Analysis Seminar, 2024--2025

• September 9, 23: Peter Burton (University of Wyoming)
  Title: Furstenberg's ×2 ×3 conjecture and group theory
  Abstract: Furstenberg's ×2 ×3 conjecture is a well-known problem at the intersection of ergodic theory, topological dynamics and number theory. In this talk we will present the background needed to formulate the conjecture and describe various surrounding facts that illuminate its significance. We will also discuss ongoing work of the speaker and Kate Juschenko that connects this conjecture to a ridigity property for representations of a particular semidirect-product group.
• October 7, 21: Irina Holmes Fay (University of Wyoming)
  Title: About some weak-type inequalities for the dyadic square function operator
  Abstract: The weak-type inequality for the dyadic square function with an A2 weight is a longstanding open problem in modern harmonic analysis. I will discuss the resolution of two other weak-type inequalities for this square function operator, in the unweighted setting, using the Bellman function method. The first talk will introduce the dyadic setting and the dyadic square function operator, motivate the weighted problem, then discuss the unweighted L^2 case. The second talk will discuss the unweighted L^1 case, and discuss possible insights this can provide to the open problem of A2 weights.
• November 4, 11, 18: Zhuang Niu (University of Wyoming)
  Title: Structure of transformation group C*-algebras: Z-stability and small boundary property
  Abstract: Consider a free and minimal action on a compact metrizable space by of a discrete amenable group, and let us study the structure of the associated crossed-product C*-algebra (the transformation group C*-algebra). I will survey on the Rokhlin property, comparison property, and the implications on tracial approximation of the C*-algebra. And, eventually, I will discuss the Z-stability of the transformation group C*-algebras and its relation to the mean dimension and the small boundary property of the dynamical system.

• January 27: Peter Burton (University of Wyoming)
  Title: Furstenberg's ×2 ×3 conjecture and complex analysis
  Abstract: Furstenberg's ×2 ×3 conjecture has remained a central open problem in ergodic theory for over 50 years, and it serves as the basic test case for a broad class of rigidity phenomena which are believed to hold in number-theoretic dynamics. More recently, two related statements have appeared in the literature: a question about periodic approximation raised by Levit and Vigdorovich in the context of approximate group theory and a periodic equidistribution conjecture formulated by Bourgian and Lindenstrauss. In the first part of this talk, we will present results of the speaker and Jane Panangaden which provide an equivalent formulation of each these three conjectures in terms of certain holomorphic functions on the unit disk. (The article is https://arxiv.org/abs/2410.22701.) In the second part, we will invite a friendly and open discussion about possible further directions in this project, in particular extensions to higher-dimensional rigidity conjectures in ergodic theory.