In many different areas of mathematics (such as number theory, discrete geometry, and combinatorics), one is often presented with a large "unstructured" object, and asked to find a smaller "structured" object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and present some recent progress on one of the central problems in the area: bounding the so-called "diagonal" Ramsey numbers. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

[1] 인간의 질병 발생과 예방을 근본적으로 파악하려면 '인간에 대한 이해'가 필요합니다. 인간의 몸은 물질이며, 물질은 특성상 물리적 및 화학적 반복자극에 반드시 손상됩니다. 부모님께 몸을 받아 수십 년간 살다 보면 인간 내부의 태생적-구조적 요인과 외부의 환경적 요인에 의하여 부지불식중 가해지는 내부 및 외부 자극에 반복적으로 노출될 수밖에 없습니다. 이에 그와 같이 질병으로 진행될 수 밖에 없는 인간의 특성을 명화(그림)을 통하여 소개할 예정입니다. [2] 인간의 또 다른 이해로 과학적 혹은 수학적 평가로 가시화(객관화)하기 어려운 부문에 대한 내용을 역시 명화를 통하여 논의할 예정입니다.

Knowledge graphs represent human knowledge as a directed graph, representing each fact as a triplet consisting of a head entity, a relation, and a tail entity. Knowledge graph embedding is a representation learning technique that aims to convert the entities and relations into a set of low-dimensional embedding vectors while preserving the inherent structure of the given knowledge graph. Once the entities and relations in a knowledge graph are represented as a set of feature vectors, those vectors can be easily integrated into diverse downstream tasks. This talk introduces a new concept of knowledge graph called a bi-level knowledge graph, where the higher-level relationships between triplets can be represented. Learning representations on a bi-level knowledge graph, machines are allowed to solve problems requiring more advanced reasoning than simple link prediction. Also, as a practical example of knowledge graph embedding, how one can utilize the knowledge representations to operate a real robot is briefly explained. This talk discusses how knowledge graph embedding models effectively deliver human knowledge to machines, which is critical in many AI applications.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701 ACMseminar mailing list registration: https://mathsci.kaist.ac.kr/mailman/listinfo/acmseminar

We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the unit circle. We show that this system is locally well-posed for L^p data as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for almost bounded logarithmic spirals and give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.

We first survey on nodal solutions for coupled elliptic equations, using results from nonlinear scalar field equations as motivations. Then we discuss work for constructing multiple nodal solutions using various variational methods. In particular we discuss in some details the results about solutions having componentwisely-shared nodal numbers of coupled elliptic systems. These works are done by further developing minimax type critical point theory with built-in flow invariance of the associated gradient or parabolic flows, which has been a useful tool to give locations of critical points via minimum methods, also revealing complex dynamic behavior of the flow.

다양한 소비재 중에서 유독 패션 카테고리는 그간 디지털 전환이 느린 분야였으며, 유통과 마케팅의 영역에서만 데이터를 주로 활용하는 양상을 보여왔다. 비정형적, 주관적인 의사 결정이 주를 이루는 '패션'에서 수학은 어떤 의미와 효용이 있는지 제조업을 운영하는 디자이너의 관점에서 실무 사례 위주로 이야기하려 한다.

In this talk, we provide an overview of the historical development of fast solution methods for partial differential equations, as well as their current status and potential for future advancements. We first begin with a historical survey and describe recent advances in efficient techniques, such as multigrid and domain decomposition methods. In addition, we will explore the potential of emerging methods in the realm of scientific machine learning.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

In this talk, we look at the results of various studies in which computational mathematics is used in medical imaging. Through the various scope of research from mathematical modeling to data-based methodology, we can think about the future direction by examining what we can do in data science can contribute and what contribution we can make to medical imaging.

Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for highdimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. By reframing the setup of B-PINNs as a traditional Bayesian inverse problem, we can take advantage of EKI’s key features: (1) gradient-free, (2) computational complexity scales linearly with the dimension of the parameter spaces, and (3) rapid convergence with typically O(100) iterations. We demonstrate the applicability and performance of the proposed method through various types of numerical examples. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications.

