Variable selection is an approach to identifying a set of covariates that are truly important to explain the feature of a response variable. It is closely connected or belongs to model selection approaches. This talk provides a gentle introduction to Bayesian variable selection methods. The basic notion of variable selection is introduced, followed by several Bayesian approaches with a simple application example.

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomassé, and Watrigant [FOCS 2020] defined the twin-width of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has maximum degree at most $k$. For any graph parameter $f$, we define the reduced-$f$ of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has $f$ at most $k$. Our focus is on graph classes with bounded reduced-bandwidth, which implies and is stronger than bounded twin-width (reduced-maximum-degree). We show that every proper minor-closed class has bounded reduced-bandwidth, which is qualitatively stronger than a result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least $2^{1000}$. We show that planar graphs have reduced-bandwidth at most $466$ and twin-width at most $583$; moreover, the witnessing reduction sequence can be constructed in polynomial time. We show that $d$-powers of graphs in a proper minor-closed class have bounded reduced-bandwidth (irrespective of the degree of the vertices). This is joint work with Édouard bonnet and David Wood.

This seminar has two purposes. One is to understand the nature of finance and the current international monetary order that contributed to the root cause of the global financial crisis. They are the financial system risk caused by maturity transformation, which is the core of the financial activity, the procyclicality of finance in which credit expands in an asset market boom and shrinks in a recession, and safe assets, etc. Another is to understand the ripple effect of the 21st century digital technology revolution. Just as cryptocurrencies cause conflicts with states monopolizing creating money, currencies based on digital platforms are causing conflicts with existing finance by shifting the core function of finance from intermediation to payment and settlement. Moreover, the democratization of finance thanks to the development of digital technology is another conflicting factor against the existing elite-centered finance, where the financial elite evaluates risks and receives rewards. Along with asset inflation, the pandemic has significantly increased debt on the one hand and accelerated the digitalization of finance on the other. The challenge in the post-pandemic era is to normalize increased debt and liquidity to a sustainable level while minimizing economic costs, and to come up with an alternative solution to the dilemma of the dollar-centered international monetary order. The first day of the seminar will focus on the attributes of finance, and the second day will discuss the challenges facing finance in the post-pandemic era and the digitalization of finance.

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2개 세션으로 나누어 진행

This seminar has two purposes. One is to understand the nature of finance and the current international monetary order that contributed to the root cause of the global financial crisis. They are the financial system risk caused by maturity transformation, which is the core of the financial activity, the procyclicality of finance in which credit expands in an asset market boom and shrinks in a recession, and safe assets, etc. Another is to understand the ripple effect of the 21st century digital technology revolution. Just as cryptocurrencies cause conflicts with states monopolizing creating money, currencies based on digital platforms are causing conflicts with existing finance by shifting the core function of finance from intermediation to payment and settlement. Moreover, the democratization of finance thanks to the development of digital technology is another conflicting factor against the existing elite-centered finance, where the financial elite evaluates risks and receives rewards. Along with asset inflation, the pandemic has significantly increased debt on the one hand and accelerated the digitalization of finance on the other. The challenge in the post-pandemic era is to normalize increased debt and liquidity to a sustainable level while minimizing economic costs, and to come up with an alternative solution to the dilemma of the dollar-centered international monetary order. The first day of the seminar will focus on the attributes of finance, and the second day will discuss the challenges facing finance in the post-pandemic era and the digitalization of finance.

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2개 세션으로 나누어 진행

본 발표에서는 다양한 기초 생명-의학 분야에서 생성되고 있는 오믹스 자료의 연구 개발 현황에 대해서 다룰 예정이다. 보다 큰 규모로, 보다 빠르게, 보다 정확하게, 보다 정밀하게 라는 궁극적인 목표하에 이뤄지고 있는 오믹스 자료의 진화에 발맞춰, 이를 분석하는 수리통계적 모형 역시 진화하고 있다. 그 중, 이번 발표에서는 미국의 초 대형 정밀의료 프로젝트인 TopMed에서 진행하고 있는 COPD에 관한 다중 오믹스 자료의 통합 분석 방법 및 결과에 대해서 자세히 다룰 예정이다. 아울러 정밀의료라는 목표를 달성하기 위해 반드시 모형에서 고려해야 하는 “환경 특이적 효과”에 대해 강연할 예정이다.

