Abstracts

 Khovanov homology, as a categorification of the Jones polynomial, is a powerful invariant of knots and links. Various attempts have been made to establish geometric realizations of Khovanov homology. Ultimately, Lipshitz and Sarkar constructed spectra \( \mathcal{X}_{Kh}^{j}(L) \) whose reduced singular cohomology is isomorphic to the Khovanov homology of a link $L.$ Meanwhile, Gonz\'alez-Meneses, Manch\'on, and Silvero showed that the (hypothetical) extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex of its Lando graph. In this talk, we construct explicit geometric realizations of the real-extreme Khovanov homology of certain families of links, including pretzel links and torus links. This is joint work with Mark H. Siggers, Jinseok Oh, and Hongdae Yun.

 Yang-Baxter Operators are bilinear endomorphisms from the tensor product of a vector space to itself which satisfy the Yang-Baxter Equation. They can be seen to generalize biquandles, and thus are of general interest to knot theorists, though the particular class of Yang-Baxter Operators studied here ($R_{(m)}$) were of particular interest to Vaughan Jones and are related to the famous Jones polynomial. Following a standard technique, these Yang-Baxter Operators yield a homology theory ($H_n(R_{(m)})$) based on that of pre-cubic modules. 

In this talk, we present a general result for $H_n(R_{(m)})$ that depends only on the explicit computation of a finite number of initial conditions. We then produce explicit formulas for the third and fourth homology by computing the requisite initial conditions.

 In 1992, Thomas Bier introduced a combinatorial construction that yields many simplicial $(m-2)$-dimensional $PL$-spheres on $2m$ vertices. The study of full subcomplexes of a simplicial complex is important for understanding the structure of simplicial complexes, or its associated topological spaces. In this talk, we will discuss the homological types of full subcomplexes of Bier spheres.

 In this talk, we introduce the rational tangle theory which is related to the knot theory. The knot theory is one of popular research areas in studying Topology. It is related to the classification of closed 3-dimensional manifold. One of the methods to classify knots is n-bridge decomposition of knots. We cut a knot in S^3 into two collections of n arcs by a sphere. If the collections of the arcs satisfies a certain condition, then we call it rational n-tangle. So, by gluing two rational n-tangle, we can construct a knot. There are many application problems related to this tangle theory and we would introduce some of them.

 Ideal triangulation is a useful tool to study 3-manifolds, translating topological information to combinatorial data. It allows us to compute some 3-manifold invariants (e.g, the volume) efficiently, but some intriguing invariants (e.g. the Alexander polynomial and the Jones polynomial) are not included in the list. In this talk, I would like to explain how the Alexander polynomial as well as twisted ones are related to Neumann-Zagier matrices, encoding combinatorial data of ideal triangulations. This is joint work with Stavros Garoufalidis.

 A Bott tower is a sequence of projective bundles starting from a point, and its total space is known as a Bott manifold. In this talk, we present a classification theorem stating that two Bott manifolds are diffeomorphic if and only if their integral cohomology rings are isomorphic as graded rings. This result provides strong supporting evidence for the cohomological rigidity conjecture for toric manifolds, a central problem in toric topology. This talk is based on joint work with Taekgyu Hwang and Hyeontae Jang. 

 Toric varieties are classical objects in algebraic geometry, widely used as testing grounds for new ideas, since their topological structure can be explicitly described by the corresponding fans. However, the real loci of toric varieties, known as real toric varieties, are much more difficult to study, as their structure differs significantly from the complex case. Thus, finding families of real toric varieties with well-behaved topological and combinatorial properties is an important direction of research. In this talk, we introduce real toric varieties associated with graphs. We examine how their Betti numbers are determined by the combinatorial structure of the graphs and explore several notable properties of these Betti numbers.

