2026년 2학기 건국대학교수리과학연구소 콜로퀴움

Classical epidemic models treat transmission parameters as fixed and populations as well mixed. In practice, however, epidemic dynamics are shaped by population structure, behavioural decision-making, and adaptive learning. In this talk, I present a mathematical framework that integrates these components within a unified dynamical perspective.

We begin with a structured SEIRL model and revisit the basic reproduction number as the spectral radius of a next-generation operator. Estimating early exponential growth from incidence data allows us to connect observed growth rates to transmission parameters and highlight the sensitivity of epidemic dynamics to structural heterogeneity.

We then introduce a game-theoretic viewpoint in transmission dynamics, specifically touching on how vaccination may lend itself to be modelled by classic Nash games, and then how compliance with non-pharmaceutical interventions across interacting population groups may arise as Nash equilibria of a multi-player game in which perceived infection risk and behavioural discomfort compete. Embedding these equilibrium compliance levels into the transmission rate produces a coupled feedback system in which disease prevalence and behavioural decisions co-evolve.

Finally, we consider adaptive ensemble forecasting methods formulated as constrained optimization problems over expanding time windows. These provide a learning layer that dynamically updates predictive weights in response to past performance. In order to bring forecasting insights to the public, we put together the AI4Casting Hub, which we will showcase at the end of the talk.

Taken together, these components suggest a broader mathematical view: epidemic systems can be understood as nonlinear dynamical systems coupled with equilibrium and learning mechanisms. I conclude by discussing open questions concerning stability, equilibrium selection, and the analysis of adaptive epidemic models.

Biological systems often exhibit strongly nonlinear dynamics, such as slow–fast manifold oscillations, making accurate state-space estimation from noisy measurements particularly challenging. Classical state-space estimation methods, such as Kalman–Bucy and particle filters, either rely on restrictive Gaussian assumptions or become computationally prohibitive in high-dimensional settings. In this talk, I present two complementary approaches to nonlinear, high-dimensional data assimilation. First, I introduce an adaptive Gaussian mixture filter that splits and merges Gaussian particles to efficiently capture asymmetric and highly nonlinear state distributions. Second, I present a score-based filter that integrates score-based generative modeling with a forward process derived from the measurement equation, enabling analytically tractable likelihood updates and improved stability in high dimensions. Together, these methods provide scalable and accurate frameworks for state estimation in complex biological dynamical systems.

이번 강연에서는 미분기하학의 한 분야인 공형기하학의 세계를 탐구합니다. 특히 2차원의 공형기하를 탐구하는 복소해석학과 리만곡면론에 초점을 맞추어 설명할 예정입니다. (시간이 허락하면 3차원 이상 공형기하학의 주요 주제 중 하나인 야마베 문제에 대해서 잠깐 짚어볼 수도 있습니다.)

이 주제들을 통해 복소해석학, 기하학, 편미분방정식론, 위상수학 사이에 숨겨진 깊은 연결성을 발견하고, 수학의 우아함과 통일성을 감상할 것입니다. 강연은 대부분의 학부생들이 이해할 수 있도록 엄밀성보다는 직관적인 설명을 중심으로 진행할 계획입니다.

Biological systems, including drug action and viral dynamics, do not always respond only to their current state. Drug action depends on past exposure, and immune responses depend not only on infection levels but also on how infection has been changing. Yet these models assume instantaneous effects or rapidly fading exponential memory. This talk introduces a simple framework that embeds memory directly into model structure without adding extra states. In pharmacology, the basic target-mediated drug disposition model is extended to incorporate memory, and a reduced model is derived together with a clear validity condition. In viral dynamics, heterogeneity in immune responses is modeled using a memory structure, allowing cumulative effects to be incorporated into infected-cell growth and accounting for early suppression, delayed peaks, and persistent viral tails. The central message is that some biological memory can be modeled in a simple and structured way.

In this talk, we introduce the concept of t-core partitions. We discuss the generating function and modularity, along with some results and applications of t-core partitions. Recent results on simultaneous core partitions will also be presented. Toward the end of the talk, we introduce numerical semigroups and explore connections between numerical semigroups (or numerical sets) and partitions. Additionally, we present some open problems related to these topics.

정수들을 이용하여 정의할 수 있는 두 수학적 개념인 partition과 numerical semigroup에 관해 알려진 정리들을 소개한다. 또한 partition과 numerical semigroup간의 연결성에 대해 알아보고 이와 관련된 결과들을 설명한다.

• 학계와 업계의 차이. 연구와 공부의 차이.
• 업계로 가면 어떤 고민들을 해야 할지.
• 업계에서 쓰는 전공수학은 어떤 것들이 있을지.
• 수학 유튜브에 대한 질문 등
• 여러 주제들을 가지고 자유롭게 쌍방소통하는 시간이 되길 희망합니다.

어려운 수학 문제가 풀리는 데엔 그만한 사연이 있다. 그 예시 중 하나로 소파 옮기기 문제가 어떻게 풀리게 됐는지를 발표자의 개인적 여정을 통해 알아본다.

Generative models have achieved remarkable success across a wide range of machine learning tasks. This talk presents a unified perspective on generative modeling through the lens of optimal transport, viewing learning as an optimization problem over the space of probability measures. We will explore how this framework inspires connections between diffusion models, energy-based models, and GANs. Along the way, we will highlight recent advances and discuss how these ideas extend new directions for advancing generative models.