For given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a 'colorful' copy of $H$, i.e. a copy of $H$ containing at most one edge from each $G_i$? Keevash, Saks, Sudakov, and Verstraëte defined $\operatorname{ex}_k(n,H)$ to be the maximum total number of edges of the graphs $G_1,\dots, G_k$ on a common vertex set of size $n$ having no colorful copy of $H$. They completely determined $\operatorname{ex}_k(n,K_r)$ for large $n$ by showing that, depending on the value of $k$, one of the two natural constructions is always the extremal construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for $3$-color-critical graphs. We prove their conjecture for $4$-color-critical graphs and for almost all $r$-color-critical graphs when $r > 4$. Moreover, we show that for every non-color-critical non-bipartite graphs, none of the two natural constructions is extremal for certain values of $k$. This is a joint work with Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, and Hong Liu.

메타버스 웹3.0 NFT 등 미래를 이끌어갈 새로운 기술들이 개발되면서 많은 사람들의 관심을 끌고있다. 본 강연에서는 이러한 기술의 근간이 되는 블록체인의 기본 개념과 암호화폐 및 스마트 컨트랙트의 의미와 속성에 대하여 설명한다. 또한, 이러한 기술이 Defi로 대변되는 탈중앙화 금융에 어떻게 활용되는지 설명한다.

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2회에 걸처 세미나 진행

메타버스 웹3.0 NFT 등 미래를 이끌어갈 새로운 기술들이 개발되면서 많은 사람들의 관심을 끌고있다. 본 강연에서는 이러한 기술의 근간이 되는 블록체인의 기본 개념과 암호화폐 및 스마트 컨트랙트의 의미와 속성에 대하여 설명한다. 또한, 이러한 기술이 Defi로 대변되는 탈중앙화 금융에 어떻게 활용되는지 설명한다.

A geometric transversal to a family of convex sets in $\mathbb R^d$ is an affine flat that intersects the members of the family. While there exists a far-reaching theory concerning 0-dimensional transversals (intersection patterns of convex sets), much less is known when it comes to higher dimensional transversals. In this talk I will present some new and old results and problems regarding geometric transversals, based on joint work with Otfried Cheong and Xavier Goaoc.

The ions in a fully ionized electrostatic plasma are described by the Euler-Poisson system with the Boltzmann relation. We numerically observe the behavior of solutions to the 1D Euler-Poisson system for various initial data and how solitary waves and singularities are developed. We also introduce some results on the linear stability of solitary waves and the formation of singularities. These are joint works with Junho Choi and Bongsuk Kwon.

Inside living cells, chemical reactions form a large web of networks. Understanding the behavior of those complex reaction networks is an important and challenging problem. In many situations, it is hard to identify the details of the reactions, such as the reaction kinetics and parameter values. It would be good if we can clarify what we can say about the behavior of reaction systems, when we know the structure of reaction networks but reaction kinetics is unknown. In these talks, I plan to introduce two approaches in this spirit. Firstly, we will discuss the sensitivity analysis of reaction systems based on the structural information of reaction networks [1]. I will give an introduction to the method of identifying subnetworks inside which the effects of the perturbation of reaction parameters are confined. Secondly, I will introduce the reduction method that we developed recently [2]. In those two methods, an integer determined by the topology of a subnetwork, which we call an influence index, plays a crucial role. [1] T. Okada, A. Mochizuki, “Law of Localization in Chemical Reaction Networks,” Phys. Rev. Lett. 117, 048101 (2016); T. Okada, A. Mochizuki, “Sensitivity and network topology in chemical reaction systems,” Phys. Rev. E 96, 022322 (2017). [2] Y. Hirono, T. Okada, H. Miyazaki, Y. Hidaka, “Structural reduction of chemical reaction networks based on topology”, Phys. Rev. Research 3, 043123 (2021).

