Derived/spectral algebraic geometry is a relatively new area which features homotopy theory in algebraic geometry. I’ll take deformation theory and intersection theory to provide some flavor of these new fields. There are no prerequisites required other than ordinary algebraic geometry, so everyone is welcome to attend. (There are two lectures; I and II. This is the first of them.)

Modern deep learning (DL) algorithms rely extensively on large amounts of annotated data. Even when a large dataset is available, DL algorithms often fail miserably when deployed to settings with data characteristics significantly differing from those used for training. Domain adaptation (DA) and domain generalization (DG) algorithms aim to mitigate the gap between source (train) and target (test) distributions by learning domain-agnostic features or minimizing the discrepancy in the model’s predictions between the source and target distributions. This issue is prevalent in practical medical imaging settings, as the cost of obtaining both images and annotations is extremely expensive, limiting data accessibility to only a bulk of images collected from a few hospitals or detector devices, but a model must be suitable for multi-center, multi-device settings. In this seminar, we will cover existing literature on DA and DG, discussing their capabilities, assumptions, methodologies, along with their limitations. The session will conclude with research directions relevant to pragmatic industrial settings.

With the goal of reducing the number of annotated data necessary for current deep learning (DL) algorithms, semi-supervised learning (SSL) algorithms use unlabeled data which is vastly more accessible than their labeled counterpart to enhance the performance of deep neural networks (DNNs) when trained on a small number of labeled data. As an example, state-of-the-art SSL algorithms can achieve up to ~84% accuracy on the CIFAR10 dataset using 1 image per class, as long as the single image is of “prototypical” quality. This session will introduce common SSL settings considered in recent works and cover DL-based SSL algorithms in a chronological fashion. While existing SSL algorithms are mainly heuristics (they lack theoretical justifications), the intuition underlying such algorithms will also be discussed in relation to the merging consensus in DL-based generalization theory/studies.

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.

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Meeting ID: 914 3828 0517 Password: 633013

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)

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.

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

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.