권도현 (서울시립대학교)

Title: Crystalline mean curvature flow with a volume constrain

Abstract: Atoms and molecules tend to minimize their surface energy, while crystals exhibit a preference for specific directions. The motion of sets by crystalline curvature arises from these physical phenomena, with sets evolving to reduce their anisotropic perimeter. In this talk, we discuss the crystalline mean curvature flow with nonlocal forcing given by a volume constraint. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using a discrete-in-time approximation, we establish the global-in-time existence and regularity for a class of initial data with the reflection property. This talk is based on joint work with Inwon Kim (UCLA) and Norbert Pozar (Kanazawa University).

 

권혁준 (고등과학원)

Title: Quasi-probabilities in quantum information theory

Abstract: 

We review various types of quasi-probabilities introduced in quantum information theory. In particular, we discuss how negativity appearing in these quasi-probabilities can be understood as non-classical characteristics of quantum systems.

 

김도현 (성균관대학교)

Title: Asymptotic convergence of heterogeneous first-order aggregation models: from the sphere to the unitary group

Abstract: 

In this talk, we introduce heterogeneous first-order aggregations models on the unit sphere and the unitary group, and provide detailed asymptotic behavior for the models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that relative distances converge to definite positive values, and furthermore that each oscillator converges to a possibly different stationary point. In order to establish the desired results, we introduce a novel method, called dimension reduction method that can be applied to a specific situation when the degree of freedom of the natural frequency is one. In this way, we can say that although a small perturbation is allowed, convergence toward an equilibrium of the gradient flow is still guaranteed. Several first-order aggregation models are provided as concrete examples by using the dimension reduction method to study the structure of the equilibrium. This talk is based on joint work with Dr. Hansol Park (Simon Fraser University). 

 

김민현 (한양대학교)

Title: Robust near-diagonal Green function estimates

Abstract: 

We study sharp near-diagonal pointwise bounds for the Green function for nonlocal operators of fractional order $alpha in (0,2)$. The novelty of our results it two-fold: the estimates are robust as $alpha to 2-$ and we prove the bounds without making use of the Dirichlet heat kernel. In this way we can cover cases, in which the Green function satisfies isotropic bounds but the heat kernel does not. This talk is based on a joint work with Moritz Kassmann and Ki-Ahm Lee.

 

김상집 (고려대학교)

Title: FFT in IT

Abstract: 

고전 불변다항식론은 주어진 군의 작용 아래 변하지 않는 다항식들이 이루는 환에 대해 공부합니다. 우리가 관심있는 대부분의 군에 대해 이러한 환이 유한개의 원소들만으로 생성될 수 있다는 사실이 잘 알려져 있습니다만, 그럼에도 불구하고 불변다항식들을 잘 기술하는 일은 간단하지 않기에, 고전 불변다항식론은 다양한 분야의 수학자들에게 나름의 관심사에 부합하는 흥미로운 연구 주제를 지속적으로 제공하고 있습니다. 이 강의에서는 고전 선형군의 작용 아래 불변인 다항식들에 대해 소개하겠습니다.

 

박현준 (고등과학원)

Title: A Darboux theorem and virtual Lagrangian cycles for (-2)-shifted symplectic fibrations

Abstract: 

There are two crucial discoveries for (-2)-shifted symplectic derived schemes: (1) Darboux theorem of Brav-Bussi-Joyce/Bouaziz-Grojnowski (2) virtual Lagrangian cycles of Borisov-Joyce/Oh-Thomas. In this talk, we extend these to families of (-2)-shifted symplectic derived schemes. The key result is that these (-2)-shifted symplectic fibrations are (-1)-shifted Lagrangians on the derived critical loci. As an application, we prove that the Donaldson-Thomas invariants of Calabi-Yau 4-folds are invariant along the deformations for which the (0,4)-Hodge pieces of the second Chern characters remain zero. This assures that the reduced virtual cycles for counting surfaces, developed in the joint work with Bae-Kool, detect the variational Hodge conjecture. We also propose a refined Donaldson-Thomas theory of Calabi-Yau 4-folds in terms of the vanishing cycle cohomology of the Hodge loci, relying on the Joyce conjecture for (-1)-shifted Lagrangians. This is work in progress.

 

서성미 (충남대학교)

Title: Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

Abstract: 

In this talk, I will introduce the two-dimensional Coulomb gas model, where particles interact logarithmically with each other in the presence of an external field. This talk focuses on the asymptotic behavior of the partition function as the number of particles tends to infinity. We find that the certain correction terms in the asymptotic expansion contain topological information about the droplet, the region where the particles accumulate. 

This is based on joint work with Sung-Soo Byun and Nam-Gyu Kang. 

