In this talk, we study the behaviour of rational points on the expanding horospheres in the space of unimodular lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira (2016). Their proof uses techniques from homogeneous dynamics and relies particularly on measure-classification theorems due to Ratner. We pursue an alternative strategy based on Fourier analysis, Weil's bound for Kloosterman sums, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums, Roger's formula, and the spectral theory of automorphic functions. Our methods yield an effective estimate on the rate of convergence for a specific horospherical subgroup in any dimension. This is a joint work with D. El-Baz, B. Huang, J. Marklof and A. Strömbergsson.

Given a set $E$ and a point $y$ in a vector space over a finite field, the radial projection $\pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly, $|pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss several results on the general question: For how many points $y$ can $|\pi_y(E)|$ be much smaller than this maximum? This is motivated by an analogous question in fractal geometry. The Hausdorff dimension of a radial projection of a set $E$ in $n$ dimensional real space will typically be the minimum of $n-1$ and the Hausdorff dimension of $E$. Several recent papers by authors including Matilla, Orponen, Liu, Shmerikin, and Wang consider the question: How large can the set of points with small radial projections be? This body of work has several important applications, including recent progress on the Falconer distance conjecture. This is joint with Thang Pham and Vu Thi Huong Thu.

In this talk, we discuss the fluctuation of f(X) as a matrix, where  X is a large square random matrix with  centered, independent, identically distributed entries and f is an analytic function. In particular, we show that for a generic deterministic matrix A of the same size as X, the trace of f(X)A is approximately Gaussian which decomposes into two independent modes corresponding to tracial and traceless parts of A. We also briefly discuss the proof that mainly relies on Hermitization of X and its resolvents. 

Over the recent years, various methods based on deep neural networks have been developed and utilized in a wide range of scientific fields. Deep neural networks are highly suitable for analyzing time series or spatial data with complicated dependence structures, making them particularly useful for environmental sciences and biosciences where such type of simulation model output and observations are prevalent. In this talk, I will introduce my recent efforts in utilizing various deep learning methods for statistical analysis of mathematical simulations and observational data in those areas, including surrogate modeling, parameter estimation, and long-term trend reconstruction. Various scientific application examples will also be discussed, including ocean diffusivity estimation, WRF-hydro calibration, AMOC reconstruction, and SIR calibration.

In addition to diffusive signals, cells in tissue also communicate via long, thin cellular protrusions, such as airinemes in zebrafish. Before establishing communication, cellular protrusions must find their target cell. In this talk, we demonstrate that the shapes of airinemes in zebrafish are consistent with a persistent random walk model. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive search (highly curved, random). We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding that there is a theoretical trade-off between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of non-canonical cellular protrusions in general.

심사위원장 : 백형렬, 심사위원 : 박정환, 최서영, 김상현(겸직교수), 이계선(서울대학교)

For the last decade, there have been a number of studies reporting that certain surface singularities give rise to vector bundle on their smoothing. The first result is by Hacking, who studies this correspondence for Wahl singularities. I am going to introduce a generalization of Hacking's result to singularities of class T, which is a natural extension of Wahl singularities. Also, if time permits, I will introduce a recent result of Tevelev-Urzua which generalizes this to arbitrary cyclic quotient surface singularities.

심사위원장 : 이창옥, 심사위원 : 임미경, 김동환, 예종철(겸임교수), 김윤호(UNIST 자연과학부)

Given a space, one can study its singularities. The converse direction is called reconstruction problem: How to reconstruct spaces from given singularity information? In this talk, by introducing a notion called a semicascade we derive a bound of Picard number for toric log del Pezzo surfaces in terms of the singular points generalizing some results of Dais and Suyama, which solves the reconstruction problem with the help of computer. We also discuss Kähler-Einstein toric log del Pezzo surfaces as an application of semicascades.

