated spaces which reflect certain expectations coming from ”arithmetic topology“. In addition to some Diophantine utility, this should be viewed as foundational towards a ”TQFT“ approach to L-functions and related invariants analogous to a parallel story producing knot invariants from structures on character varieties which will be elaborated upon.

Self-testing is the strongest form of quantum functionality verification, which allows one to deduce the quantum state and measurements of an entangled system from its classically observed statistics. From a mathematical perspective (which will be the perspective of this talk), self-testing is an intriguing uniqueness phenomenon, pertaining to functional analysis, moment problems, convexity and representation theory. This talk addresses basic motivation and ideas behind self-testing, and discusses which states and measurements can be self-tested. In particular, the talk focuses on how tuples of projections adding to a scalar multiple of identity, and Jordan algebras find its way into this corner of quantum information theory. Based on joint work with Ranyiliu Chen and Laura Mančinska.