Abstract: Inside living cells, chemical reactions form a large web of networks. Understanding the behavior of those complex reaction networks is an important and challenging problem. In many situations, it is hard to identify the details of the reactions, such as the reaction kinetics and parameter values. It would be good if we can clarify what we can say about the behavior of reaction systems, when we know the structure of reaction networks but reaction kinetics is unknown. In these talks, I plan to introduce two approaches in this spirit. Firstly, we will discuss the sensitivity analysis of reaction systems based on the structural information of reaction networks [1]. I will give an introduction to the method of identifying subnetworks inside which the effects of the perturbation of reaction parameters are confined. Secondly, I will introduce the reduction method that we developed recently [2]. In those two methods, an integer determined by the topology of a subnetwork, which we call an influence index, plays a crucial role. References [1] T. Okada, A. Mochizuki, “Law of Localization in Chemical Reaction Networks,” Phys. Rev. Lett. 117, 048101 (2016); T. Okada, A. Mochizuki, “Sensitivity and network topology in chemical reaction systems,” Phys. Rev. E 96, 022322 (2017). [2] Y. Hirono, T. Okada, H. Miyazaki, Y. Hidaka, “Structural reduction of chemical reaction networks based on topology”, Phys. Rev. Research 3, 043123 (2021).

In adult tissues, stem cells undergo clonal competition because they proliferate while the stem cell niche provides limited space. This clonal competition allows the selection of healthy stem cells over time as unfit stem cells tend to lose from the competition. It could also be a mechanism to select a supercompetitor with tumorigenic mutations, which may lead to tumorigenesis. I am going to explain general concepts of clonal competition and how a simple model can explain the behaviour of adult stem cells. I will also show how geometric constraints affect the overall dynamics of stem cell competition.

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infinitely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $\frac{2}{d+2}$ for every $d\geq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $d\geq 3$, the facet hypergraph $\mathcal{F}(\mathbf{P})$ of a $(d+1)$-dimensional simplicial polytope $\mathbf{P}$ has the chromatic number $\chi(\mathcal{F}(\mathbf{P})) \in O(n^{\frac{\lceil d/2\rceil-1}{d}})$, where $n$ is the number of vertices of $\mathbf{P}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz. This is a joint work with Joseph Briggs and Michael Gene Dobbins.

In this talk, we prove a generalization of the del Pezzo-Bertini classification of varieties of minimal degree to higher secant varieties of minimal degree. It states that higher secant varieties of minimal degree are mostly divided into two classes: scroll type and Veronese type. Its proof is based on methods of gluing some 1-generic matrices. We also present some simple examples to explain our result. This is a joint work with Prof. Sijong Kwak.

Free-by -cyclic groups have been studied as algebraic counterparts of cusped hyperbolic mapping torus groups. Free-by-cyclic groups and cusped hyperbolic mapping torus groups share many algebraic properties. Nonetheless, free-by-cyclic groups are more complicated because not every free-by-cyclic group is realized as a cusped hyperbolic mapping torus group. In this talk, I explain some basic concepts and summarize some previous results related to free-by-cyclic groups. Also, I discuss some problems about free-by-cyclic groups.

The Yau-Zaslow formula describes the number of rational curves in a linear system on a smooth projective K3 surface in terms of a modular form. In this talk, I will review the Yau-Zaslow formula with some examples and then discuss an equivariant version of the formula for K3/abelian surfaces. When the K3/abelian surface admits a finite group G-action, we can consider a linear system with the induced action. It turns out that the equivariant version of the formula will count G-rational curves and it will also provide interesting modular forms.

심사위원장 : 김동환, 심사위원 : 강완모, 이창옥, 임미경, 윤세영(AI 대학원)

심사위원장 : 강완모, 심사위원 : 황강욱, 김동환, 윤세영(AI대학원), 전광성(Assistant Professor, Department of Computer Science, University of Arizona)

심사위원장 : 한상근, 심사위원 : 곽도영(명예교수), 황강욱, 강완모, 조현숙(이사장, 코드게이트 보안포럼)

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming space and symmetric groups decays exponentially as their centers drift apart. To verify condition (iii), we prove some deviation inequalities `on the slice’ for functions with Lipschitz conditions. We then use these estimates on intersection volumes to obtain a sharp lower bound on list-decodability of random q-ary codes, confirming a conjecture of Li and Wootters [IEEE Trans. Inf. Theory 2021]; and improve sphere-covering bound from the 70s on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy [IEEE Trans. Inf. Theory 2004]. Our probabilistic point of view also offers a unified framework to obtain improvements on other sphere-covering bounds, giving conceptually simple and calculation-free proofs for q-ary codes, permutation codes, and spherical codes. This is joint work with Jaehoon Kim and Hong Liu.

