(전체일정: 7/28, 7/29, 8/3, 8/5) In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang. Lec 2: Phase transition and the shape theorem

Analyzing group behavior of systems of interacting variables is a ubiquitous problem in many fields including probability, combinatorics, and dynamical systems. This problem also naturally arises when one tries to learn essential features (dictionary atoms) from large and structured data such as networks. For instance, independently sampling some number of nodes in a sparse network hardly detects any edges between adjacent nodes. Instead, we may perform a random walk on the space of connected subgraphs, which will produce more meaningful but correlated samples. As classical results in probability were first developed for independent variables and then gradually generalized for dependent variables, many algorithms in machine learning first developed for independent data samples now need to be extended to correlated data samples. In this talk, we discuss some new results that accomplish this including some for online nonnegative matrix and tensor factorization for Markovian data. A unifying technique for handling dependence in data samples we develop is to condition on the distant past, rather than the recent history. As an application, we present a new approach for learning "basis subgraphs" from network data, that can be used for network denoising and edge inference tasks. We illustrate our method using several synthetic network models as well as Facebook, arXiv, and protein-protein interaction networks, that achieve state-of-the-art performance for such network tasks when compared to several recent methods.

Certain zero range processes on a finite set exhibits metastability. Most of the time nearly all particles of the zero range process are at one single site, and the site of condensate asymptotically behaves as a Markov chain. In this talk, we prove the metastability of zero range processes on a finite set with an approach using the Poisson equation. This approach doesn't need precise estimates of capacities and can be applied for both reversible and non-reversible cases. This talk is based on the joint work with F. Rezakhanlou.

(전체일정: 7/28, 7/29, 8/3, 8/5) In 2d first-passage percolation, we let (t_e) be a family of i.i.d. weights associated to the edges of the square lattice, and let T = T(x,y) be the induced weighted graph metric on Z^2. If we let p be the probability that a weight takes the value 0, then there is a transition in the large-scale behavior of T depending on the value of p. Specifically, when p < 1/2, T grows linearly, but when p > 1/2, T is stochastically bounded. In these lectures, I will describe some of the standard questions of FPP in the case p < 1/2, and then focus on the "critical" case, where p = 1/2. Regarding this case, I will show some of my work over the last few years including exact asymptotics for T, universality results, and recent work on a dynamical version of the model. The work I will present was done in collaboration with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang. Lec 1: Introduction and the passage time to infinity

Most organisms exhibit various endogenous oscillating behaviors, which provides crucial information about how the internal biochemical processes are connected and regulated. Along with physical experiments, studying such periodicity of organisms often utilizes computer experiments relying on ordinary differential equations (ODE) because configuring the internal processes is difficult. Simultaneously utilizing both experiments, however, poses a significant statistical challenge due to its ill behavior in high dimension, identifiability, and numerical instability. This article devises a new Bayesian calibration strategy for oscillating biochemical models. The proposed methodology can efficiently estimate the computer experiments’ (ODE) parameters that match the physical experiments. The proposed framework is illustrated with circadian oscillations observed in a model filamentous fungus, Neurospora crassa.

TBA

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This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)

(전체일정: 7/26, 7/27, 8/2, 8/4) Lec 2: Introduction to cut-off phenomenon In this second lecture, we introduce the so-called cut-off phenomenon and look at several examples exhibiting this phenomenon. Then, we rigorously prove the cut-off phenomenon for the simplest possible model, the lazy random walk on hypergraphs.

Lec 1: Introduction to Markov chain mixing theory In this lecture, we introduce the Markov chain mixing theory by assuming that the attendances have no background on this field. The precise definition of the mixing time and several basic techniques estimating mixing time will be discussed.

In this talk, we will introduce some applications of currents. Thanks to results of Demailly--Paun or Collins--Tosatti, we have a decent understanding of how (cohomology class of) currents are found to be nef and big, the positivity notions that succeed Kahler-ness. With the K3 surface example, where its intersection theory is well-known, we then sketch on how all these machinery are applied to give some result (of Filip--Tosatti) on dynamical rigidity on K3 surfaces.

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Zoom 회의 참가 https://zoom.us/j/92837213232?pwd=Y1RTODFITGFHQWNRbTRTeTlNakZsQT09 회의 ID: 928 3721 3232 암호: barH1Y

In this talk, we will introduce the notion of currents, a generalization of differential forms, allowing distributions (dual of smooth compactly supported functions) as coefficients. With this, we introduce their examples, derivatives, cohomology, "de Rham" type groups, and how they interplay with divisors of complex varieties.