The connection between deep neural networks and ordinary differential equations (ODEs) is an active field of research in machine learning. In this talk, we view the hidden states of a neural network as a continuous object governed by a dynamical system. The underlying vector field is written using a dictionary representation motivated by the equation discovery method. Within this framework, we develop models for two particular machine learning tasks: time-series classification and dimension reduction. We train the parameters in the models by minimizing a loss, which is defined using the solution to the governing ODE. To attain a regular vector field, we introduce a regularization term measuring the mean total kinetic energy of the flow, which is motivated by optimal transportation theory. We solve the optimization problem using a gradient-based method where the gradients are computed via the adjoint method from optimal control theory. Through various experiments on synthetic and real-world datasets, we demonstrate the performance of the proposed models. We also interpret the learned models by visualizing the phase plots of the underlying vector field and solution trajectories.

A monotone symplectic manifold is a symplectic analogue of a smooth Fano variety and it provides an important classes of objects, called monotone Lagrangian tori, in view of mirror symmetry. In this talk, I will explain a way of producing monotone Lagrangian tori in a given smooth Fano variety using toric degeneration. Using this technique, we prove that there exist infinitely many monotone Lagrangian tori not Hamiltonian isotopic to each other in a full flag variety. This is based on joint work with Myungho Kim, Yoosik Kim, Jaehoon Kwon, and Euiyong Park at Center for Quantum Structures in Modules and Spaces (QSMS).

Twin-width is a recently introduced graph parameter based on vertex contraction sequences. On classes of bounded twin-width, problems expressible in FO logic can be solved in FPT time when provided with a sequence witnessing the bound. Classes of bounded twin-width are very diverse, notably including bounded rank-width, $\Omega ( \log (n) )$-subdivisions of graphs of size $n$, and proper minor closed classes. In this talk, we look at developing a structural understanding of twin-width in terms of induced subdivisions. Structural characterizations of graph parameters have mostly looked at graph minors, for instance, bounded tree-width graphs are exactly those forbidding a large wall minor. An analogue in terms of induced subgraphs could be that, for sparse graphs, large treewidth implies the existence of an induced subdivision of a large wall. However, Sintiari and Trotignon have ruled out such a characterization by showing the existence of graphs with arbitrarily large girth avoiding any induced subdivision of a theta ($K_{2,3}$). Abrishami, Chudnovsky, Hajebi and Spirkl have recently shown that such (theta, triangle)-free classes have nevertheless logarithmic treewidth. After an introduction to twin-width and its ties to vertex orderings, we show that theta-free graphs of girth at least 5 have bounded twin-width. Joint work with Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé and Rémi Watrigant.

We consider an optimal transport problem where the cost depends on the stopping time of Brownian motion from a given distribution to another. When the target measure is fixed, it is often called the optimal Skorokhod embedding problem in the literature, a popular topic in math finance. Under a monotonicity assumption on the cost, the optimal stopping time is given by the hitting time to a space-time barrier set. When the target measure is optimized under an upper bound constraint, we will show that the optimal barrier set leads us to the Stefan problem, a free boundary problem for the heat equation describing phase transition between water and ice. This is joint work with Young-Heon Kim at UBC.

In this talk, we introduce a various methods of representations of graphs which are mathematical objects expressing a variety of non-Euclidean data such as Molecules, social networks, genes, transportation networks, citation networks of papers and so on. Graph representation as a Euclidean vector is inevitable in machine learning for classifications for graphs which is closely related to graph neural network in computer science. We would like to introduce a few literatures, Weisfeiler-lehman algortihm, random walks, graph convolution whci are commonly used techniques and explain the result of combining them with topological invarints of graphs

