In this talk, I will talk about my experience of being a postdoc in Korea. In addition, I will discuss how the job process in Korea is going and share some useful advice to increase the possibility of getting a tenure-track job even though I have not made it yet.

이 발표에서는 학술적 성취를 효과적으로 표현하고 홍보하기 위한 방법들을 소개합니다. 학술적 웹사이트 생성 방법, CV와 Research statement 작성 및 유의점, 학회에서의 네트워킹 및 학술적 발표 구성 방법 등을 다룹니다.

In this talk, I will introduce fundamental theorems in convexity problems of discrete geometry, including Radon's theorem, Tverberg's theorem, Carathéodory's theorem along Helly's theorem along with their colorful versions. I will cover some proofs and techniques such as linear dependency, Sarkaria's tensor product method, configuration space/test map scheme and discrete Morse theory. I will also discuss some transversal theorems, including our own result and the ideas of proof, if time permits.

It is well-known that a tournament always contains a directed Hamilton path. Rosenfeld conjectured that if a tournament is sufficiently large, it contains a Hamilton path of any given orientation. This conjecture was approved by Thomason, and Havet and Thomassé completely resolved it by showing there are exactly three exceptions.

We generalized this result into a transversal setting. Let $\mathbf{T} = \\{T_1,\dots,T_{n-1}\\}$ be a collection of tournaments on a common vertex set $V$ of size $n$. We showed that if $n$ is sufficiently large, there is a Hamilton path on $V$ of any given orientation which is obtained by collecting exactly one arc from each $T_i$. Such a path is said to be \emph{transversal}.

It is also a folklore that a strongly connected tournament always contains a directed Hamilton cycle. Rosenfeld made a conjecture for arbitrarily oriented Hamilton cycles in tournaments as well, which was approved by Thomason (for sufficiently large tournaments) and Zein (by specifying all the exceptions). We also showed a transversal version of this result. Together with the aforementioned result, it extends our previous research, which is on transversal generalizations of existence of directed paths and cycles in tournaments.

This is a joint work with Debsoumya Chakraborti, Jaehoon Kim, and Hyunwoo Lee.

A given graph is called Toeplitz if it has the Toeplitz matrix as its adjacency matrix. I proposed a problem that how can we determine a given graph is a Toeplitz graph or not. For this purpose, I give a talk about Toeplitz graphs that might be related to this problem. Some simple observations and some previous results about Toeplitz graphs will be covered.