심사위원장 : 김용정, 심사위원 : 강문진, 김재경, 임미경, 안인경(고려대 세종캠퍼스 데이터계산학과)

심사위원장 : 배성한, 심사위원 : 김완수, 임보해, 임수봉(성균관대 수학교육학과), 최도훈(고려대 수학과)

Liouville quantum gravity (LQG) surfaces are random topological surfaces which are important in statistical mechanics and have deep connections to other mathematical objects such as Schramm–Loewner evolution and random planar maps. These random surfaces are too singular and fractal in the sense that the Hausdorff dimension, viewed as a metric space equipped with its intrinsic metric, is strictly bigger than two. I will talk about the interesting geometric structure and recent progress on LQG surfaces.

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(KAIST 입시일정과 겹쳐 1주 연기합니다)

하나금융 융합기술원은 국내 금융그룹 최초의 AI 연구소로 2018년부터 지난 4년 간 다양한 금융서비스에 현행 AI 응용기술들을 접목시키고 금융사 내 기술 전파에 큰 성과를 올려왔다. 그 중에서도 융합기술원이 연구/개발하는 신용평가 기술은 업계를 선도하고 있으며 그런 선도 기술을 만들어나가는 과정을 소개하려 한다. 또한, 응용기술 뿐만 아니라 향후 다양한 분야의 원천기술 연구를 위해 국내 유수 산업/학계 인재들이 모이는 조직으로 변형해가는 노력을 소개할 예정이다.

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온라인, 오프라인 동시진행

In this presentation, I will present me, Daeyeol Jeon, and Chang Heon Kim's construction of certain points on $X_1(N)$ over ring class fields (and therefore construction of points on the abelian varieties associated to newforms of level $\Gamma_1(N)$). Our work generalizes Bryan Birch's Heegner points on $X_0(N)$. Then, we show that these points form Euler systems (like the Heegner points), and we improve Kolyvagin's Euler system techniques to show that for our point $P_{\tau_K/c}$ and any ring class character $\chi$ of the extended ring class field of conductor $c$ satisfying $\chi=\overline{\chi}$, if $P_{\tau_K/c}^\chi$ is non-torsion and $G_K \to \operatorname{Aut} A_f[\pi]$ is surjective, then the corank of $\Sel(A_\chi/K)$ is 1, which implies the rank of $A_f(K)^\chi$ is 1. (Please contact Bo-Hae Im if you want to join the seminar.)

The independent domination number of a graph $G$, denoted $i(G)$, is the minimum size of an independent dominating set of $G$. In this talk, we prove a series of results regarding independent domination of graphs with bounded maximum degree. Let $G$ be a graph with maximum degree at most $k$ where $k \ge 1$. We prove that if $k = 4$, then $i(G) \le \frac{5}{9}|V(G)|$, which is tight. Generalizing this result and a result by Akbari et al., we suggest a conjecture on the upper bound of $i(G)$ for $k \ge 1$, which is tight if true. Let $G'$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq 3$. We prove that $i(G')\le \frac{k-1}{2k-1}|V(G')|$, which is tight for $k \in \{3, 4\}$, generalizing a result by Lam, Shiu, and Sun. This result also answers a question by Goddard et al. in the affirmative. In addition, we show that $\frac{i(G')}{\gamma(G')} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, \v Skrekovski, and Tepeh, where $\gamma(G')$ is the domination number of $G'$. Moreover, if we restrict $G'$ to be a cubic graph without $4$-cycles, then we prove that $i(G') \le \frac{4}{11}|V(G')|$, which improves a result by Abrishami and Henning. This talk is based on joint work with Ilkyoo Choi, Hyemin Kwon, and Boram Park.

심사위원장 : 폴정, 심사위원 : 이지운, 남경식, 양홍석(전산학부), Greg Markowsky(Senior Lecturer, School of Mathematics, Monash University)

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model to describe the dynamics of Newtonian stars is given by the gravitational Euler-Poisson system, which admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In particular, using numerics, the Euler-Poisson system in the super-critical regime has been widely used inastrophysics literature todescribe the gravitational collapse, but its rigorous proof has been established only recently. The main challenge comes from thepressure, which actsagainstgravitational force. In this talk, I will discuss some recent progress on Newtonian dust-like collapse and self-similar collapse.

This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.

This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.