Fano varieties are among the most significant families of varieties in the classification of algebraic varieties. They have been the subject of extensive study for the last century and continue to be a crucial area of research in algebraic geometry today. This talk aims to provide a brief history of Fano varieties, shedding light on their evolution in the field of algebraic geometry. In addition, it will present some contemporary research issues on Fano varieties, highlighting the recent advancements in this field.

최근 GPT, Stable Diffusion 등과 같이 다양한 인공지능 분야 (영상, 언어, 음성 등)에서 생성 인공지능 모델과 관련 기술이 크게 각광을 받고 있다. 본 강연에서는 이러한 생성 모델의 기본이 되는 수학적인 원리를 소개하고, 이들을 활용한 다양한 응용 사례를 소개한다. 또한 이러한 생성 인공지능 기술이 인간수준의 지능을 가진 인공지능을 달성하기 위해 가지는 역할에 대해서 이야기한다.

The renormalized volume is an invariant for convex cocompact hyperbolic 3-manifolds and the Liouville action is an invariant for conformal boundary Riemann surface of the convex cocompact hyperbolic 3-manifolds. In this talk, some basic results including their relationship are explained.

The mean curvature flow is an evolution of surfaces satisfying a geometric heat equation. The flow develops singularities, and changes the topology of the surfaces through singularities. Thus, the flow should be well-defined even in space-time neighborhoods of singularities. In this talk, we discuss about how to define the flow at singularities by using surgery and weak solutions.

Among many different ways to introduce derived algebraic geometry is an interplay between ordinary algebraic geometry and homotopy theory. The infinity-category theory, as a manifestation of homotopy theory, supplies better descent results even for ordinary algebro-geometric objets, not to mention objects of interest in the derived setting. I’ll explain what this means in the first half. The second half will be devoted to my recent work on some excision and descent results for commutative ring spectra, generalizing Milnor excision for perfect complexes of ordinary commutative rings and $v$-descent for perfect complexes of locally noetherian derived stacks by Halpern-Leistner and Preygel, respectively. No prior experience on derived algebraic geometry is required for the talk.

In an algebraic view point, a Hamiltonian system is an evolution equation described by a Poisson algebra structure. Particularly, in infinite dimensional cases, Hamiltonian systems can be investigated via a special type of Poisson algebras called Poisson vertex algebras. Moreover, if the system additionally has supersymmetries, we need to consider Poisson vertex algebra with supersymmetries, namely SUSY Poisson vertex algebra. In this talk, I will explain basic notions and properties of super Hamiltonian systems using the language of Poisson algebras.

One of the central topics in abstract harmonic analysis is Fourier analysis. The Fourier transforms on groups can be explained using convolution operators, and their analytic properties are heavily influenced by the underlying spaces and their algebraic structures. For example, the Fourier coefficients of integrable functions on non-abelian compact groups are no longer numbers, but they are matrices. The main aim of this talk is to provide an introduction to abstract harmonic analysis on groups, including the Plancherel theorem, the Hausdorff-Young inequality, Hardy-Littlewood inequalities, and Sobolev embedding properties.

In this talk, the dispersion managed nonlinear Schrödinger equations and their averaged equations are considered. They arise naturally in modeling fiber-optics communication systems with periodically varying dispersion profile. We discuss the well-posedness, homogenization, existence and properties of ground states, and orbital stability of the set of ground states.

In this talk, we review the surface theories in the 6 subspaces of the four-dimensional Lorentz space $\mathbb{L}^4$, which are the Euclidean three-space $\mathbb{E}^3$, hyperbolic three-space $\mathbb{H}^3(-1)$, Lorentzian three-space $\mathbb{L}^3$, de Sitter three-space $\mathbb{S}^3_1(1)$, the isotropic three-space $\mathbb{I}^3$, and the light cone three-space $\mathbb{Q}^3+$. They belong to $$ \{ Riemannian, \ Lorentzian, \ degenerate \} \times \{ \ hyperplane, \ hypersphere \}. $$ In particular we examine the structure of the various Bjorling representation formula for CMC 0 or 1 surfaces in them.

