A digital twin is a virtual representation of real-world physical objects. In this talk, I will briefly introduce how Physics can be encoded into Neural Networks such as PINN and Operator Learning. Then we will explore real-world applications of AI-based partial differential equation (PDE) solvers in various fields.

In the 4th century BC, the ancient Mediterranean world saw the emergence of a distinctive form of writing devoted to the study of magnitudes, ratios, and proportional relations, eventually recognized as ancient Greek mathematics. Their geometry we still study in schools today is both familiar and markedly different from Our geometry. This lecture is designed to uncover the diverse aspects of ancient Greek geometry, blending the familiar with the unfamiliar. It unfolds in two primary parts. The first part explores the early landscape of Greek mathematics, with a focus on Euclid's Elements, while the subsequent part highlights the joy of mathematical discoveries that Archimedes was eager to share through his extensive works. Although the famous maxim that "there is no royal road to geometry" originates from later traditions, it underscores the significant implications of the birth and development of ancient Greek mathematics. This lecture seeks to unearth the pivotal reasons behind the invention of mathematics in ancient Greece, found in the culture of free discussion and persuasion among the citizens of the democratic societies of the Mediterranean world. 기원전 4세기 고대 지중해 세계에서 크기와 비율과 비례관계를 고민했던 특별한 형식의 글들이 고대 그리스 수학이라는 이름으로 탄생하게 되었다. 우리가 여전히 학교에서 배우고 있는 “그들의 기하학”은 여전히 친숙한 것이기도 한 동시에 “우리의 기하학”과 큰 차이를 보이기도 한다. 본 강연은 이처럼 우리에게 친숙하기도 하고 낯설기도 한 고대 그리스 기하학의 다양한 면모를 소개하고자 한다. 강연은 크게 두 부분으로 구성된다. 강연의 첫 번째 부분은 에우클레이데스의 기하학 원론을 중심으로 초기 그리스 수학의 풍경을 소개할 것이고, 두 번째 부분은 아르키메데스의 여러 작품들 속에서 그가 독자들과 공유하고 싶어 했던 수학적 발견의 기쁨에 대해 다룰 것이다. 비록 “기하학에는 왕도가 없다”라는 에우클레이데스의 대답은 후대의 전승에서 비롯된 일화이지만, 고대 그리스 수학의 탄생과 비약적인 발전에 대해 함의하는 바가 있다. 본 강연은 고대 그리스에서 수학이 발전하게 된 중요한 이유를 당시 지중해 세계의 민주정 사회 시민들의 자유로운 토론과 설득의 문화 속에서 찾고 있기 때문이다.

In this talk, I will discuss Q-Gorenstein deformations, especially Q-Gorenstein smoothings of normal rational elliptic surfaces, and its applications to construction of surfaces with $p_g=q=0$ and to compactifying moduli space of surfaces with $p_g=q=0$.

There are many important arithmetic invariants such class numbers, ranks of elliptic curves, and so on. The problem is that it is very hard to compute them. Instead of treating a single object one by one, we study their distribution and call it arithmetic statistics. Bhargava and Shankar's work on the average rank of elliptic curves would be one of the most famous results in arithmetic statistics. In this talk, we introduce several results in flavor of arithmetic statistics using L-functions. In the case of L-functions, we will explain what we have to know to work on a problem in arithmetic statistics.

Over the last few decades, zeta functions have become important tools in various areas of algebra beyond number theory. In this talk, I will introduce some recent developments in the theory of zeta functions of groups, rings, and other algebraic structures. In particular, I will show how these recent developments gave rise to new perspectives on many central questions in algebra.

I present in this talk a wide-ranging discussion of important topics in the theory of partial differential equations, with emphasis on the question of regularity of solutions.

In this talk, we consider the regularity property of a restricted X-ray transform. We prove sharp $L^p$ Sobolev regularity estimates for restricted X-ray transforms in all dimensions, which extends the result of Pramanik--Seeger in $\mathbb R^3$. This is achieved by constructing inductive argument and making use of the decoupling inequality for curves. This is joint work with Sanghyuk Lee and Sewook Oh.