We will first introduce Homogeneous dynamics, especially the mixing property of flows on spaces of hyperbolic nature. We will then survey applications of homogeneous dynamics to various problems in Number theory. (Part of the talk is based on joint work with Keivan Mallahi-Karai and Jiyoung Han.)

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The date has been postponed from March 16 to March 30.

Modeling mass or heat transfer near a wall is of broad interest in various fluid flows. Specifically, in cardiovascular flows, mass transport near the vessel wall plays an important role in cardiovascular disease. However, due to very thin concentration boundary layers, accurate computational modeling is challenging. Additionally, experimental approaches have limitations in measuring near-wall flow metrics such as wall shear stress (WSS). In this talk, first, I will briefly review the complex flow physics near the wall in diseased vascular flows and introduce the concept of WSS manifolds in near-wall transport. Specifically, I will talk about stable and unstable manifolds calculated for a surface vector field. Next, I will discuss reduced-order data assimilation modeling as well as physics-informed neural network (PINN) approaches for obtaining WSS from measurement data away from the wall. Finally, I present a boundary-layer PINN (BL-PINN) approach inspired by the classical perturbation theory and asymptotic expansions to solve challenging thin boundary layer mass transport problems. BL-PINN demonstrates how classical theoretical approaches could be replicated in a deep learning framework.

In this lecture we introduce some challenging problems in the mathematical fluid mechanics. Although fluid mechanics is one of the most important physical phenomena we experience in everyday life, and has been studied for long time in history by top class mathematicians, still there are many problems which are open even at the fundamental level. We explain these problems and briefly review some of the recent progress.

Symmetric spaces from Lie theory and differential geometry are often represented by special set of structured matrices. The Cartan decomposition and its generalization of symmetric spaces and classical Lie groups recover many of the known matrix factorizations in numerical linear algebra, such as the singular value decomposition, CS decomposition, generalized SVD and many more. We discuss a blueprint for generating fifty-three matrix factorizations from the generalized Cartan decomposition, most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978). This is joint work with Alan Edelman.

Many differential equations and partial differential equations (PDEs) are being studied to model physical phenomena in nature with mathematical expressions. Recently, new numerical approaches using machine learning and deep learning have been actively studied. There are two mainstream deep learning approaches to approximate solutions to the PDEs, i.e., using neural networks directly to parametrize the solution to the PDE and learning operators from the parameters of the PDEs to their solutions. As the first direction, Physics-Informed Neural Network was introduced in (Raissi, Perdikaris, and Karniadakis 2019), which learns the neural network parameters to minimize the PDE residuals in the least-squares sense. On the other side, operator learning using neural networks has been studied to approximate a PDE solution operator, which is nonlinear and complex in general. In this talk, I will introduce these two ways to approximate the solution of PDE and my research related to them.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

This talk reviews two notable papers in self-supervised graphical neural networks; they are "Graph contrastive learning with augmentations" presented at NeurIPS 2020 and "Contrastive multi-view representation learning on graphs" presented at ICML 2020. This will be an introduction of self-supervised graphical neural networks that has emerged as one of the hottest research fields in artificial intelligence, which requires mathematical methodology across all fields of mathematics, including graph theory, algebra, topology, analysis, and geometry.

In this talk, we study the dissipative structure for the linear symmetric hyperbolic system with general relaxation. If the relaxation matrix of the system has symmetric properties, Shizuta and Kawashima(1985) introduced the suitable stability condition, and Umeda, Kawashima and Shizuta(1984) analyzed the dissipative structure. On the other hand, Ueda, Duan and Kawashima(2012,2018) focused on the system with non-symmetric relaxation and got partial results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, this talk aims to extend the stability theory introduced by Shizuta and Kawashima(1985) and Umeda, Kawashima and Shizuta(1984) to our general system. Furthermore, we will consider the optimality of the dissipative structure. If we have time, I would like to discuss some physical models for its application and new dissipative structures.