Sunday Evening
Kye, Seung-Hyeok (SNU): Schmidt number witnesses and exactly k-entangled subspaces
Schmidt numbers of bi-partite states play the roles to measure the degree of entanglement which is the central theme in the current quantum information theory, and Schmidt number witnesses are essential to detect Schmidt numbers. We investigate global locations of Schmidt number witnesses which are outside of the convex set of all bi-partite states. Their locations are classified by interiors of faces of the convex set of all states, by considering the line segments from them to the maximally mixed state. In this way, a nonpositive Hermitian matrix of trace one is located outside of one and only one face. Faces of the convex set of all states are classified by subspaces, which are range spaces of states belonging to specific faces. A subspace of the tensor product of two Hilbert spaces will be called exactly k-entangled subspaces when k+1 is the minimum of Schmidt ranks of vectors in it. We show that a subspace is exactly k-entangled if and only if k is the greatest number such that there exist Schmidt number k+1 witnesses outside of the face associated with its orthogonal complement. When this is the case, there exist Schmidt number 2, 3, ... , k witnesses outside of the face. This talk is based on the joint paper (arXiv 2505.10288) with Kyung Hoon Han.
Jung, Mingu (KIAS/Hanyang Univ.): Group actions on Banach spaces
Given a Banach space X, the group Iso(X) of isometries on X plays a significant role for various reasons—one of which traces back to the famous Mazur rotation (open) problem. In this talk, we will briefly review some classical results along this line and discuss potential generalizations of well-known theorems in Banach space theory to settings that are compatible with group actions.
Monday Afternoon
Lee, Woo Young (KIAS): Almost invariant subspaces of the shift
In this talk we formulate the almost invariant subspaces of backward shift operators in terms of the ranges or kernels of product of Toeplitz and Hankel operators. This approach simplifies and gives more explicit forms of these almost invariant subspaces which are derived from related nearly backward shift invariant subspaces with finite defect. Furthermore, this approach also leads to the surprising result that the almost invariant subspaces of backward shift operators are the same as the almost invariant subspaces of forward shift operators which were treated only briefly in literature.
Yoo, Hyun Jae (Hankyong National Univ.): Potential theory for quantum random walks associated with transition operation matrices
We discuss the potential theory for quantum random walks associated with transition operation matrices. Both the well-known open quantum random walks and the unitary quantum random walks fall into this framework. The dual operation for a transition operation matrix defines a quantum Markov operator. We develop the potential theory for such quantum Markov operators. Using this approach, we characterize the reducibility/irreducibility and recurrence/transience of the quantum Markov operators associated with transition operation matrices. We then apply these results to open quantum random walks and unitary quantum random walks providing with some examples.
Jeong, Ja A (SNU): Recursive subhomogeneity of orbit-breaking subalgebras of C*-algebras associated to minimal homeomorphisms twisted by line bundles
We construct recursive subhomogeneous decomposition for the Cuntz-Pimsner algebras obtained from breaking the orbit of a minimal Hilbert C(X)-bimodule at a closed subset Y of X with non-empty interior. This generalizes the known recursive subhomogeneous decomposition for orbit-breaking subalgebras of crossed products by minimal homeomorphisms.
Monday Evening
Hwang, Byung-Hak (KIAS): Formalizing mathematics: why and how
Formalizing mathematics involves translating mathematical statements from natural language into a precise formal language that computers can understand. As modern mathematics becomes deeper and more complex, the importance of formalization has grown significantly. In this talk, I will introduce the concept of formalization, explore its significance, and highlight current successful and ongoing projects in the field.
Kim, Minhyun (Hanyang Univ.): Nonlocal potential theory
Nonlocal potential theory, as an extension of classical potential theory, investigates harmonic functions associated with nonlocal operators modeled on the fractional Laplacian. In this talk, I will present recent advances in nonlocal potential theory, focusing on three central aspects: the local, boundary, and global behavior of harmonic functions. The results are based on joint work with Anders Björn, Jana Björn, Ki-Ahm Lee, Se-Chan Lee, and Marvin Weidner.
Tuesday Afternoon
Lim, Tongseok (Purdue Univ.): Monotone Curve Estimation via Convex Duality
A principal curve serves as a powerful tool for uncovering underlying structures of data through 1-dimensional smooth and continuous representations. On the basis of optimal transport theories, this paper introduces a novel principal curve framework constrained by monotonicity with rigorous theoretical justifications. We establish statistical guarantees for our monotone curve estimate, including expected empirical and generalized mean squared errors, while proving the existence of such estimates. These statistical foundations justify adopting the popular early stopping procedure in machine learning to implement our numeric algorithm with neural networks. Comprehensive simulation studies reveal that the proposed monotone curve estimate outperforms competing methods in terms of accuracy when the data exhibits a monotonic structure. Moreover, through two real-world applications on future prices of copper, gold, and silver, and avocado prices and sales volume, we underline the robustness of our curve estimate against variable transformation, further confirming its effective applicability for noisy and complex data sets. We believe that this monotone curve-fitting framework offers significant potential for numerous applications where monotonic relationships are intrinsic or need to be imposed.