We consider axisymmetric incompressible inviscid flows without swirl. When the axial vorticity is non-positive in the upper half space and odd in the last coordinate, we call the flow anti-parallel and we may expect a head-on collision of anti-parallel vortex rings. By establishing monotonicity and infinite growth of the vorticity impulse on the upper half-space, we obtain infinite growth of vorticity maximum at infinite time for certain classical vorticity. On the other hand, a finite but faster growth for some smooth vorticity is obtained thanks to the stability of Hill's vortex.

The Lipschitz-free space $\mathcal{F}(M)$ over a metric space $M$ is a unique Banach space that contains an isometric copy of $M$ that is linearly dense. It also satisfies a crucial universal property: any Lipschitz mapping from $M$ into a Banach space $X$ extends to a bounded linear operator from $\mathcal{F}(M)$ into $X$. This universal property gives rise to a functor from the category of metric spaces to the category of Banach spaces, emphasizing the foundational role of Lipschitz-free spaces in studies of the nonlinear geometry of Banach spaces. In this talk, we examine some geometric and structural properties of Lipschitz-free spaces.

On a compact Riemannian manifold with or without boundary, a Laplacian, which is imposed with an elliptic boundary condition if the boundary is non-empty, has a small time heat trace asymptotic expansion, each of whose coefficients has some geometric informations, for example the volume, scalar curvature and principal curvatures, etc. Along the same line, on a compact Riemannian manifold with boundary, a Dirichlet-to-Neumann operator defined on the boundary has a small time heat trace asymptotic expansion, whose coefficients also contain some geometric informations. The Dirichlet-to-Neumann operator appears in the Steklov eigenvalue problem and the BFK-gluing formula for the zeta-determinants of Laplacians. In this talk, I’m going to review some basic facts about the heat trace asymptotic expansions of Laplacians and Dirichlet-to-Neumann operators. And then, I’m going to discuss the zeta-determinants of a one parameter family of Dirichlet-to-Neumann operators and its asymptotic expansion as the parameter goes to infinity. Finally, I’m going to discuss the relation between the coefficients of this asymptotic expansion and curvature tensors including the scalar curvature and principal curvatures, etc.

Let $G$ be a semisimple Lie group, $T$ a maximal torus of $G$, and $B$ a Borel subgroup of $G$ containing $T$. For a parabolic subgroup $P$ containing $B$, the homogeneous space $G/P$ is a smooth projective algebraic variety, called a flag variety. If $P$ is minimal, that is, $P = B$, then we call $G/B$ a full flag variety. When $P$ is maximal, we call $G/P$ a Grassmannian variety. The left multiplication of $T$ on $G$ induces that on $G/P$. Considering $T$-orbit closures, we obtain lots of toric varieties including toric Schubert varieties and toric Richardson varieties. In this talk, we study topological and geometric properties of these toric varieties. This talk is based on joint work with Mikiya Masuda and Seonjeong Park.

For a compact (weakly) special square complex, the intersection complex which is a certain complex-of-group decomposition structure of its fundamental group is a well-defined invariant under quasi-isometry. In this talk, we use this invariant to classify quasi-isometry types of 2-braid groups of circumference 1 graphs. As an application, we also classify quasi-isometry types of 4-braid groups of trees. This is a joint work with Sangrok Oh(University of the Basque Country).

In this talk, I will introduce how methods from complex analysis and combinatorics can be applied in the context of random matrix theory. As a prominent example, I will discuss a random point process on a one-dimensional discrete lattice, known as the q-deformed Gaussian Unitary Ensemble (GUE). Utilizing the Flajolet-Viennot theory on the combinatorics of orthogonal polynomials, I will introduce the spectral moments of the q-deformed GUE and its genus one expansion, which provides a quantization of the Harer-Zagier formula. Combined with complex analytic tools such as the Stieltjes transform, I will then introduce the scaled limiting density, which provides the q-analog of Wigner’s celebrated semi-circle law. This talk is based on a joint work with Jaeseong Oh and Peter J. Forrester.

