Month: March 2024

The 30th KMGS will be held on March 28th, Thursday, at Natural Science Building (E6-1) Room 1501.
We invite a speaker Dongha Lee from the Dept. of Mathematical Sciences, KAIST.
The abstract of the talk is as follows.

Slot (AM 11:50~PM 12:30)
[Speaker] 이동하 (Dongha Lee) from Dept. of Mathematical Sciences, KAIST, supervised by Prof. 박진성, Jinsung Park (KIAS) and Prof. 백형렬, Hyungryul Baik
[Title] The Renormalization of Volume and Chern-Simons Invariant for Hyperbolic 3-Manifolds
[Discipline] Differential Geometry
[Abstract]
For hyperbolic manifolds, many interesting results support a deep relationship between hyperbolic volume and the Chern-Simons invariant. In this talk, we consider noncompact hyperbolic 3-manifolds having infinite volume. For these manifolds, there is a well-defined invariant called the renormalized volume which replaces classical volume. The talk will start from a gentle introduction to hyperbolic geometry and reach the renormalization of the Chern-Simons invariant, which has a close relationship with the renormalized hyperbolic volume.
[Language] English

The 29th KMGS will be held on March 14th, Thursday, at Natural Science Building (E6-1) Room 1501.
We invite a speaker Hyunwoo Lee from the Dept. of Mathematical Sciences, KAIST.
The abstract of the talk is as follows.

[Speaker] 이현우 (Hyunwoo Lee) from Dept. of Mathematical Sciences, KAIST, supervised by Prof. 김재훈, Jaehoon Kim
[Title] Towards a high-dimensional Dirac’s theorem
[Discipline] Combinatorics
[Abstract]
Dirac’s theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles.
In this paper, we consider another natural generalization of the perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs.
We prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6},$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879… + o(1))n$ contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs.
We conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac’s theorem.
[Language] Korean