---

https://sites.google.com/view/mwagaag

Circadian rhythm is a robust internal 24 hours timekeeping mechanism maintained by the master circadian pacemaker Suprachiasmatic Nuclei (SCN). Numerous mathematical models have been proposed to capture SCN’s timekeeping mechanism and predict the circadian phase. There has been an increased demand for applying these models to the various unexplored data sets. One potential application is on data from commercially available wearable devices, which provide the noninvasive measurements of physiological proxies, such as activity and heart rate. Using these physiological proxies, we can estimate the circadian phase of the central and peripheral circadian pacemakers. Here, we propose a new framework for estimating the circadian phase using wearable data and the Level Set Kalman Filter on the nonlinear state-space model of the human circadian pacemaker. Analysis of over 200,000 days of wearable data from over 3,000 subjects using our framework successfully identified misalignment in central and peripheral pacemakers with a significantly smaller uncertainty than previous methods.

심사위원장 : 정연승, 심사위원 : 김성호(명예교수), 박철우, 전현호, 김윤희(The University of Tokyo)

The introduction for the framework of geometric deep learning will be explained in the perspective of new methodology of A.I. and data analysis. Various applications can be discussed by utilizing geometry, algebra, topology.

---

https://sites.google.com/view/mwagaag

In this talk, we consider the Ising and Potts model defined on large lattices of dimension two or three at very low temperature regime. Under this regime, each monochromatic spin configuration is metastable in that exit from the energetic valley around that configuration is exponentially difficult. It is well-known that, under the presence of external magnetic fields, the metastable transition from a monochromatic configuration to another one is characterized solely by the appearance of a critical droplet. On the other hand, for the model without external field, the saddle structure is no longer characterized by a sharp droplet but has a huge and complex plateau structure. In this talk, we explain our recent research on the analysis of this energy landscape and its application to the demonstration of Eyring-Kramers formula for models on fixed two or three dimensional lattice (cf. https://arxiv.org/abs/2102.05565) or models on growing two-dimensional lattice (cf. https://arxiv.org/abs/2109.13583).

Abstract: We discuss a new application of (a part of) the Iwasawa main conjecture to the non-triviality of Kato's Kolyvagin systems and a structural refinement of Birch and Swinnerton-Dyer conjecture. In particular, the structure of Selmer groups is completely determined by certain modular symbols for a large class of elliptic curves. (Please contact Wansu Kim at for Zoom meeting info or any inquiry.)

---

(Please contact Wansu Kim at for Zoom meeting info or any inquiry.)

In this talk, I will describe the large deviation asymptotic of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^\alpha$ in the Poisson $k$-nearest neighbor graph in $\mathbb R^d$. While the case $\alpha < d$ can be treated through classical methods from large deviations theory, an interesting dichotomy occurs if $\alpha > d$. Rare events in the lower tail can still be explained by subtle changes in the Poisson process throughout the sampling window. However, the most likely cause for rare events in the upper tail is a condensation phenomenon: the excess edge weight is caused by a negligible portion of Poisson points whose configuration can be described through a concrete geometric optimization problem. After presenting the general proof strategy, I will also elucidate on the prospects and limits of generalizing our approach to other spatial networks.

논문은 저자와 독자 사이의 학문적 소통을 위한 논리적인 글이다. 이번 강연을 통해 논문을 작성하기 위한 기본 원리를 배울 수 있다. 특히, 연구가 거의 마무리되는 시점에 (After completing your research), 연구 결과를 그림과 표로 잘 정리한 다음에 (Based on well-organized figures and tables), 본격적으로 논문 작성을 시작하는 (Compose your manuscript from a title to a conclusion) ‘ABC 논문 작성법’을 소개한다. 논문 작성의 준비 과정으로 (A와 B의 과정), 연구 노트 작성 방법, 저널 클럽 운영 방법, 한 페이지 활용 방법을 설명한다. 논문 작성 준비가 완료되면, 제목부터 결론까지 순서대 로 논문 원고를 작성할 수 있다 (C의 과정). 이렇게 하면, 단기간에 집중하여 효율적으 로 논문을 작성할 수 있다. *참고: 원병묵 교수의 과학 논문 쓰는 법

---

2022년 6월 3일 오후 4시 – 5시, 성균관대 신소재공학부의 원병묵 교수님의 논문 글쓰기 워크샵 강의가 있습니다.