Currently, millions of individuals utilize wearable devices like the Apple Watch to track their physical activity, heart rate, and other physiological signals, resulting in a vast amount of wearable data. This influx of data provides a unique opportunity for digital medicine to advance precision healthcare. However, the inherent noise in this data poses a challenge, rendering it seemingly unusable without the development of new mathematical techniques for signal extraction. In this talk, I will present several techniques we have devised for analyzing this noisy time-series data. These include the Kalman filter-based data assimilation method, which serves as a novel state space estimation technique capable of estimating phases of circadian rhythms. Additionally, I will introduce a Kalman filter-assisted autoencoder for anomaly detection in time-series data, along with feature engineering methods grounded in persistent homology and mathematical modeling. These techniques offer practical applications such as sleep quality assessment, early detection of physiological changes linked to fever and daily mood prediction.

The study of gradient flows holds significant importance across various fields, including partial differential equations, optimization, and machine learning. In this talk, we aim to explore the relationship between gradient flows and their time-discretized formulations, known as De Giorgi's minimizing movements scheme. We focus on how De Giorgi's minimizing movements coincide with gradient flows in two different spaces: the space of sets and the space of probability measures called Wasserstein space. Then, we discuss their implications for free boundary problems, optimal transport, and generative models.

We consider an optimal investment problem to maximize expected utility (CRRA) of the terminal wealth, in a market with search frictions and transaction costs. In the market model, an investor's attempt of transaction is successful only at arrival times of a Poisson process, and the investor pays proportional transaction costs. The optimal trading strategy is described by the no-trade region. We provide asymptotic analysis of the value function and the no-trade boundaries, for small search frictions and transaction costs at the same time.

In this talk, the generalized $H_2$ controller synthesis problem of sampled-data systems is concerned with, in which the induced norm from $L_2$ to $L_\infty$ is minimized. We first take an operator-based approach to sampled-data systems via the lifting treatment. We next develop a framework for piecewise constant approximation in the context of the generalized $H_2$ controller synthesis problem. An optimal controller for the approximate treatment is also shown to achieve the generalized $H_2$ performance for the sampled-data system that is close enough to its optimal generalized $H_2$ performance, if the corresponding parameter $N$ is large enough. This is established by deriving upper and lower bounds on the resulting sampled-data generalized $H_2$ performance, where their gap tends to $0$ at the rate of $1/N$. Finally, numerical examples are given to validate the overall arguments.

2015 수학과 교육과정의 6가지 수학 교과 역량은 문제해결, 추론, 창의･융합, 의사소통, 정보처리, 태도 및 실천이었다. 2022 교육과정에서는 문제해결, 추론, 의사소통, 연결, 정보처리의 다섯 개로 설정되며, 이를 통해 학생들은 복잡하고 전문화되어 가는 미래 사회에서 사회 구성원의 역할을 성공적으로 수행할 수 있고, 개인의 잠재력과 재능을 발현할 수 있으며, 수학의 필요성과 유용성을 이해하고, 수학 학습의 즐거움을 느끼며, 수학에 대한 흥미와 자신감을 기를 수 있게 하였다. 교과 역량의 변화에 따른 교과서의 변화에 대하여 함께 알아보고, 교과서 저자로서 몇 가지 제언을 하고자 한다.

