◾Schedule:

◾Titles and Abstracts

Boo Rim Choe (Korea Univ): The Shapiro-Sundberg Problems

Abstract: The Shapiro-Sundberg Problems in the title refers to the two problems, concerning composition operators acting on the Hardy space, raised by Shapiro and Sundberg in 1990. One is the path component problem and the other is the compact difference problem. For the past three decades these problems have been studied by many experts. I, together with collaborators, have quite recently succeeded in obtaining two different solutions to the compact difference problem, but the path component problem is still open. In this talk I will present background, motivation, progress, our contributions and further problems related to the Shapiro-Sundberg problems.

Hyungwoon Koo (Korea Univ): Composition operators in several variables

Abstract: In this talk, we survey the progress in the research on composition operators in several variables and talk about unsettled problems. We consider Bergman, Hardy and holomorphic Sobolev spaces over several domains: the unit ball, the unit polydisk, strictly pseudo-convex domains and the finite type domains. We focus on the case when the symbol map is smooth up to the boundary and discuss several jump phenomena. We also give some recent results on the unit polydisk.

Hun Hee Lee (Seoul Nat’l Univ): Lie group representations and operators in abstract harmonic analysis

Abstract: Lie groups are central objects in many areas of pure mathematics. In this talk we focus on its connection to functional analysis through its representation theory. When the group is non-compact we often witness a rich structure of operators acting on infinite dimensional Hilbert spaces. We will examine various operators appearing in the study of weighted Fourier algebras on Lie groups including the "ax+b" group.

Ja A Jeong (Seoul Nat’l Univ): Z-stability of simple crossed products of commutative C*-algebras by Hilbert bimodules

Abstract: We show that the Cuntz--Pimsner algebras associated to minimal homeomorphisms twistd by line bundles, along with their orbit-breaking subalgebras, are Z-stable whenever the underlying dynamical system has mean dimension zero. This entails that this class is classified by the Elliott invariant. If time permits, we will also discuss a recursive homogeneous algebra structure of the orbit-breaking subalgebras.

Kyung Hoon Han (Univ of Suwon): Duality and geometry in quantum states and positive maps

Abstract: We will survey the basic notions in entanglement theory to include the following:
- separable states / entangled states
- Schmidt number
- block positive matrices
- Choi-matrices
- positive partial transpose
- k-positive maps / k-superpositive maps
We focus on mathematical aspects without physical meaning, in particular, their duality and geometric aspects.

Seung-Hyeok Kye (Seouul Nat’l Univ): Choi matrices; infinite dimensional cases and examples

Abstract: The Choi matrix of a linear map between matrix algebras gives rise to a block matrix, which is a bi-partite states with normalization if and only if the original map is completely positive. In this talk, we consider possible variants of Choi matrices, and infinite dimensional analogues as follows: For a given normal completely bounded map between von Neumann factor which acts on a Hilbert space H with a separating and cyclic vector, we define the linear map which plays ampliation. When this map extends to the whole B(H), we associate a bounded operator and a trace class operator, which play the role of Choi matrix. In the case of type I factor, we use this construction to characterize the infinite dimensional analogues of Schmidt numbers, entanglement breaking maps and superpositive maps. We also give concrete examples to distinguish k-positivity. This is infinite dimensional analogues of Tomiyama maps, which play important roles to analyze isotropic state and Werner states. This talk is based on the papers:
- Choi matrices revisited, J. Math. Phys. 63 (2022), 092202.
- Choi matrices revisited. II, J. Math. Phys. 64 (2023), 102202. (with K. H. Han)
- Infinite dimensional analogues of Choi matrices, arXiv 2311.18240. (with K. H. Han and E. Stormer)
- Positive maps on the infinite dimensional type I factors (in preparation)

In Sung Hwang (Sungkyunkwan Univ): TBA

Abstract: TBA

Sumin Kim (Sungkyunkwan Univ): TBA

Abstract: TBA

Il Bong Jung (Kyungpook Nat’l Univ): Some aspects of the invariant subspace problem

Abstract: TBA

Jaeseong Heo (Hanyang Univ): TBA

Abstract: TB

Hyoung Joon Kim (Seoul Nat’l Univ): TBA

Abstract: TBA

In Hyoun Kim (Incheon Nat’l Univ): TBA

Abstract: TBA

Dong-O Kang (Chungnam Nat’l Univ): TBA

Abstract: TBA

Inyoung Park (Ehwa Womans Univ): TBA

Abstract: TBA

Koeun Choi (Ehwa Womans Univ): TBA

Abstract: TBA

Ji Eun Lee (Sejong Univ): On Toeplitz operators on the Newton space

Abstract: In this talk, we study the Newton space N^2(P) which has the Newton polynomials as an orthonormal basis. We first explore some relations between the orthonormal basis {z^n} of the Hardy space H^2(D) and the orthonormal basis {N_n} of the Newton space N^2(P). Also, we investigate the truncated Toeplitz operator on N^2(P) and we examine the product of N_m and N_n and the orthogonal projection P of overline{N_n}N_m. Finally, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N^2(P). Furthermore, we investigate properties of expansive and contractive Toeplitz operators with analytic and co-analytic symbols on a Newton space. (This is jointed work Eungil Ko and Jongrak Lee.)

Yoenha Kim (Ajou Univ): TBA

Abstract: TBA

Mee-Jung Lee (Kookmin Univ): TBA

Abstract: TBA

Jongrak Lee (Sungkyunkwan Univ): TBA

Abstract: TBA