이 ‘영어 논문 쓰기’ 워크숍은 현재의 영어 수준이나 과거의 영어 학습 경험과 상관 없이 누구나 영어 논문 쓰기를 시작할 수 있는 방법에 관한 것입니다. 우선 영어에 대한 막연한 두려움, 과거 경험으로 인한 자신감 부족, 게다가 ‘쓰기’라는 쉽지 않은 인지 활동, 심지어 논문이라는 큰 벽, 혹은 섣부른 자신감 등 ‘영어 논문 쓰기’를 방해하는 요인을 생각해 봅니다. 이런 요인을 자세하게 들여다보면 각각의 걸림돌을 넘어갈 방법도 명쾌하게 발견할 수 있습니다. 이 세미나에서 다룰 구체적 내용은 다음과 같습니다. - 한국인들이 ‘영어’와 ‘영어 쓰기’를 어렵게 느끼는 원인 이해하고 극복하기 - ‘읽기’와 ‘쓰기’의 서로 다른 두 가지 인지활동의 관련성 이해하고 적용하기 - 논문에 적합한 단어, 시제, 구두점 선택하고 사용하기 - 영문초록, 제목, 소제목 스타일 이해하고 작성하기 - 영어논문 쓰기를 돕는 디지털 도구 선택하고 활용하기 이 워크숍에서 안내될 몇 가지 방법은 영어 논문 쓰기가 더 이상 두려운 것이 아닌 연구자로서의 목표를 이루는 디딤돌이 되도록 할 것입니다.

---

2022년 6월 3일 오후 5시 – 6시, 부산교육정책연구소 박영민 연구원님의 논문 글쓰기 워크샵 강의가 있습니다.

Transport properties of Gibbs and Gaussian measures under different transformations have been studied in probability theory. In this talk, I will discuss the invariance and quasi-invariance of Gaussian type measures on functions/distributions under the flow of Hamiltonian PDEs.

---

[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

심사위원장 : 전현호, 심사위원 : 김용정, 박철우, 정연승, 이주호(김재철AI대학원)

심사위원장 : 최건호, 심사위원 : 곽도영(명예교수), 김동석(경영공학부), 권순식, 김완수

One of the important work in graph theory is the graph minor theory developed by Robertson and Seymour in 1980-2010. This provides a complete description of the class of graphs that do not contain a fixed graph H as a minor. Later on, several generalizations of H-minor free graphs, which are sparse, have been defined and studied. Also, similar topics on dense graph classes have been deeply studied. In this talk, I will survey topics in graph minor theory, and discuss related topics in structural graph theory.

---

ZOOM Meeting ID: 873 7478 2790 Direct link: https://kaist.zoom.us/j/87374782790

Quantitative characterization of biomolecular networks is important for the analysis and design of network functionality. Reliable models of such networks need to account for intrinsic and extrinsic noise present in the cellular environment. Stochastic kinetic models provide a principled framework for developing quantitatively predictive tools in this scenario. Calibration of such models requires an experimental setup capable of monitoring a large number of individual cells over time, automatic extraction of fluorescence levels for each cell and a scalable inference approach. In the first part of the talk we will cover our microfluidic setup and a deep-learning based approach to cell segmentation and data extraction. The second part will introduce moment-based variational inference as a scalable framework for approximate inference of kinetic models based on single cell data.

