KAIST Math Graduate Student Seminar (KMGS)

The 33rd KMGS will be held on May 2nd, Thursday, at Natural Science Building (E6-1) Room 1501.
We invite a speaker Eunchan Jeon from the Dept. of Mathematical Sciences, KAIST.
The abstract of the talk is as follows.

Slot (AM 11:50~PM 12:30)
[Speaker] 전은찬 (Eunchan Jeon) from Dept. of Mathematical Sciences, KAIST, supervised by Prof. 변재형, Jaeyoung Byeon
[Title] The Hardy type inequality on the bounded domain with mean zero condition
[Discipline] Analysis, PDE
[Abstract]
This talk aims to consider the attainability of the Hardy-type inequality in the bounded smooth domain with average-zero type constraint. Since the criteria of the attainability depends to the concentration-compactness type arguments, we will briefly introduce the results for some classical Hardy-type inequalities and the concentration-compactness arguments. Subsequently, we propose new function spaces that well define the new inequalities. Finally, we will discuss the attainability of the optimal constant of the inequality in the general smooth domain.
[Language] Korean

The 32nd KMGS will be held on April 25th, Thursday, at Natural Science Building (E6-1) Room 1501.
We invite a speaker Sungho Han from the Dept. of Mathematical Sciences, KAIST.
The abstract of the talk is as follows.

Slot (AM 11:50~PM 12:30)
[Speaker] 한성호 (Sungho Han) from Dept. of Mathematical Sciences, KAIST, supervised by Prof. 강문진, Moon-Jin Kang
[Title] Long-time behavior of viscous-dispersive shock for the Navier-Stokes-Korteweg equations
[Discipline] Analysis
[Abstract]
We consider the Naiver-Stokes-Korteweg(NSK) equations for the dynamics of compressible barotropic viscous fluids with internal capillarity. We handle the time-asymptotic stability in 1D of the viscous-dispersive shock wave that is a traveling wave solution to NSK as a viscous-dispersive counterpart of a Riemann shock. More precisely, we prove that when the prescribed far-field states of NSK are connected by a single Hugoniot curve, then solutions of NSK tend to the viscous-dispersive shock wave as time goes to infinity. To obtain the convergence, we extend the theory of a-contraction with shifts, used for the Navier-Stokes equations, to the NSK system. The main difficulty in analysis for NSK is due to the third-order derivative terms of the specific volume in the momentum equation. To resolve the problem, we introduce an auxiliary variable that is equivalent to the derivative of the specific volume.
[Language] Korean but English if it is requested

The 31st KMGS will be held on April 4th, Thursday, at Natural Science Building (E6-1) Room 1501.
We invite a speaker Lucas MacQuarrie from the Dept. of Mathematical Sciences, KAIST.
The abstract of the talk is as follows.

Slot (AM 11:50~PM 12:30)
[Speaker] Lucas MacQuarrie from Dept. of Mathematical Sciences, KAIST, supervised by Prof. 김재경, Jaekyoung Kim
[Title] The Asymptotic Iteration Method and Hanh Difference Equations
[Discipline] Analysis (Differential and Difference Equations)
[Abstract]
The Asymptotic Iteration Method (AIM) is a relatively unknown method for solving second order linear homogeneous ordinary differential equations analytically and numerically computing eigenvalues. With a little work, The AIM theory can be extended to difference equations and $q$-differential equations, which can be combined into a generalized operator called the Hanh operator. In this talk, I will introduce the AIM theory and its use cases as well as show how it extends to the previously mentioned operators. I will also discuss some of the theory around these operators, their use cases, and further work that can be done on AIM.
[Language] English

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