 

이기암 (서울대학교)

Title: Homogenizations of Nonlinear Partial Differential Equations

Abstract: 

In this talk, we are going to consider recents developments in the theory of Homogenizations for the nonlinear partial differential equations, 

which is concerned with the derivation of homogenized or effective equations satisfied by the limit of solutions of differential equations with rapidly oscillating data. 

Mainly we will discuss convergence rates and higher order convergence, nonvariational problems, lower order oscillations and so on.

 

이동건 (IBS-CCG)

Title: Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

Abstract: 

An alternative way to describe the twin manifold of a regular semisimple Hessenberg variety is as a manifold of staircase Hermitian matrices with a given spectrum. One interesting aspect of this manifold is its inherent symmetric group action on its cohomology. In this talk, I will show that certain triples of twin manifolds are related by blowups and smooth fibrations, and establish relations, commonly known as the modular law, between the induced representations on their cohomology. As a corollary, we provide a direct proof of that the Frobenius characteristics of these representations coincide with the associated unicellular LLT polynomials, which are certain symmetric functions introduced in the study of quantum affine algebras. This talk is based on a joint work with Prof. Young-Hoon Kiem.

 

이동헌 (고려대학교)

Title: Regularizing Knowledge Gradient for Online Learning

Abstract: 

강화학습과 많은 연결점을 가지고 있는 sequential decision making 문제와, 이를 해결하기 위해 고안된 knowledge gradient (KG) 기법에 대해서 간단히 살펴보도록 하겠습니다. 또한, 원래 offline learning에 대응하는 KG 알고리즘을 online learning에 대응하도록 개량하는 방법으로서 regularization을 사용한 연구에 대해서 논의합니다. 이 연구를 통하여 고안된 ORKG알고리즘은, KG계열의 알고리즘 중에서 일반적인 sublinear regret bound를 최초로 보였으며, 여타 KG알고리즘 대비 online learning대응 성능 또한 향상되어 다양한 multi-armed bandit(MAB)대응 알고리즘들과 Gaussian MAB에서 대등한 성능을 달성하였습니다.

 

이중경 (고등과학원)

Title: Mathematical study of metastability

Abstract:

Metastability is a phenomenon observed in various physical models in the low temperature regime.

Its description can be approached from two primary perspectives.

The first involves estimating the exit time from a metastable state, which is known as the Eyring-Kramers formula.

The second perspective entails using Markov chain to provide a comprehensive description of successive transitions.

In this talk, we will focus on introducing the metastability of overdamped Langevin dynamics, which represents one of the most significant models where metastability occurs.

 

 

임성빈 (고려대학교)

Title: Advances in the Score-based Generative Models: Theory and Application

Abstract: 

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. 

 

정민구 (고등과학원)

Title: On norm attaining mapping

Abstract: 

In this talk, we look at the behavior of norm attaining mappings between Banach spaces and other Banach space-related topics. The presentation will be structured to help the audience understand from the fundamentals of the subject to the recent developments.

 

지운식 (충북대학교)

Title: Analytic Representation of CAP Operators 

Abstract: 

Let $X$ be a random variable with finite moments of all orders.

By the three-term recurrence relation of the orthogonal polynomials associated with the distribution of $X$,

the random variable $X$ as a multiplication operator is represented as a sum of three linear operators 

called creation, annihilation and preservation operators (CAP-operators). 

In this talk, we discuss analytic representations of the CAP operators in terms of differential operators with polynomial coefficients.

This approach extends the usual Boson quantum mechanics corresponding to the Gaussian measure 

to the quantum mechanics associated with random variables.  

This talk is based on a joint work with A. Ebang Ella, L. Accardi and Y. G. Lu.

 

최인혁 (고등과학원)

Title: Geometry and dynamics of a finitely generated group

Abstract: 

A finitely generated group G comes with a homogeneous and locally compact simplicial graph called the Cayley graph of G. Since G naturally acts on its Cayley graph, one can discuss the dynamics of each element of G. In this talk, I will explain how one can detect the hyperbolicity of G via the dynamics of a typical element of G. Joint work with Kunal Chawla and  Giulio Tiozzo.

 

홍영준 (성균관대학교)

Title: Recent advancements on machine learning for scientific computing

Abstract: 

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 scientific machine learning for solving PDEs and generative models in computational materials. Specifically, my research presents a novel machine learning-based numerical solver for solving parametric PDEs without harnessing data based on spectral methods or Finite element methods. Our proposed framework combines the advantages of numerical methods, including high accuracy, efficiency, generalization, and exact satisfaction of boundary conditions, with the capabilities of deep neural networks. It can effectively learn and predict solutions of complex parametric PDEs, including singularly perturbed convection-diffusion equations and Navier-Stokes equations.