---

This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, just from input and output pairs where inputs are given as images. In this paper, we show how to improve the learning of SATNet by exploiting symmetries in the target rules of a given but unknown logical puzzle or more generally a logical formula. We present SymSATNet, a variant of SATNet that translates the given symmetries of the target rules to a condition on the parameters of SATNet and requires that the parameters should have a particular parametric form that guarantees the condition. The requirement dramatically reduces the number of parameters to learn for the rules with enough symmetries, and makes the parameter learning of SymSATNet much easier than that of SATNet. We also describe a technique for automatically discovering symmetries of the target rules from examples. Our experiments with Sudoku and Rubik’s cube show the substantial improvement of SymSATNet over the baseline SATNet. This is joint work with Sangho Lim and Eungyeol Oh.

Offline reinforcement learning (RL) refers to the problem setting where the agent aims to optimize the policy solely from the pre-collected data without further environment interactions. In offline RL, the distributional shift becomes the primary source of difficulty, which arises from the deviation of the target policy being optimized from the behavior policy used for data collection. This typically causes overestimation of action values, which poses severe problems for model-free algorithms that use bootstrapping. To mitigate the problem, prior offline RL algorithms often used sophisticated techniques that encourage underestimation of action values, which introduces an additional set of hyperparameters that need to be tuned properly. In this talk, I present OptiDICE, an offline RL algorithm that prevents overestimation in a more principled way. OptiDICE directly estimates the stationary distribution corrections of the optimal policy and does not rely on policy-gradients, unlike previous offline RL algorithms. Using an extensive set of benchmark datasets for offline RL, OptiDICE is shown to perform competitively with the state-of-the-art methods. This is a joint work with Jongmin Lee (UC Berkeley), Wonseok Jeon (Qualcomm), Byung-Jun Lee (Korea U.), and Joelle Pineau (MILA)

The law of iterated logarithm (LIL) is a crowning achievement in classical probability theory that gives the sharp upper bound for the magnitude of fluctuations of a random walk. If each step has mean zero and variance one, then the upper bound (in certain sense) is given by \sqrt{2n\log\log n}, hence the name “iterated logarithm.” Despite being considered the “third fundamental limit theorem in probability” by some probabilists after the law of large numbers and the central limit theorem, its proof is not so accessible to non-experts. For instance, most textbooks either only consider special cases or use sophisticated machineries in their proofs. The purpose of this talk is to provide a relatively simple and elementary proof of the so-called Hartman—Wintner LIL. The idea is to generalize a proof of the central limit theorem (CLT), which will be also presented, to obtain a result on the rate of convergence in the CLT. First principles in probability (e.g. the second Borel—Cantelli lemma) are the only technical prerequisites.

Around early 2010, there was a huge success in understanding the spectrum of large random matrices, in other words, large random graphs. It was only for large but dense random graphs at first. However, as random matrix theory has been developed, there is some progress in sparse cases. In this short talk, I will review a series of results for spectral statistics of sparse random graphs and explain their implications.

---

[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

Randomness of biochemical reactions is inherent in various biological systems, from DNA to organs and the human body. These stochastic dynamics are frequently modeled using a continuous-time Markov chain (CTMC). Its long-term behavior is described by a stationary distribution, corresponding to its deterministic counterpart called a steady state. Stationary distribution can be derived analytically only in limited systems such as linear or finite-state systems. In this talk, I will introduce a recent result by Anderson, Craciun, and Kurtz deriving stationary distribution from the underlying graph structure of a reaction network and how we can extend it. For those who are first told the word 'Mathematical Biology,' I will briefly introduce mathematical biology before going into the detailed topic.

---

[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

심사위원장 : 권순식, 심사위원 : 배명진, 남경식, Yoshio Tsutsumi(Kyoto University), Mamoru Okamoto(Osaka University)

Inside living cells, chemical reactions form a large web of networks and they are responsible for physiological functions. Understanding the behavior of complex reaction networks is a challenging and interesting task. In this talk, I would like to illustrate how the methods of algebraic topology can shed light on the properties of chemical reaction systems. In particular, we discuss the following two problems: (1) response of reaction systems to external perturbations and (2) simplification of complex reaction networks without altering the behavior of the system.

---

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tian’s partial C0-estimate.

