Thursday, May 27, 2021

2021-06-03 / 09:30 ~ 10:30

학과 세미나/콜로퀴엄 - 박사학위심사: 데이터 기반 시뮬레이션 모델링, 불확실성 정량화 그리고 최적화

by 김태호(KAIST)

심사위원장 김경국, 심사위원 황강욱, 전현호, 정연승, 송은혜(Department of Industrial & Manufacturing Engineering, Pennsylvania State University)

2021-05-27 / 14:00 ~ 15:00

학과 세미나/콜로퀴엄 - 박사학위심사: 랜덤 행렬의 합과 곱의 스펙트럼

by 지홍창(KAIST)

심사위원장 이지운, 심사위원 강완모, 황강욱, 윤세영(AI 대학원), 강남규(고등과학원)

2021-05-27 / 13:00 ~ 14:00

학과 세미나/콜로퀴엄 - 박사학위심사: 확률적 블록 모형의 스펙트럼에 관한 연구

by 양우석(KAIST)

심사위원장 이지운, 심사위원 강완모, 황강욱, 윤세영(AI 대학원), 강남규(고등과학원)

2021-06-01 / 16:30 ~ 17:30

학과 세미나/콜로퀴엄 - Discrete Math:

by 고두원()

Let $\mathbb{F}_q$ be a finite field of order $q$ which is a prime power. In the finite field setting, we say that a function $\phi\colon \mathbb{F}_q^d\times \mathbb{F}_q^d\to \mathbb{F}_q$ is a Mattila-Sjölin type function in $\mathbb{F}_q^d$ if for any $E\subset \mathbb{F}_q^d$ with $|E|\gg q^{\frac{d}{2}}$, we have $\phi(E, E)=\mathbb{F}_q$. The main purpose of this talk is to present the existence of such a function. More precisely, we will construct a concrete function $\phi: \mathbb{F}_q^4\times \mathbb{F}_q^4\to \mathbb{F}_q$ with $q\equiv 3 \mod{4}$ such that if $E\subset \mathbb F_q^4$ with $|E|>q^2,$ then $\phi(E,E)=\mathbb F_q$. This is a joint work with Daewoong Cheong, Thang Pham, and Chun-Yen Shen.

2021-05-28 / 10:00 ~ 12:00

학과 세미나/콜로퀴엄 - SAARC 세미나:

by ()

One famous conjecture in quantum chaos and random matrix theory is the so-called phase transition conjecture of random band matrices. It predicts that the eigenvectors' localization-delocalization transition occurs at some critical bandwidth $W_c(d)$, which depends on the dimension $d$. The well-known Anderson model and Anderson conjecture have a similar phenomenon. It is widely believed that $W_c(d)$ matches $1/\lambda_c(d)$ in the Anderson conjecture, where $\lambda_c(d)$ is the critical coupling constant. Furthermore, this random matrix eigenvector phase transition coincides with the local eigenvalue statistics phase transition, which matches the Bohigas-Giannoni-Schmit conjecture in quantum chaos theory. We proved the eigenvector's delocalization property for most of the general $d>=8$ random band matrix as long as the size of this random matrix does not grow faster than its bandwidth polynomially. In other words, as long as bandwidth $W$ is larger than $L^\epislon$ for some $\epislon>0$, and matrix size $L$. It is joint work with H.T. Yau (Harvard) and F. Yang (Upenn).

2021-06-03 / 10:00 ~ 11:00

학과 세미나/콜로퀴엄 - 정수론:

by 차병철()

Diophantine approximation is a branch of number theory where one studies approximation of irrational numbers by rationals and quality of such approximations. In this talk, we will consider intrinsic Diophantine approximation, which deals with approximating irrational points in a closed subset $X$ in $\mathbb{R}^n$ via rational points lying in $X$. First, we consider $X = S^1$, the unit circle in $\mathbb{R}^2$ centered at the origin. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Additionally, we present similar results for approximations of $S^1$ by a few different sets of rational points. This is joint work with Dong Han Kim (Dongguk University, Seoul). (Contact Bo-Hae Im if you plan to join the seminar.)

2021-06-03 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄: Infinite order rationally slice knots

by 박정환(KAIST)

A knot is a smooth embedding of an oriented circle into the three-sphere, and two knots are concordant if they cobound a smoothly embedded annulus in the three-sphere times the interval. Concordance gives an equivalence relation, and the set of equivalence classes forms a group called the concordance group. This group was introduced by Fox and Milnor in the 60's and has played an important role in the development of low-dimensional topology. In this talk, I will present some known results on the structure of the group. Also, I will talk about a knot that has infinite order in the concordance group, though it bounds a smoothly embedded disk in a rational homology ball. This is joint work with Jennifer Hom, Sungkyung Kang, and Matthew Stoffregen.

2021-05-27 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄: Bayesian statistical methods of analyzing complex count data with applications to microbiome study

by 이주희(UC 산타크루즈)

The rapid development of high-throughput sequencing technology in recent years is providing unprecedented opportunities to profile microbial communities from a variety of environments, but analysis of such multivariate taxon count data remains challenging. I present two flexible Bayesian methods to analyze complex count data with application to microbiome study. The first project is to develop a Bayesian sparse multivariate regression method that model the relationship between microbe abundance and environmental factors. We extend conventional nonlocal priors, and construct asymmetric non-local priors for regression coefficients to efficiently identify relevant covariates and their effect directions. The developed Bayesian sparse regression model is applied to analyze an ocean microbiome dataset collected over time to study the association of harmful algal bloom conditions with microbial communities. For the second project, we develop a Bayesian nonparametric regression model for count data with excess zeros. The approach provides straightforward community-level insights into how characteristics of microbial communities such as taxa richness and diversity are related to covariates. The baseline counts of taxa in samples are carefully constructed to obtain improved estimates of differential abundance. We apply the model to a chronic wound microbiome dataset, comparing the microbial communities present in chronic wounds versus in healthy skin.

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