In 1976, Cappell and Shaneson constructed an infinite family of smooth homotopy 4-spheres, called Cappell-Shaneson homotopy 4-spheres. Cappell-Shaneson homotopy 4-spheres are the most notable potential counterexamples to the smooth 4-dimensional Poincare conjecture and they are related to other important conjectures including the Gluck, Schoenflies, slice-ribbon conjectures. In this talk, I will give a survey on Cappell-Shaneson homotopy 4-spheres.

Diffusion models have recently acquired significant attention in the field of generative modeling of machine learning research due to their various theoretical advantages and remarkable applications in artificial intelligence, such as Stable Diffusion and DALL-E. In this presentation, we first introduce the theoretical background of the score-based diffusion models and present the latest results of their applications to machine learning. We also present advanced score-based generative models based on the time reversal theory of Lévy processes and diffusion theory in Hilbert space. 

In this talk, we will present the Active Neuron Least Squares (ANLS), an efficient training algorithm for neural networks(NNs). ANLS is designed from the insight gained from the analysis of gradient descent training of NNs, particularly, the analysis of Plateau Phenomenon. The core mechanism is the option to perform the explicit adjustment of the activation pattern at each step, which is designed to enable a quick exit from a plateau. The performance of ANLS will be demonstrated and compared with existing popular methods in various learning tasks ranging from function approximation to solving PDEs, and operator learning.

This lecture explores the topics and areas that have guided my research in computational mathematics and deep learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and deep neural networks have achieved state-of-the-art results in a range of fields. The rapid expansion and outstanding success of deep learning and scientific computing have led to their applications across multiple disciplines. In this lecture, I will focus on connecting machine learning with applied mathematics, specifically discussing topics such as adversarial examples, generative models, and scientific machine learning.

Abduction, the process of making based on observations, is a critical component of mathematics education. It enables students to formulate conjectures or theories that can be tested and proven through deductive reasoning. This is essential for mathematical modeling and proving, as abduction is often used to explore and discover the underlying structure of a problem, while deduction is used to confirm the results. However, students often struggle with abductive reasoning, as it requires a high level of cognitive flexibility and creativity. Therefore, it is important for educators to emphasize the importance of abductive reasoning in mathematics education and provide students with opportunities to develop this skill. Students who can do abductive inferences are more likely to have higher mathematics achievement in tests, as well as develop problem-solving skills and a sense of curiosity and inquiry. In addition, epistemic actions, such as exploring possibilities, and making hypotheses, are considered to be the top of the hierarchy of epistemic actions. By incorporating these actions into mathematics education, educators can help students develop a deeper understanding of mathematical concepts and become more effective problem solvers.

The nerve complex of a family of convex sets is an abstract simplicial complex that records the intersection patterns of the family, and the f-vector of the family is the face vector of the nerve complex. In the context of geometric transversal theory one can similarly define a chain of nerve complexes which gives us a collection of f-vectors. In this talk we will discuss results and open questions related to f-vectors and nerve complexes of families of convex sets.

1994년에 P. Shor가 소개한 양자 알고리즘은 양자컴퓨팅을 사용하여 다항식 시간 내에 인수분해나 이산대수 문제를 해결할 수 있는 것으로 알려져 있어 양자컴퓨터가 실용화된다면 기존 공개키암호에 심각한 위협이 될 수 있다. 양자컴퓨팅을 이용한 공격에도 안전할 것으로 기대되는 암호를 ‘양자내성암호’라고 한다. 2017년 시작된 미국 표준화 기구 NIST의 표준화 공모사업을 필두로 많은 국가에서 양자내성암호의 개발 및 표준화에 힘쓰고 있으며, 국내에서도 이와 관련된 분야의 활성화가 필요한 실정이다. 본 발표에서는 국가보안기술연구소가 국가정보원의 후원으로 2021년에 발족한 ‘양자내성암호연구단’의 설립 목적을 소개하고 현재까지의 연구단 주요 활동 내용과 진행사항 및 추후 계획에 대해 소개하고자 한다.