---

https://zoom.us/j/98194255942?pwd=cWpLM0c1T2U2OG9MR0VJNHpOTFBrdz09 아이디: 981 9425 5942 비밀번호: 373452

I will introduce mathematical and computational methods for spatio-temporal modelling in molecular and cell biology, including all-atom and coarse-grained molecular dynamics (MD), Brownian dynamics (BD), stochastic reaction-diffusion models and macroscopic mean-field equations. Microscopic (BD, MD) models are based on the simulation of trajectories of individual molecules and their localized interactions (for example, reactions). Mesoscopic (lattice-based) stochastic reaction-diffusion approaches divide the computational domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment, while macroscopic models are often written in terms of mean-field reaction-diffusion partial differential equations for spatially varying concentrations.

---

This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

I will discuss the development, analysis and applications of multi-resolution methods for spatio-temporal modelling of intracellular processes, which use (detailed) Brownian dynamics or molecular dynamics simulations in localized regions of particular interest (in which accuracy and microscopic details are important) and a (less-detailed) coarser model in other regions in which accuracy may be traded for simulation efficiency. I will discuss the error analysis and convergence properties of the developed multi-resolution methods, their software implementation and applications of these multiscale methodologies to modelling of intracellular calcium dynamics, actin dynamics and DNA dynamics. I will also discuss the development of multiscale methods which couple molecular dynamics and coarser stochastic models in the same dynamic simulation.

---

This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

심사위원장 : 김용정, 심사위원 : 강문진, 김재경, 임미경, 안인경(고려대학교)

심사위원장 : 김용정, 심사위원 : 강문진, 김재경, 임미경, 안인경(고려대학교)

This talk is concerned with the bifurcation and stability of the compresible Taylor vortex. Consider the compressible Navier-Stokes equations in a domain between two concentric infinite cylinders. If the outer cylinder is at rest and the inner one rotates with sufficiently small angular velocity, a laminar flow, called the Couette flow, is stable. When the angular velocity of the inner cylinder increases, beyond a certain value of the angular velocity, the Couette flow becomes unstable and a vortex pattern, called the Taylor vortex, bifurcates and is observed stably. This phenomena is mathematically formulated as a bifurcation and stability problem. In this talk, the compressible Taylor vortex is shown to bifurcate near the criticality for the incompressible problem when the Mach number is sufficiently small. The localized stability of the compressible Taylor vortex is considered under sufficiently small axisymmetric perturbations; and it is shown that the large time behavior of solutions around the Taylor vortex is described by solutions of a system of diffusion equations.

Despite of great progress over the last decades in simulating complex problems with the numerical discretization of (stochastic) partial differential equations (PDEs), solving high-dimensional problems governed by parameterized PDEs remains challenging. Machine learning has emerged as a promising alternative in scientific computing community by enforcing the physical laws. We review some of machine learning approaches and present a novel algorithm based on variational inference to solve (stochastic) systems. Numerical examples are provided to illustrate the proposed algorithm.

Modular curves for Hecke congruence groups, or more generally, for Fuchsian groups of the first kind can be seen as the moduli spaces of the isomorphism classes of elliptic curves with torsion data in some sense. In this talk, I will introduce the notion of modular curves and modular forms. If time permits, I will also introduce their applications to some transcendental questions.

---

[Zoom 링크] Zoom 회의 참가 https://kaist.zoom.us/j/2655728482?pwd=OXpJeFdDcWliSG51WUp0N1Nad2JHdz09 회의 ID: 265 572 8482 암호: 2AHRKr [Gather Town 링크] https://gather.town/app/ffr2PVibAWRIyXWO/kaistmath 두 발표 세션이 끝나면 Gather Town으로 옮겨와 대학원생들간에 자유롭게 이야기를 나누는 시간을 가질 계획입니다. Gather Town은 크롬이 깔려있는 기기(노트북, 패드, 스마트폰 등)에서 모두 접속 가능합니다. 별도의 회원가입 없이도 개별 캐릭터 설정 후 접속하면 주변의 다른 캐릭터들과 대화할 수 있는 플랫폼입니다.