Nam, Kyeongsik (KAIST): Optimal transport in free probability
We develop a free‐probabilistic analogue of optimal transport by replacing classical probability measures with non‐commutative laws. In particular, I will talk about the Monge–Kantorovich duality in this setting: A new class of non‐commutative convex functions, which characterize optimal couplings, will be introduced. Joint work with Wilfrid Gangbo, David Jekel and Dimitri Shlyakhtenko.
Seo, Seong-Mi (Chungnam National Univ.): Free energy expansions of 2D Coulomb gas ensembles
The free energy expansion of a Coulomb gas ensemble is one of the most fundamental topics in this area and has received considerable attention in recent years. The coefficients in the expansion not only contain electrostatic quantities but also provide topological and geometric information about the system. In this talk, I will present a precise free energy expansion for a class of 2D Coulomb gas models with determinantal or Pfaffian structures. Our results provide explicit coefficients beyond the entropy term, revealing their dependence on the topological properties of the system and verifying Zabrodin-Wiegmann conjecture regarding the spectral determinant emerging at the O(1) term in the free energy expansion. This talk is based on joint works with Sung-Soo Byun, Meng Yang, and Nam-Gyu Kang.
Tuesday Evening
Choi, Jae-Hwan (KIAS): Sobolev Regularity Theory for Time-Fractional Evolution Equations with Anisotropic Diffusion
In this talk, we introduce a Sobolev space framework for time-space nonlocal evolution equations arising from anomalous random walk. The nonlocality in time is captured by the Caputo fractional derivative, while the spatial operator is given by the infinitesimal generator of a vector of independent subordinate Brownian motions, leading to anisotropic diffusion. These equations naturally appear in various applied contexts, and our approach provides a rigorous analytic foundation for their study.
Lee, Jung-Kyoung (KIAS): Metastability of Langevin Dynamics and Its Application
Metastability is a phenomenon observed in various physical models. While it is an interesting topic in its own right, it can also be applied to various problems. In this talk, we will introduce the metastability of overdamped Langevin dynamics and present results related to the corresponding parabolic equation and slow mixing. This talk is based on joint with Claudio Landim(Instituto de Matemática Pura e Aplicada) and Insuk Seo(Seoul National University).
Wednesday Morning
Youn, Sang-Gyun (SNU): The Day-Dixmier similarity property for locally compact quantum groups
The celebrated work of Day and Dixmier in 1950 states that, if G is an amenable locally compact group (LCG), then every uniformly bounded representation of G is similar to a unitary representation. Furthermore, amenability implies the Day-Dixmier property with degree less than or equal to 2. This property has been studied in the broader category of locally compact quantum groups (LCQG). In particular, all Kac compact quantum groups and all Kac amenable discrete quantum groups are known to satisfy the Day-Dixmier property with degree less than or equal to 2. We propose a unified approach encompassing the known results by showing that the inner co-amenability of the dual of G implies the Day-Dixmier property with degree less than or equal to 2. Moreover, we prove that most of the non-Kac discrete quantum groups do not satisfy the Day-Dixmier property with degree less than or equal to 2. On the other hand, for discrete groups G, Pisier proved that the Day-Dixmier property with degree less than or equal to 2 implies amenability. We extend this conclusion to all discrete quantum groups.
Han, Kyung Hoon (The Univ. of Suwon): Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses
Entanglement has become indispensable in quantum information theory, and the Schmidt number of a bipartite state quantifies its degree of entanglement. The class of k-blockpositive matrices serves as entanglement witnesses and helps to determine Schmidt numbers. In this talk, we explore these concepts from a geometric perspective. We consider the compact convex set of all bipartite states of Schmidt number less than or equal to k, together with that of k-blockpositive matrices of trace one. The focus of the talk is to identify hyperplanes that support these convex sets and are perpendicular to a one-parameter family passing through the maximally mixed state. The results will be illustrated using one-parameter families that connect the maximally mixed state to either a general rank-one pure state or its partial transpose. When the Schmidt coefficients of the pure state are uniformly distributed, this construction yields isotropic states and Werner states, respectively. This talk is based on joint work with Seung-Hyeok Kye (SNU).
Ji, Un Cig (Chungbuk National Univ.): Quantum Dynamical Semigroups Induced by Gaussian Generalized Mehler Semigroups
In this talk, we first discuss the Gaussian generalized Mehler semigroups (on white noise functionals), along with their fundamental properties: invariant measures, invariant white noise operators, and mean ergodicity. Then, utilizing canonical topological isomorphisms between the spaces of two-variable white noise functionals and the spaces of white noise operators, we formulate a quantum analogue of the Gaussian generalized Mehler semigroups and discuss their fundamental properties. By establishing the notion of the positivity of white noise operators, we introduce the concept of quantum dynamical (Markov) semigroups acting on the space of white noise operators. Finally, we discuss the Gaussianity of the quantum dynamical semigroups induced by the Gaussian generalized Mehler semigroups. The construction of the quantum dynamical semigroups leads the general notion of quantum dynamical semigroups on generalized operators.