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.)

심사위원장 : 정 연 승, 심 사 위 원 : 김성호, 황강욱, 전현호, 이은지(충남대 통계학과)

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.

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온라인으로 진행예정

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.

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.

심사위원장 : 강 완 모, 심 사 위 원 : 황강욱, 김동환, 윤세영(AI대학원), 조경현(뉴욕대 전산학과)

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.

심사위원장 : 안드레아스 홈슨, 심 사 위 원 : 김동수, 엄상일, 김재훈, 마틴 지글러(전산학부)

We establish general central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018. In our new work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces. We will see how our techniques replace the classical thermodynamic formalism and allow us to provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. Joint work with I. Gekhtman and S. Taylor.

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심사위원장: 한상근, 심사위원: 김용정, 황강욱, 김동환, 김용대(전기및전자)

In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.

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To try and understand the group of symmetries of a surface, its mapping class group, it is useful to look at its action on the first homology of the surface. For finite-type surfaces this action is fairly well understood. I will recall what happens in this case, introduce infinite-type surfaces (surfaces whose fundamental group is not finitely generated) and discuss joint work with Sebastian Hensel and Nick Vlamis in which we describe the action on homology for these surfaces.

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In 1752, Euler first formulated the system of equations describing the dynamics of a perfect fluid. This system was complemented by Clausius in the 19th century, by introducing the concept of entropy of thermodynamics. This self-contained system is called compressible Euler system (CE). The most important feature of CE is the finite-time breakdown of smooth solutions, that is, the formation of shock as severe singularity due to irreversibility and discontinuity. Therefore, a fundamental question (since Riemann 1858) is on what happens after a shock occurs. This is the problem on well-posedness (that is, existence, uniqueness, stability) of weak solutions satisfying the 2nd law of thermodynamics, which is called entropy solution. This issue has been conjectured as follows: Well-posedness of entropy solutions for CE can be obtained in a class of vanishing viscosity limits of solutions to the Navier-Stokes system. This conjecture for the fundamental issue remains wide open even for the one-dimensional CE. My recent result (arXiv:1902.01792) provides a first answer to the conjecture in the case of the 1D isentropic CE starting from a shock. The proof crucially uses our new methodology (arXiv:1712.07348) to get the contraction of any large perturbations from viscous shock to the Navier-Stokes. This will be a main part of my talk.

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학생 참여 불가(Students cannot join this meeting.). 모든 참석자 마스크 착용 필수(All meeting participants should wear a face mask.)

There are two different but closely related perspectives in low dimensional topology. Both are motivated by the fact that it is often easier to understand manifolds when broken into smaller pieces. Given a closed 3-manifold, it is natural to ask which compact 4-manifolds can it bound. More concretely, one can ask whether it bounds a compact 4-manifold with simple homology. I will talk about some recent developments in this direction including joint work with Aceto and Celoria. Another perspective is to consider knots in a 3-manifold which arises as the boundary of a 4-manifold and ask what kind of surfaces can the knots bound in the 4-manifold. A commonly studied special case is the 3-sphere and the 4-ball. I will talk about a result joint with Hom and Kang where we study the complexity of disks embedded in the 4-ball.

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온라인 콜로퀴엄(Online Colloquium). 학생 참여 불가(Students cannot join this meeting.). 모든 참석자 마스크 착용 필수(All meeting participants should wear a face mask.)

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map.

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