Helly-type theorems and problems form a nice area of discrete geometry. I will start with the notable theorems of Radon and Tverberg and mention the following conjectural extension.

For a set X of points x(1), x(2),...,x(n) in some real vector space V we denote by T(X,r) the set of points in X that belong to the convex hulls of r pairwise disjoint subsets of X.
We let t(X,r) = 1 + dim(T(X,r)).

Radon's theorem asserts that
If t(X,1) < |X| then t(X, 2) > 0.

The first open case of the cascade conjecture asserts that
If t(X,1) + t(X,2) < | X | then t(X,3) >0.

In the lecture I will discuss connections with topology and with various problems in graph theory.
I will also mention questions regarding dimensions of intersection of convex sets.

Some related material:
1) A lecture (from 1999): An invitation to Tverberg Theorem: https://youtu.be/Wjg1_QwjUos
2) A paper on Helly type problems by Barany and me https://arxiv.org/abs/2108.08804
3) A link to Barany's book: Combinatorial convexity https://www.amazon.com/Combinatorial-Convexity-University-Lecture-77/dp/1470467097

---

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

심사위원장 : 임미경, 심사위원 : 김동환, 김용정, 권기운(동국대 수학과), 이현대(인하대 수학과)

Counting the number of points on a variety is a historical method for investigating the variety, for example, in the Weil conjecture. Nowadays, it is known that the point count helps us determine the E-polynomial. This E-polynomial, in turn, gives arithmetic-geometric information on the variety such as the dimension, the number of irreducible components and Euler characteristic. In this talk, we will consider a specific type of variety, the character variety associated to the fundamental group of a surface. In short, we will discuss this variety for a punctured surface, with regular semisimple or regular unipotent monodromy at the punctures. This variety plays a crucial role in diverse areas of mathematics, including non-abelian Hodge theory, geometric Langlands program and mathematical physics. The complex representation theory of finite groups will be used to compute the number of points on such a variety.

Hadwiger’s transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals in R d by Goodman, Pollack, and Wenger. Here we establish a colorful extension of their theorem, which proves a conjecture of Arocha, Bracho, and Montejano. The proof uses topological methods, in particular the Borsuk-Ulam theorem. The same methods also allow us to generalize some colorful transversal theorems of Montejano and Karasev.

9:30-10:30am Title: Equations in Simple Groups Abstract: Given a word w in a free group on variables x_1,...,x_n, a finite group G, and an element g in G, we consider the question of whether the equation w = g has solutions where the x_i take values in G, and if so, how many. I am particularly interested in what happens when the word is fixed and G is a large finite simple groups. I will say something about the ideas which have led to progress for certain families of words, with emphasis on open problems. 10:50-11:50 Title: Elliptic curves and field arithmetic Abstract: Let E be an elliptic curve over a field K. When K is a number field, Mordell's theorem says that the points of E over K form a finitely generated group. We say a field is "anti-Mordellic" if the opposite is true for all E/K. I will discuss what is known about anti-Mordellic fields, with emphasis on a longterm joint project with Bo-Hae Im to understand the relation between the anti-Mordellic property and the absolute Galois group of K.

The boundary of melting ice forms a random interface. So does the frontier of slowing burning pieces of paper. As time changes, the interface evolves in a random fashion. In probability theory, a collection of models often exhibits universal behaviors when the system size or time becomes large. The KPZ universality class comprises 1+1 dimensional probability models that mimic the random growth behavior mentioned above and display particular universal fluctuations.We will overview some of the development in this class that started about two decades ago.