 Dehn surgery is a central operation in 3-manifold topology, offering a powerful method for constructing and classifying 3-manifolds. By removing a tubular neighborhood of a knot or link in a 3-manifold and gluing it back in a different way, one can generate a wide array of new manifolds. This process not only lies at the heart of the celebrated Lickorish-Wallace theorem — asserting that every closed, orientable 3-manifold can be obtained from the 3-sphere via Dehn surgery on a link — but also plays a key role in the study of knot complements, contact structures, and hyperbolic geometry. In this talk, we will introduce the basics of Dehn surgery, explore illustrative examples, and discuss its applications in both classical and modern 3-manifold theory, including connections to the geometrization conjecture and the role of exceptional surgeries.

 It is well-known that the rational cohomology ring of a toric variety with orbifold singularities, say toric orbifolds, behaves similarly to the integral cohomology ring of smooth toric varieties. However, the information about the integral cohomology ring of a (underlying topological spaces of) toric orbifold or a singular toric variety in general is somewhat restrictive compared to the smooth case. In this talk, we consider real 4-dimensional toric varieties, which have at worst orbifold singularities. The main result determines their integral cohomology ring structure in terms of “bases” and “relations”, which can be easily read off from the underlying combinatorial data. This is a joint work with X. Fu and T. So.

 Matsumoto-Montesinos showed that there is a one to one correspondence, up to equivalence, between pseudo-periodic maps on Riemann surfaces and surface fibrations over disks with one singular fiber (called the degeneration of Riemann surfaces), whose each monodromy is the corresponding pseudo-periodic map. If we factorize such map into the product of positive Dehn twists, there corresponds to the Lefschetz fibration over 2-disk, but the total space may not be diffeomorphic.

 In this talk, I introduce my research to find out the factorization of two maps of order 3 on genus 4 surface into positive Dehn twists such that the total space of the corresponding Lefschetz fibration is diffeomorphic to the degeneration of Riemann surfaces.

In this talk, we discuss lens spaces that bound smooth 4-manifolds with second Betti number one under various conditions. In particular, we show that there are infinite families of lens spaces that bound smooth, compact, simply-connected 4-manifolds with second Betti number one but cannot bound a 4-manifold with a single 0- and 2-handle. (This is joint work with Jongil Park and Kyungbae Park.)

The study of smooth 4-manifolds occupies a central place in low-dimensional topology, exhibiting phenomena that do not appear in any other dimension. In this talk, I will provide an introduction to the topology of 4-manifolds. We will discuss basic tools such as handle decompositions, intersection forms, and Kirby calculus, and explore how these techniques reveal the subtle differences between smooth and topological categories in dimension four. 

Spatial graph theory is an area of knot theory which has a close connection with molecular biology and chemistry. Spatial graph theory is the study of graphs embedded in S^3. 

Most of the work in this area is rooted in the result of Conway and Gordon’s intrinsic properties of graphs. In this talk, I introduce intrinsic properties and their result.

Topological Data Analysis (TDA), particularly persistent homology, provides a powerful framework for extracting intrinsic geometric and topological features from complex data. In this talk, we present a novel application of TDA to object detective problems, focusing on the analysis of Ground Penetrating Radar (GPR) data. We propose a shape-aware topological representation derived from B-scan GPR images, where persistent homology captures essential structures of buried objects. These topological features are integrated into a deep neural network architecture for object detection, demonstrating how topological invariants can enhance robustness and generalization under domain shifts.To address the scarcity of real-world labeled data, we employ a Sim2Real strategy by generating synthetic GPR datasets and transferring topological representations to real data. Our experimental results show that TDA-guided feature fusion improves detection accuracy and geometric interpretability.

 Topological Data Analysis (TDA) has proven invaluable for revealing hidden structures in large or high-dimensional data, yet its application often faces computational bottlenecks. In 2025, Choi et al. introduced the Characteristic Lattice Algorithm (CLA), a method that reduces dataset size while maintaining geometric and topological features. However, CLA struggles to handle datasets with noise that is widely dispersed.

 To overcome this limitation, we propose the Refined Characteristic Lattice Algorithm (RCLA) and present analysis results obtained with it. This is joint work with Seonmi Choi, Jeong Rye Park, Seung Yeop Yang.