부산대학교 기하학, 위상수학 세미나
(PNU Geometry and Topology seminar)

Title. An introduction to the minimal surface system

Abstract. Minimal surface system is a quasilinear PDE system whose graph describes a geometric object called a minimal submanifold. This PDE system is a generalization of the classical minimal surface equation to the higher codimension setting. For this PDE system, one can ask about its existence, uniqueness, stability (area-minimizing) and regularity problem. In this talk, I will introduce a historical work of minimal surface system by Lawson-Osserman and compare it to the classical hypersurface setting. Then I will mention more recent works by Lee, Wang, Tsui and also a joint-work of mine on the uniqueness problem with Yng-Ing Lee and Mao-Pei Tsui.

Title. Variation formulas in minimal submanifolds and their related results

Abstract. In this talk, we study variation formulas in minimal submanifolds and their related results. More specifically, we define the first variation formula of the area functional and show that a minimal submanifold satisfies the Euler-Lagrange equation. And then we define the second variation formula for a given minimal submanifold and a concept of stability. In this setting, we introduce several important results for stable minimal submanifolds in Euclidean space. Finally, we are going to generalize these results to translators of MCF.

Title. A few approaches to the Fraser-Li conjecture

Abstract. In this talk we introduce the well known Fraser-Li conjecture and look into historical remarks and different approaches to get close to the conjecture.

Title. A quick tour to the vanishing garden (theorem) on complex manifolds

Abstract. I will motivate the study of complex manifolds via the embedding problem, which leads us to the construction of holomorphic functions, or, more generally, sections of holomorphic line bundles on a complex manifold. We will see how Čech and Dolbeault cohomologies come into play and how the vanishing of cohomology groups guarantees the existence of the desired holomorphic sections. The goal is to introduce the celebrated Kodaira vanishing theorem. If time permits, we will also see the Nadel (or equivalently, Kawamata-Viehweg) vanishing theorem.

Title. Knot Concordance and H-cobordism

Abstract. The knot concordance group has played a central role in low dimensional topology since its introduction by Fox and Milnor in the 1960’s. However, a little was known about the H~-cobordism group, which was introduced by Kawauchi in 1976. In this talk, I will present known results about the structure of the knot concordance group, and the relation between two groups.

Title. Asymptotic homology of graph braid groups

Abstract. We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.

Title. How to define knot invariants from Gauss diagrams

Abstract. One-dimensional manifolds (knots) in a three-dimensional space have been distinguished through various invariants. In this talk, I would like to introduce the invariants defined from Gauss diagrams. One of famous invariants is the intersection index defined by Henrich [2], Im, Lee and Lee [4]. Some of its applications are Manturovs index parity [5], Chengs transcendental function [1], and Nikonovs a various of invariants [6]. Prof. Im and I [3] also investigated the applications of the intersection index via the Gauss diagram. The talk will show you how to define these invariants via knot diagrams and Gauss diagrams. And I will introduce new invariants for knot diagrams on the cylinder: a group presentations and the enhanced winding index through its Gauss diagram.

[1] Z. Cheng, A transcendental function invariant of virtual knots, J. Math. Soc. Japan, 69 (4) (2017) 1583-1599.
[2] A. Henrich, A sequence of degree one vassiliev invariants for virtual knots, J. Knot Theory Ramifications, 19 (4) (2010) 461-487.
[3] Y.H. Im, S. Kim, A sequence of polynomial invariants for Gauss diagrams, J. Knot Theory Ramifications 26 (7) (2017) 1750039.
[4] Y.H. Im, K. Lee, S.Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications 19 (2010) 709-725.
[5] V. O. Manturov. Parity in knot theory, Mat. Sb. 201(5) (2010), 65-110
[6] I. Nikonov, Intersection formulas for parities on virtual knots, Arxiv:submit/3980787

Title. Limits of Bergman kernels on a tower of coverings of compact K¨ahler manifolds

Abstract. Click here

Title. Classifications of self-shrinkers under some conditions

Abstract. Self-shrinkers arise as special solutions of the mean curvature flow that preserves the shape of the evolving submanifold. In 1990 and 1993, Huisken classified mean convex (i.e., $H \geq 0$) self-shrinkers with a priori bound on the second fundamental form $|A|$. We first show that Huisken's classification holds even without the $|A|$ bound. Secondly, we will characterize $\mathcal{F}$-stable self-shrinkers where $\mathcal{F}$ is a weighted functional. Furthermore,