Ellipsoidal BGK model (ES-BGK) is a generalized version of the Boltzmann-BGK model. In this model, the local Maxwellian in the relaxation operator is extended to an ellipsoidal Gaussian with a Prandtl parameter ν, so that the correct Prandtl number can be computed in the Navier-Stokes limit. In this talk, we review some of the recent results on ES-BGK model, such as the existence (stationary or non-stationary) theory and the entropy-entropy production estimates. A dichotomy is observed between −1/2 < v < 1 and ν=−1/2. In the former case, an equivalence relation between the local temperature and the temperature tensor enables one to apply theories developed for the original BGK model in a modified form. In the critical case (ν=−1/2), where the correct Prandtl number is achieved, such equivalence breaks down, and the structure of the flow has to be incorporated to estimate the temperature tensor from below. This is from joint works with Stephane Brull, Doheon Kim, and Son Sung Jun.

The idea of balance between excitation and inhibition is central in the theory of biological neural networks. I will give a brief introduction to the concept of such balance, and an overview of the mathematical ideas that can be used to study it.

---

This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

I will first describe how to extend the theory of balanced networks to account for synaptic plasticity. This theory can be used to show when a plastic network will maintain balance, and when it will be driven into an unbalanced state. I will next discuss how this approach provides evidence for a novel form of rapid compensatory inhibitory plasticity. Experimental evidence for such plasticity comes from optogenetic activation of excitatory neurons in primate visual cortex (area V1) which induces a population-wide dynamic reduction in the strength of neuronal interactions over the timescale of minutes during the awake state, but not during rest. I will shift gears in the final part of the talk, and discuss how community detection algorithms can help uncover the large scale organization of neuronal networks from connectome data.

---

This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

심사위원장 : 황강욱, 심사위원 : 강완모, 정연승, 전현호, 문일철(산업및시스템공학과)

심사위원장 : 이지운, 심사위원 : 강남규(겸직교수), 서인석(서울대학교 수리과학부), 폴정, 남경식

A subset V of a domain Ω has the extension property if for every holomorphic function p on V there is a bounded holomorphic function φ on Ω that agrees with p on V and whose sup-norm on Ω equals the sup-norm of p on V. Within the talk, we shall study mutual relations between extension property and interpolations problems.

---

Zoom link https://zoom.us/j/95969402165?pwd=MUhVTUQ5azZSOW1EdkRMMFRVM1R4QT09 ID: 959 6940 2165 PW: 962685

The upper tail problem for subgraph counts in the Erdos-Renyi graph, introduced by Janson-Ruciński, has attracted a lot of attention. There is a class of Gibbs measures associated with subgraph counts, called exponential random graph model (ERGM). Despite its importance, lots of fundamental questions have remained unanswered owing to the lack of exact solvability. In this talk, I will talk about a brief overview on the upper tail problem and the concentration of measure results for the ERGM. Joint work with Shirshendu Ganguly and Ella Hiesmayr.

심사위원장 : 폴정, 심사위원 : 이지운, 남경식, 강남규(겸직교수), 서인석(서울대학교)

---

https://us02web.zoom.us/j/2793681958?pwd=SVF2OHNkbFZweWwwK0tZb2FEU2dIdz09

We investigate in depth the behaviour of Monge-Ampère volumes of quasi-psh functions on a given compact hermitian manifold. We prove that the property for these Monge-Ampère volumes to stay bounded away from zero or infinity is a bimeromorphic invariant. We show in particular that a conjecture of Demailly-Paun holds true if and only if such Monge-Ampère volumes stay bounded away from infinity. This is a joint work with Vincent Guedj.

---

Zoom link https://zoom.us/j/95121528401?pwd=SXd2bnVFNW1veEJEZUJuNUhVaUdJZz09 ID: 951 2152 8401 PW: 641513

In this talk, we introduce homomorphisms between binary matroids that generalize graph homomorphisms. For a binary matroid $N$, we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial-time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem. This is joint work with Hyobin Kim and Mark Siggers at Kyungpook National University.