Artificial intelligence and mathematics have long been closely connected, each inspiring progress in the other. Mathematics offers the language and principles that guide the design of AI, while AI provides new tools and fresh perspectives to approach challenging mathematical problems. This talk reflects on the two-way relationship between AI and mathematics (AI for Mathematics & Mathematics for AI), showing how the two fields can enrich one another and open up exciting opportunities for research and innovation.

Random graph는 graph들의 모임을 확률적인 공간으로 생각하는 것으로서 1959년에 Erdős–Rényi에 의해 정의되었고, 현재는 조합론의 기초연구와 실생활로부터 정의되는 그래프 모델링 등 다양한 분야에 이용되고 있다.

본 강연에서는 먼저 random graph를 쉽게 정의한 후, 두 가지 결과를 소개하고자 한다. 첫째로는 random graph가 주어진 그래프들을 포함하는 문제인 containment problem과 universality의 개념을 설명하고 새로운 결과를 설명한다. 둘째로는 메르스의 확산 모델인 random intersection graph를 소개하고 그 모델을 random graph와 비교한 새로운 결과를 소개한다.

** 본 강연은 학부생도 쉽게 이해할 수 있는 강연입니다.

본 강연에서 물리·사회 현상에서 비선형 편미분방정식을 도출하고 해의 정칙성과 근사성에 대하여 논의한다. 특히 금융수학과 미분기하학에서의 응용문제들과 연관성에 대하여 발표하고자 한다.

In recent years, modern machine learning techniques using deep neural networks have achieved tremendous success in various fields. From a mathematical point of view, deep learning essentially involves approximating a target function, relying on the approximation power of deep neural networks. Therefore, it is important to understand the approximation and generalization properties of neural networks in high dimensions. The primary objective of this talk is to mathematically analyze the approximation of neural networks within the classical numerical analysis framework. We will explore the proper regularity of target functions which is suitable for the neural network approximation, and investigate how these properties are reflected in the approximation and learning complexity of neural networks. Next, I will apply these theories to my recent work on the operator learning method for solving parametric PDEs. I will analyze the intrinsic structure of the proposed method through the theory described above, deriving some useful results both theoretically and practically. Furthermore, I will demonstrate some relevant numerical experiments, confirming that these theory-guided strategies can be utilized to significantly improve the performance of the method.

Computational fluid dynamics (CFD) has been rapidly developed with the evolution of high-performance computing devices, such as many-core CPUs and hardware accelerators. At the same time, more complicated flow simulations are demanded for real-life applications, e.g., hypersonic reentry vehicles, magneto-hydrodynamic drive, and urban air mobility. These applications commonly contain massive flow separation, complex shock-vortex interaction, and laminar-to-turbulence transition where second-order finite volume methods (FVMs) are fundamentally limited. In this talk, the speaker introduces high-order methods (HOMs) as promising candidates to overcome the limitations of FVMs. Despite its bright outlook, however, there are still obstacles to be addressed for HOMs to reach industrial levels of applications. The speaker presents four critical challenges:

1. Preserving accuracy and cost-efficiency on hybrid meshes
2. Robust and accurate shock-capturing on multi-physical flows
3. Maximizing the computational performance on modern computing devices
4. Robust and accurate modeling for multi-physical flow simulations

Then, the speaker presents recent novel progresses to revolutionize these challenges. Finally, Deneb, a high-performance high-order CFD solver for multi-physical flow simulations, is introduced, including highlights on its attractive features.

비눗방울은 둥글다. 비눗면에 작용하는 장력과 압력 등의 평형 상태를 이루기 위해, 특정한 타원형 미분방정식을 만족하는 곡면이기 때문이다. 이는 타원형 범함수의 오일러–라그랑주 방정식으로 이해할 수 있고, 안정된 비눗방울은 그 범함수의 임계점이다. 한편, 타원형 범함수는 자연스러운 기울기 흐름을 가지며, 이는 곧 포물형 미분방정식이 된다. 포물형 미분방정식의 일종인 열방정식과 마찬가지로 극한에서 평형 상태를 이루며 특정한 모양을 취한다.

이를 통해서 곡면의 에너지가 단조 감소하는 기울기 흐름인 평균 곡률 흐름이 가지는 특이점의 형태를 관찰할 수 있고, 특이점을 통한 곡면의 위상 변화를 연구할 수 있다. 이 강연에서는 곡면의 에너지를 정의하고, 그 평형 상태의 안정성을 오일러–라그랑주 방정식의 선형화 연산자가 가지는 모스 인덱스로서 관찰하여, 작은 에너지를 가지는 단순하고 특별한 특이점에서 위상이 변화하게 강제하는 방법에 대해서 이야기한다. 시간이 있으면 이를 바탕으로 4차원 부드러운 푸앵카레 추측 연구와의 관계도 소개한다.

In this talk, we will explore various approaches to extending fundamental concepts and theorems of graph theory to signed graphs, focusing particularly on the connections with spectral graph theory and matroid theory. We will discuss the unique properties and key concepts of signed graphs, examining how familiar results in graph theory reveal new insights when applied to signed graphs. Through this exploration, we aim to highlight the potential applications and expanded perspectives that signed graph theory offers.