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Zoom 회의 참가 https://zoom.us/j/92837213232?pwd=Y1RTODFITGFHQWNRbTRTeTlNakZsQT09 회의 ID: 928 3721 3232 암호: barH1Y

The studies on the fibers of the Hitchin map are equivalent to those on spectral data for Higgs bundles. In this talk, I will introduce spectral data for SL(2, C)- Higgs bundles over a smooth curve and then discuss how to describe spectral data for SL(2, C)-Higgs bundles over an irreducible nodal curve.

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Zoom ID: 352 730 6970, PW: 9999.

Organisms have evolved an internal biological clock which allows them to temporally regulate and organize their physiological and behavioral responses to cope in an optimal way with the fundamentally periodic nature of the environment. It is now well established that the molecular genetics of such rhythms within the cell consist of interwoven transcriptional-translational feedback loops involving about 15 clock genes, which generate circa 24-h oscillations in many cellular functions at cell population or whole organism levels. We will present statistical methods and modelling approaches that address newly emerging large circadian data sets, namely spatio-temporal gene expression in SCN neurons and rest-activity actigraph data obtained from non-invasive e-monitoring, both of which provide unique opportunities for furthering progress in understanding the synchronicity of circadian pacemaking and address implications for monitoring patients in chronotherapeutic healthcare.

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This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)

TBA

MUltiple SIgnal Classification (MUSIC) is a well-known, non-iterative imaging technique in inverse scattering problem. Throughout various researches, it has been confirmed that MUSIC is very fast, effective, and stable. Due to this reason MUSIC has been applied to various inverse scattering problems however, it has not yet been designed and used to identify unknown anomalies from measured scattering parameters (S-parameters) in microwave imaging. In this presentation, we apply MUSIC in microwave imaging for a fast identification of arbitrary shaped anomalies from real-data and establish a mathematical theory for illustrating the feasibilities and limitations of MUSIC. Simulations results with real-data are shown for supporting established theoretical results. Meeting ID: 873 9069 4743 Passcode: 728543

Gromov-Witten invariants are some rational numbers roughly counting curves inside a Calabi-Yau manifold. These numbers have some recurvsive structure on the genus of the curve. I will explain how to study this recursive structure throughholomorphic anomaly equation. For a semi-simple Gromov-Witten theory, I will explain the method of proof of holomorphic anomaly equation for several examples using Givental-Teleman's classification thoerem.If I have more time, I will discuss how to generalize this method to a non-semi simple Gromov-Witten theory.

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Zoom ID: 352 730 6970, Password: 9999

Abstract: We consider the problem of nonparametric imputation using neural network models. Neural network models can capture complex nonlinear trends and interaction effects, making it a powerful tool for predicting missing values under minimum assumptions on the missingness mechanism. Statistical inference with neural network imputation, including variance estimation, is challenging because the basis for function estimation is estimated rather than known. In this paper, we tackle the problem of statistical inference with neural network imputation by treating the hidden nodes in a neural network as data-driven basis functions. We prove that the uncertainty in estimating the basis functions can be safely ignored and hence the linearization method for neural network imputation can be greatly simplified. A simulation study confirms that the proposed approach results in efficient and well-calibrated confidence intervals even when classic approaches fail due to severe nonlinearity and complicated interactions.

Abstract: The biological master clock, the suprachiasmatic nucleus (SCN) of a mouse, contains many (~20,000) clock cells heterogeneous, particularly with respect to their circadian period. Despite the inhomogeneity, within an intact SCN, they maintain a very high degree of circadian phase coherence, which is generally rendered visible as system-wide propagating phase waves. The phase coherence is vital for mammals sustaining various circadian rhythmic activities. It is supposedly achieved not by one but a few different cell-to-cell coupling mechanisms: Among others, action potential (AP)-mediated connectivity is known to be essential. However, due to technical difficulties and limitations in experiments, so far, very little information is available about the (connectome) morphology of the AP-mediated SCN neural connectivity. With that limited amount of information, here we exhaustively and systematically explore a large (~25,000) pool of various model network morphologies to come up with the most realistic case for the SCN. All model networks within this pool reflect an actual indegree distribution as well as a physical range distribution of afferent clock cells, which were acquired in earlier optogenetic connectome experiments. Subsequently, our network selection scheme is based on a collection of multitude criteria, testing the properties of SCN circadian phase waves in perturbed (or driven) as well as in their natural states: Key properties include, 1) degree of phase synchrony (or dispersal) and direction of wave propagation, 2) entrainability of the model oscillator networks to an external circadian forcing (mimicking the light modulation subject to the geophysical circadian rhythm), and 3) emergence of “phase-singularities” following a global perturbation and their decay. The selected network morphologies require several common features that 1) the indegree – outdegree relation must have a positive correlation; 2) the cells in the SCN core region have a larger total (in+out) degree than that of the shell region; 3) core to shell (or shell to core) connections should be much less than core to core (and shell to shell) connections. Taken all together, our comprehensive test results strongly suggest that degree distribution over the whole SCN is not uniform but position-dependent and raise a question of whether this inhomogeneous degree distribution is related to the distribution of known subpopulations of SCN cells.

