We introduce elliptic curves, their Mordell-Weil group structure, and isogenies over number fields. At the last of the talk, some results on the torsion subgroups of Mordell-Weil groups of elliptic curves defined over a number field will be given. The results are joint works with my advisor Bo-Hae Im.

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Discipline of talk : Number Theory / Advisor: 임보해(Bo-Hae Im)

To understand nonparametric regression, we should know first what the parametric model is. Simply speaking, the parametric regression model consists of many assumptions and the nonparametric regression model eases the assumptions. I will introduce what assumptions the parametric regression model has and how the nonparametric regression model relieves them. In addition, their pros and cons will be also presented.

TBA

TBD

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

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심사위원장: 엄상일, 심사위원 : 안드레아스 홈슨, 김재훈, 권오정(한양대학교), 오은진(POSTECH)

A graph class $\mathcal{G}$ has the strong Erdős-Hajnal property (SEH-property) if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq \left\lfloor c|V(G)| \right\rfloor$. We prove that the class of chordal graphs satisfies SEH-property with constant $c = 2/9$. On the other hand, a strengthening of SEH-property which we call the colorful Erdős-Hajnal property was discussed in geometric settings by Alon et al.(2005) and by Fox et al.(2012). Inspired by their results, we show that for every pair $F_1, F_2$ of subtree families of the same size in a tree $T$ with $k$ leaves, there exist subfamilies $F'_1 \subseteq F_1$ and $F'_2 \subseteq F_2$ of size $\theta \left( \frac{\ln k}{k} \left| F_1 \right|\right)$ such that either every pair of representatives from distinct subfamilies intersect or every such pair do not intersect. Our results are asymptotically optimal. Joint work with Andreas Holmsen, Jinha Kim and Minki Kim.

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심사위원장 : 김재경 / 심사위원 : 김용정, 정연승, 김진수(POSTECH), 이승규(고려대학교)

TBA

TBA

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree more than the number of atoms in the universe ($\delta(G)\ge 10^{10^{10}}$), then it contains a pillar, which is a graph that consists of two vertex-disjoint cycles of the same length, $s$ say, along with $s$ vertex-disjoint paths of the same length which connect matching vertices in order around the cycles. Despite the simplicity of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In this talk, we will try to give an idea of the tools used in the proof of this conjecture, which consists of building a pillar (algorithmically) in sublinear expanders. This is joint work with Hong Liu.

Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph $G$, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. In 2008, Butler, Hajiaghayi, Kleinberg, and Leighton asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively, showing that for sufficiently large $n$, $HG(K_{n,n})\ge n^{0.5-o(1)}$. Our guessing strategy is based on some ideas from coding theory and probabilistic method. Based on a joint work with Noga Alon, Omri Ben-Eliezer, and Itzhak Tamo.

The well-known 1-2-3 Conjecture by Karoński, Łuczak and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1, 2 or 3 so that for each edge $uv$ the sums of the weights at $u$ and at $v$ are distinct. The list version of the 1-2-3 Conjecture by Bartnicki, Grytczuk and Niwczyk states that the same holds if each edge $e$ has the choice of weights not necessarily from $\{1,2,3\}$, but from any set $\{x(e),y(e),z(e)\}$ of three real numbers. The goal of this talk is to survey developments on the 1-2-3 Conjecture, especially on the list version of the 1-2-3 Conjecture.

In a rainbow variant of the Turán problem, we consider $k$ graphs on the same set of vertices and want to determine the smallest possible number of edges in each graph, which guarantees the existence of a copy of a given graph $F$ containing at most one edge from each graph. In other words, we treat each of the $k$ graphs as a graph in one of the $k$ colors and consider how many edges in each color force a rainbow copy of a given graph $F$. In the talk, we will describe known results on the topic, as well as present recent developments, obtained jointly with Sebastian Babiński and Magdalen Prorok, solving the rainbow Turán problem for a path on 4 vertices and a directed triangle with any number of colors.