# 세미나 및 콜로퀴엄

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Zoom Meeting
PDE 세미나
최범준 (Univ. of Toronto)
On classification of translating solitons to powers of Gauss curvature flow

Zoom Meeting

PDE 세미나

A classical result in Monge-Ampere equation states the paraboloids are the only convex entire solutions to $\det D^2 u = 1$. In this talk, we discuss a recent progress on the generalization of this classification in 2-dimension when the right-hand side is $(1+|Dx|^2)^{\beta}$. This corresponds to the classification of translating solitons to the flow by power of the Gauss curvature.
Our proof combines spectral analysis from the linear theory and the theory of Monge-Ampere equation. This is a joint work with Kyeongsu Choi and Soojung Kim.

Meeting ID: 914 3828 0517 Password: 633013

Meeting ID: 914 3828 0517 Password: 633013

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zoom online
정수론
김서영 박사 (Queen\'s University, Canada)
From the Birch and Swinnerton-Dyer conjecture to Nagao\'s conjecture

zoom online

정수론

Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant $\Delta_E$. For primes $p$ of good reduction, let $N_p$ be the number of points modulo $p$ and write $N_p=p+1-a_p$. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies
$$\lim_{x\to\infty}\frac{1}{\log x}\sum_{\substack{p\leq x\\ p\nmid \Delta_{E}}}\frac{a_p\log p}{p}=-r+\frac{1}{2},$$
where $r$ is the order of the zero of the $L$-function $L_{E}(s)$ of $E$ at $s=1$, which is predicted to be the Mordell-Weil rank of $E(\mathbb{Q})$. We show that if the above limit exits, then the limit equals $-r+1/2$. We also relate this to Nagao's conjecture. This is a recent joint work with M. Ram Murty.
(If you would like to join this online seminar, please email me (Bo-Hae Im) to get a link.)

We study probabilistic behaviors of elliptic curves with torsion points. First, we compute the probability for elliptic curves over the rationals with a non-trivial torsion subgroup $G$ whose size $\leq 4$ to satisfy a certain local condition.
We have a good interpretation of the probabilities we obtain, and for multiplicative reduction case, we have a heuristic to explain the probability. Furthermore, for $G=\mathbb{Z}/ 2\mathbb{Z} $ or $ \mathbb{Z} /2 \mathbb{Z} \times \mathbb{Z} /2 \mathbb{Z} $, we give an explicit upper bound of the $n$-th moment of analytic ranks of elliptic curves with a torsion subgroup $G$ for every positive integer $n$, and show that the probability for elliptic curves with a torsion group $G$ with a high analytic rank is small under GRH for elliptic $L$-function. From the results we have obtained, we conjecture that the condition of having the analytic rank $0$ or $1$ is independent of the condition of having the torsion subgroup $G= \mathbb{Z} /2 \mathbb{Z}$ or $ \mathbb{Z} /2 \mathbb{Z} \times \mathbb{Z} /2 \mathbb{Z}$.
(Send me(Bo-Hae Im) an email to get the Zoom link, if you would like to join this seminar.)

First talk: "Topics on graphons as limits of graph sequences I: Sampling"
In this penultimate talk of the Graphon Seminar, we investigate the method of sampling from a graph as a method of gathering information about very large, dense graphs.
We will talk about this method in the context of graphons and introduce the concept of a W-random graph for a graphon W.
This talk is based on chapter 10 of the book "Large networks and graph limits" by Lászlo Lovász.
Second talk: "Topics on graphons as limits of graph sequences II: Convergence of dense graph sequences"
In this final talk of the Graphon Seminar, we take a closer look at how graphons arise as the limit of convergent sequences of dense graphs.
This talk is based on chapter 11 of the book "Large networks and graph limits" by Lászlo Lovász.

온라인으로 진행예정

온라인으로 진행예정

In this penultimate talk of the Graphon Seminar, we investigate the method of sampling from a graph as a method of gathering information about very large, dense graphs.
We will talk about this method in the context of graphons and introduce the concept of a W-random graph for a graphon W.
This talk is based on chapter 10 of the book "Large networks and graph limits" by Lászlo Lovász.