Jihoon Park
Title : Embedding problem of RAAGs into combinatorial HHG
Abstract : Hierarchically Hyperbolic Space(HHS) is a large-scale geometric structure which utilize geometry of both mapping class groups of surfaces and CAT(0) cube complex. One main interest about HHS theory is to extend geometric, algebraic results of hyperbolic groups and MCG to the groups acting on HHS. In this talk, we focus on groups acting on a combinatorial HHS and its slice stabilizer property, and we will show that given a finite collection of unbounded domains of CHHS, there exists a collection of axial elements which generates a right-angled Artin subgroup of combinatorial HHG.
Kyungbae Park
Title : On lens spaces bounding small smooth 4-manifolds
Abstract : The main question addressed in this talk is: Which lens spaces can bound smooth 4-manifolds with second Betti number one under various topological conditions? Specifically, we show that there are infinite families of lens spaces that bound compact, simply-connected, smooth 4-manifolds with second Betti number one, yet cannot bound a 4-manifold consisting of a single 0-handle and 2-handle. Additionally, we establish the existence of infinite families of lens spaces that bound compact, smooth 4-manifolds with first Betti number zero and second Betti number one, but cannot bound simply-connected 4-manifolds with second Betti number one. The construction of such 4-manifolds with lens space boundaries is motivated by the study of rational homology projective planes with cyclic quotient singularities.
Sungmo Kang
Title : Introduction to Topological Data Analysis(TDA)
Abstract : I will give a talk about the introduction to Topological Data Analysis(TDA) from the unprofessional viewpoint. Topological Data Analysis is a rapidly growing field at the intersection of mathematics, computer science, and data science, offering powerful tools for analyzing complex datasets. Unlike traditional approaches that rely on metrics and geometry, TDA leverages concepts from algebraic topology to uncover the intrinsic shape and structure of data. The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types of data across many fields.
Bohyun Kwon
Title : Prime 3-bridge knot not satisfying the Casson-Gordon's rectangle condition
Abstract : The rectangle condition was introduced by Casson and Gorden to check the strong irreducibility of Heegaard splittings of 3-dimensional manifolds. The rectangle condition also can be defined on bridge decompositions of knots to check the strong irreducibility of the decompositions. If a Heegaard diagram of Heegaard splitting of a 3-manifold or a bridge decomposition diagram of a knot satisfies the rectangle condition then it would be strongly irreducible. However, we do not have enough information whether or not there exists a diagram satisfying the rectangle condition when a 3-manifold or a knot is arbitrary given. In this paper, we discuss an interesting method to trace a diagram for rectangle condition when an arbitrary 3-bridge decomposition diagram of a knot is given. Moreover, we give an example of 3-bridge knot which is prime but not satisfying the rectangle condition.(Joint work with Sungmo Kang and Jung Hoon Lee)
Hwa Jeong Lee
Title : Petal number of torus knots of type (r,r +2)
Abstract : A petal projection of a knot K is a projection of K with a single multi-crossing such that there are no nesting loops. The petal number, p(K), is the minimum number of loops among all petal projections of K, or equivalently, the minimum number of strands passing through the single multi-crossing. Let r be an odd integer, r ≥ 3. In this talk, we prove that the petal number of the torus knot of type (r,r +2) is equal to 2r + 3.
Sungjong No
Title : Stick Numbers of Knots and Spatial graphs
Abstract : In this talk, we explore the concept of 'stick number' in knot theory, which is a measure of the minimum number of straight line segments required to construct a knot. The stick number serves as an important indicator of the complexity and geometric properties of a knot. The focus will be on the upper bounds of stick numbers of general knots and spatial graphs. Additionally, we propose more efficient stick presentations for specific knot families(2-bridge knots and Montesinos knots) and use them to compute their upper bounds.
Hyungkee Yoo
Title : Introduction to quantum computation(6th Jan.)
Abstract : Quantum computation applies quantum mechanics to information processing, enabling new computational capabilities. This lecture introduces the mathematical framework of quantum computation, covering fundamental concepts such as qubits, superposition, and entanglement. It focuses on the structures and principles underlying quantum algorithms, providing a clear introduction for those new to the field.
Title : Decoding linear codes using Grover's algorithm(7th Jan.)
Abstract : In the coding theory, it is one of the most important research problems to find an efficient algorithm for the decoding process. In this talk, we find a decoding algorithm for binary linear codes using quantum circuits; this is the first time that quantum circuits are used as a main tool for decoding binary linear codes. Our decoding algorithm is mainly based on Grover's search algorithm. This is joint work with Professor Whan-Hyuk Choi (Kangwon National University) and Professor Yoonjin Lee (Ewha Womans University).
Minseo Lee
Title : Three-page indices of torus links
Abstract : An arc presentation of a link is an embedding into the open book decomposition of ℝ³ with a finite number of pages. An important rule of arc presentations is that different arcs must be placed on separate pages. In 1999, Dynnikov proposed a three-page presentation that bends this rule by restricting the total number of pages to three. Dynnikov showed that every link admits a three-page presentation. In this paper, we provide an alternative proof of this result. Also we define the three-page index α₃(L) of a link L that the minimum number of arcs needed to represent L in a three-page presentation. We examine three-page presentations for torus links, leading to the determination of the exact three-page indices for several torus links.
Seungeun Lee
Title : Monomer-dimer mixtures on a Honeycomb lattice and Symmetric Lozenge tilings
Abstract : The monomer-dimer model, originally introduced to describe diatomic molecules on crystal surfaces, has broad applications in statistical mechanics and combinatorics. In this talk, we investigate monomer-dimer mixtures on planar honeycomb lattices, with a particular focus on deriving the generating function that counts all possible mixtures with respect to monomer activity. We highlight the significance of the Hosoya index in this context and present a new expression for the matching polynomial on such lattices. Furthermore, we explore lozenge tilings of semiregular hexagons, which correspond to dimer packings on honeycomb graphs. The results include a detailed analysis of symmetric lozenge tilings of a symmetric hexagon with side lengths m, m, n, m, m, n, providing insights into the combinatorial structures relevant to graphene and topological insulators.