The linear bandit problem has received a lot of attention in the past decade due to its applications in new recommendation systems and online ad placements where the feedback is binary such as thumbs up/down or click/no click. Linear bandits, however, assume the standard linear regression model and thus are not well-suited for binary feedback. While logistic linear bandits, the logistic regression counterpart of linear bandits, are more attractive for these applications, developments have been slow and practitioners often end up using linear bandits for binary feedback -- this corresponds to using linear regression for classification tasks.In this talk, I will present recent breakthroughs in logistic linear bandits leading to tight performance guarantees and lower bounds. These developments are based on self-concordant analysis, improved fixed design concentration inequalities, and novel methods for the design of experiments. I will also discuss open problems and conjectures on concentration inequalities. This talk will be based on our recent paper accepted to ICML'21 (https://arxiv.org/abs/2011.11222).

TBA

피부암은 미국에서 가장 흔한 암 중의 하나이며 국내에서도 고령화, 오존층 변화와 자외선 노출 빈도의 증가에 따른 발병 율이 증가 하고 있다. 피부암 중의 약 2.1%인 전이 율이 높아 치료하기가 쉽지 않은 악성 흑색종(Malignant melanoma)이며 이와 달리 양성 흑색종 (NMSK: Nonmelanoma Skin Cancer)은 암으로 분류되긴 하지만, 피부 내에서만 자라고, 전이 율이 적어 초기에 발견하여 암 부위만 제거하면 10년 이상 생존율이 89%이상인 양성 암이다. 따라서, 초기에 아주 작은 수지만 위험한 악성 흑색종과 많은 수의 크게 위험하지 않은 양성 흑색종을 구분하여 적절히 치료하는 것은 피부암 치료에서 아주 중요한 부분이라고 할 수 있다. 흑색종 진단에 많이 사용되어져 왔던 전통적인 특징점(Feature)들로는 ABCD criteria(그림 1)과 같이 의심영역의 비대칭성(A: Asymmetry), 경계의 불규칙성(B:Border irregularity), 색조의 다양성(C: Color variegation) 등이 있다. 피부암 진단에 대한 연구는 많은 진전이 있었고, 2018년의 International Skin Imaging Collaboration (ISIC) 는 피부암 판별 경진대회를 열어서 사실상 이 분야의 벤치마크가 되었다. 다양한 영상 분류 알고리즘과 기법이 시도되었고 특히 CNN을 이용한 진단 결과들은 피부과 전문의 진단결과와 비슷하거나 더 좋은 결과를 내기도 했다. CNN의 이런 좋은 성과에도 불구하고 의학영역은 환자의 목숨을 다루기 때문에 높은 정확도와 함께 고려해야 할 것은 설명 가능한 심층학습의 필요성이다. 예를 들어, 만약 의학적 사고가 발생하였을 때 의료 행위에 대한 과실 책임에 대한 문제가 대두 되고 있다. 즉 의료사고에 대한 책임을 누가 질 것인가?, 의료 사고의 발생 원인이 무엇이며 어떻게 설명할 수 있는가?, 만약 결과가 좋다면 어떻게 이 방법을 발전시킬 수 있을 것인가? 의 문제들이다. 이를 위해 Grad-CAM 기법, Image Occulsion 기법, LIME (Local interpretable model-agnostic explanations)기법 등이 사용되고 있다. 이런 기법들을 이용하여 잘 알려진 진단 결과가 우수한 심층학습을 대상으로 어떤 특징 점들이 진단 결과의 우수성에 영향을 미치는 지 조사하려고 한다. 이 기법을 통해 전통적으로 알려진 특징 점인 A(비대칭성), B(경계 불규칙성), C(색조 다양성), D(크기) 와 설명 가능한 심층학습 기법으로 새로운 특징 점을 발견할 수 있을지 알아보고자 한다. Meeting ID: 857 8963 8008 Passcode: 932161

In this lecture, I will present recent works on the asymptotical stability of wave patterns for viscous conservation laws under periodic perturbations.

