일정:
12월 19일 오후 3시 - 오후 5시 (이우영 1강)
12월 20일 오전 9시 30분 - 12시 (Kalantar 1강)
12월 20일 오후 2시 30분 - 5시 (이우영 2강)
12월 21일 오전 9시 - 10시 (Kalantar 2강)
12월 21일 오전 10시 30분 - 12시 (이우영 3강)
12월 21일 오후 1시 - 오후 5시 (DIscussion)
12월 22일 오전 9시 30분 - 12시 (Kalantar 3강)
참가에 관심있는 분들은 이훈희(E-mail: hunheelee_at_snu_dot_ac_kr)에게 메일로 문의바랍니다.
강연안내
초록:
Lecture 1: Higher-order de Branges--Rovnyak spaces
Lecture 2: Higher-order sub-Bergman spaces
Lecture 3: Finite dimensional higher-order sub-Bergman spaces
In these lectures, we introduce the novel concept of higher-order de Branges--Rovnyak spaces by combining the studies of de Branges--Rovnyak spaces and hypercontractions, both topics have been studied intensively in the last several decades. We develop some basic properties of these higher-order de Branges--Rovnyak spaces. In particular, we show they can be viewed as iterated de Branges--Rovnyak spaces. We apply an abstract operator theoretic approach to the study of higher-order sub-Bergman spaces. We compute the reproducing kernels of higher-order sub-Bergman spaces and use them effectively to answer a number of questions. We identify these higher-order sub-Bergman spaces when the associated symbols are finite Blaschke products. We demonstrate that some natural function spaces are contained in higher-order sub-Bergman spaces for general associated symbols. We find finite dimensional higher-order sub-Bergman spaces and produce explicit orthonormal bases for these spaces. In comparison with the extensive theory of de Branges--Rovnyak spaces and sub-Hardy spaces, it is clear that there are many questions about higher-order de Branges--Rovnyak spaces and sub-Bergman spaces for further exploration.
Lecture 1: Furstenberg boundary and Hamana’s equivariant injective envelopes
Abstract: In the first lecture we introduce the notions of topological and measurable boundary actions in the sense of Furstenberg, and recall Hamana’s construction of equivariant injective envelopes. We will prove, for a discrete group G, a canonical identification between the G-injective envelope of the trivial G-action and the space of continuous functions on the Furstenberg boundary of G. We will use this identification to give a characterization of C*-simplicity and uniqueness of trace of the reduced C*-algebra of G. Time permitting, we conclude with concrete examples where the above characterizations may be applied.
Lecture 2: Other characterization, and generalizations
Abstract: We begin by going over details of several other approaches in applying boundary theory to C*-simplicity and uniqueness of trace. We introduce the Chaubauty space, and stationary actions of G, and give characterizations of C*-simplicity in those terms. We then discuss generalizations of these ideas and applications in similar structural questions concerning more general classes of C*-algebras, with the focus on certain quasi-regular representations..
Lecture 3: Relative boundary actions and structure of C*-algebras generated by covariant representations
Abstract: We extend the techniques mentioned in previous talks to much more general settings, giving complete characterization of simplicity and existence/uniqueness of trace on C*-algebras generated by covariant representations of minimal actions of discrete groups on locally compact spaces.
We conclude with a quick overview of some recent progress, mostly by others, on few other rigidity problems concerning group C* and von Neumann algebras in which boundary actions have played key role.