지방교육자치단체인 경상남도교육청의 수학교육정책과 구체적 운영 사례를 소개한다. 경상남도교육청은 공교육기관 최초로 일반인과 학생들을 위한 수학문화관과 수학체험센터를 2015년부터 건립하여 운영하고 있다. 뉴욕의 Momath(Museom of Mathematics), 도쿄의 동경이과대학 수학체험관에 못지않게 많은 사람들이 생활 속에서 수학을 경험하고 수학이 깃든 문화를 느낄 수 있는 공간이다. 학생들이 항상 가깝게 갈 수 있도록 경남 전역에 1개의 본원과 6개의 분원을 두어 경남수학교육체험벨트를 운영하고 있다. 풍성하고 알찬 수학문화관과 수학체험센터 운영을 위해, 또 수학에 대한 흥미를 많이 느낄 수 있도록 매년 수학교사들이 함께 연구하여 수학을 체험할 수 있는 콘텐츠를 개발하고 있어 올 때마다 새로운 수학문화관이 될 수 있도록 하며, 새로운 수학교육 콘텐츠 개발을 위해 매년 약 3~5억원 정도 투자한다. 경남수학문화관은 전문 연구인력을 두고 수학교육연구센터를 운영하며 학교수학교육을 지원할 수 있는 연구를 수행하고 있다. 가장 최근 연구로는 수학체험활동이 학생들의 수학적 성장에 도움이 되는가를 측정하기 위한 지표를 개발하였다. 학교에서 수학교육에 매진하는 초중등 수학교사를 지원하기 위해 경남수학교육소식을 초등과 중등으로 구분하여 매월 발간하고 있다. 그밖에 수학교육과정 운영 내실화, 학생중심 수학교육 활성화, 수학교육 교원 역량 강화, 수학문화 대중화, 체험으로 배우는 수학교육문화조성 등 경상남도교육청의 수학교육정책 전반에 대해 소개한다.

We discuss methods, results, and open problems on graph and hypergraph colorings, including the recent resolution of the Erd\H{o}s-Faber-Lov\'{a}sz conjecture and its generalizations for large hypergraphs.

In 1988, Macdonald introduced a remarkable new basis for the space of symmetric functions, called Macdonald polynomials. Upon the introduction, Macdonald also defined the integral form Macdonald polynomials $J_\mu (X;q,t)$ and conjectured positivity of their expansion in terms of modified Schur functions. This Macdonald positivity conjecture has been proved by Haiman in 2001 algebraically, but no combinatorial formulas are known in general. In the same vein of approaching to find combinatorial formulas, Haglund conjectured that for any $k\in \mathbb{N}$, \[ \left\langle \frac{J_\mu(X;q,q^\alpha)}{(1-q)^{|\mu|}}\right\rangle\in\mathbb{N}[q]. \] On the other hand, (integral form) Jack polynomials can be realized as \[ J^{(\alpha)}_\mu (X) =\lim_{t\rightarrow 1}\frac{J_\mu (X; t^\alpha, t)}{(1-t)^{|\mu|}}. \] Note that if we let $\displaystyle \tilde{J}^{(\alpha)}_\mu (X)= \lim_{q\rightarrow 1}\frac{J_\mu (X;q, q^\alpha )}{(1-q)^{|\mu|}}$, then $\tilde{J}^{(\alpha)}_\mu (X) = \alpha^n J_\mu ^{(\alpha^{-1})}(X)$. Their Schur coefficients are not $\alpha$-positive, however, Alexandersson-Haglund-Wang noticed that the Schur coefficients are positive in binomial bases $\left\{ \binom{\alpha+k}{n}\right\}_{0\le k\le n-1}$ and $\left\{ \binom{\alpha}{k}k!\right\}_{1\le k\le n}$. As the integral form and modified Macdonald polynomials are related via plethystic substitution, we go over some techniques to deal with such relations and present how we applied those techniques to $\alpha$-chroamtic symmetric functions in recent research. This is based on joint work with Jim Haglund and Jaeseong Oh.

In this talk, we discuss lattice-based cryptosystems (such as zero-knowledge proofs, signatures, or protocols) whose security is based on computational hardness problems with partial information leakage on the secret information (e.g. key, noise, randomness). We investigate how this information leakage affects the security, study the existing mitigation strategies, and propose new approaches to achieve better efficiency.