We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with Richard P. Thomas.

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Zoom ID: 352-730-6970, PW:9999 The above times are in Korean Standard Time. This is the same time as 9AM of June 18, 2021 (Friday) in the UK (GMT+1)

Bayesian statistical approaches have been developed for various applications due to their flexibility. I will cover different application areas of Bayesian methods including applications to analyses of complex count data, decision making problems, and analyses of survival times data. For part I, I will discuss two recently completed projects and comment on some on-going and future projects. Part II will cover a gentle introduction to survival analyses focusing on Bayesian approaches and discuss its extensions for joint analysis with recurrent events data or longitudinal data. For part III, I will cover a general Bayesian decision making framework and their applications to clinical trial design and data analysis.

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Zoom ID : 901 228 2472 (Password : 123456)

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

It is known that the stochastic Potts model exhibits exponentially slow mixing at the low temperature regime. In this talk, we explain precise quantitative results regarding this slow mixing behavior of the Potts model at large lattices. This talk is based on the joint work with Seonwoo Kim.

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

Bayesian statistical approaches have been developed for various applications due to their flexibility. I will cover different application areas of Bayesian methods including applications to analyses of complex count data, decision making problems, and analyses of survival times data. For part I, I will discuss two recently completed projects and comment on some on-going and future projects. Part II will cover a gentle introduction to survival analyses focusing on Bayesian approaches and discuss its extensions for joint analysis with recurrent events data or longitudinal data. For part III, I will cover a general Bayesian decision making framework and their applications to clinical trial design and data analysis.

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Zoom ID : 901 228 2472 (Password : 123456)

심사위원장 정연승, 심사위원 김성호, 황강욱, 전현호, 최태련(고려대 통계학과)

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

TBA

Bayesian statistical approaches have been developed for various applications due to their flexibility. I will cover different application areas of Bayesian methods including applications to analyses of complex count data, decision making problems, and analyses of survival times data. For part I, I will discuss two recently completed projects and comment on some on-going and future projects. Part II will cover a gentle introduction to survival analyses focusing on Bayesian approaches and discuss its extensions for joint analysis with recurrent events data or longitudinal data. For part III, I will cover a general Bayesian decision making framework and their applications to clinical trial design and data analysis.

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Zoom ID : 901 228 2472 (Password : 123456)

심사위원장 백형렬, 심사위원 박정환, 최서영, 임선희(서울대학 수리과학부), 김상현(고등과학원 수학부)

An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed. Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width, we introduce its directed analogue called `bi-mim-width’ and prove that various classes of reflexive intersection digraphs have bounded bi-mim-width. In particular, we show that as a natural extension of $H$-graphs, reflexive $H$-digraphs have linear bi-mim-width at most $12|E(H)|$, which extends a bound on the linear mim-width of $H$-graphs [On the Tractability of Optimization Proble

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Zoom ID: 934 3222 0374 (ibsdimag)

심사위원장 김재경, 심사위원 김용정, 황강욱, 정연승, 이승규(고려대 세종캠퍼스 응용수리과학부)

MUltiple SIgnal Classification (MUSIC) is a well-known, non-iterative imaging technique in inverse scattering problem. Throughout various researches, it has been confirmed that MUSIC is very fast, effective, and stable. Due to this reason MUSIC has been applied to various inverse scattering problems however, it has not yet been designed and used to identify unknown anomalies from measured scattering parameters (S-parameters) in microwave imaging. In this presentation, we apply MUSIC in microwave imaging for a fast identification of arbitrary shaped anomalies from real-data and establish a mathematical theory for illustrating the feasibilities and limitations of MUSIC. Simulations results with real-data are shown for supporting established theoretical results.

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갑작스러운 연사 사정으로 세미나 일정이 취소되었습니다. 양해 부탁드립니다.

We discuss sharp regularity results for a larger class of of elliptic and parabolic equations in divergence form and present new and interesting features from the view point of regularity theory

I will discuss various near-rationality concepts for smooth projective varieties. I will introduce the standard norm variety associated with a symbol in mod-l Milnor K-theory. The standard norm varieties played an important role in Vovedsky's proof of the Bloch-Kato conjecture. I will then describe known near-rationality results for standard norm varieties and outline an argument showing that a standard norm variety is universally R-trivial over an algebraically closed field of characteristic 0. The talk is based on joint work with Chetan Balwe and Amit Hogadi.

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Zoom ID: 352 730 6970, Password : 9999. All times are local in KST.

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

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.)

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.

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.