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Motivated by the Cohen-Lenstra heuristics, Friedman and Washington studied the distribution of the cokernels of random matrices over the ring of p-adic integers. This has been generalized in many directions, as well as some applications to the distribution of random algebraic objects. In this talk, first we give an overview of random matrix theory over the ring of p-adic integers, together with their connections to conjectures in number theory. After that, we investigate the distribution of the cokernels of random p-adic matrices with given zero entries. The second part of this talk is based on work in progress with Gilyoung Cheong, Dong Yeap Kang and Myungjun Yu.
Host: Bo-Hae Im     한국어 (필요한 경우 영어 가능) ( )     2024-02-18 14:15:00
In dimension 4, the works of Freedman and Donaldson led us to the striking discovery that the smooth category is drastically different from the topological category, compared to other dimensions. Since then, it has been extraordinarily successful in investigating the difference in various contexts. In contrast, our understanding of when smooth and topological categories would exhibit similarity in dimension 4 remained, at best, minimal. In this talk, we will introduce some recent progress on new “topological = smooth” results in dimension 4, focusing on embedded disks.
Host: 백형렬     영어     2024-02-22 11:15:46
Scientific knowledge, written in the form of differential equations, plays a vital role in various deep learning fields. In this talk, I will present a graph neural network (GNN) design based on reaction-diffusion equations, which addresses the notorious oversmoothing problem of GNNs. Since the self-attention of Transformers can also be viewed as a special case of graph processing, I will present how we can enhance Transformers in a similar way. I will also introduce a spatiotemporal forecasting model based on neural controlled differential equations (NCDEs). NCDEs were designed to process irregular time series in a continuous manner and for spatiotemporal processing, it needs to be combined with a spatial processing module, i.e., GNN. I will show how this can be done.
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
Host: 2734 박진현     Contact: 2734 박진현 (2734 박진현)     미정     2024-01-26 13:31:00
The Kudla-Rapoport conjecture predicts a relation between the arithmetic intersection numbers of special cycles on a unitary Shimura variety and the derivative of representation densities for hermitian forms at a place of good reduction. In this talk, I will present a variant of the Kudla-Rapoport conjecture at a place of bad reduction. Additionally, I will discuss a proof of the conjecture in several new cases in any dimension. This is joint work with Qiao He and Zhiyu Zhang.
Host: Bo-Hae Im     미정     2024-02-18 15:51:25
This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.
Host: 박진현     Contact: 박진현 (2734)     한국어     2024-02-28 14:24:00
After a brief review of the history, some applications of these models will be reviewed. This will include descriptions of rogue waves, tsunami propagation, internal waves and blood flow. Some of the theory emanaging from these applications will then be sketched.
Host: 홍영준     영어     2024-02-22 11:17:49
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
Host: 2734 박진현     Contact: 2734 박진현 (2734 박진현)     미정     2024-01-26 13:32:33
The Julia set of a (hyperbolic) rational map naturally comes embedded in the Riemann sphere, and thus has a Hausdorff dimension. But the Hausdorff dimension varies if we tweak the parameters slightly. Is there a "best" representative or more invariant dimension? One answer comes from looking at quasi-symmetries; the \emph{conformal dimension} of the Julia set is the minimum Hausdorff dimension of any metri quasi-symmetric to the original. We characterize the Ahlfors-regular conformal dimension of Julia sets of rational maps using graphical energies arising from a natural combinatorial description. (Ahlfors-regular is a dynamically natural extra condition on the metric.) This is joint work with Kevin Pilgrim.
Host: 백형렬     영어     2024-03-07 10:32:02
This talk presents mathematical modeling, numerical analysis and simulation using finite element method in the field of electromagnetics at various scales, from analyzing quantum mechanical effects to calculating the scattering of electromagnetic wave in free space. First, we discuss and analyze the Schrodinger-Poisson system of quantum transport model to calculate electron states in three-dimensional heterostructures. Second, the electromagnetic vector wave scattering problem is solved to analyze the field characteristics in the presence of stealth platform. This talk also introduces several challenging issues in these applications and proposes their solutions through mathematical analysis.
This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.
Host: 박진현     Contact: 박진현 (2734)     한국어     2024-02-28 14:22:55
A rational map, like f(z) = (1+z^2)/(1-z^2), gives a map from the (extended) complex plane to itself. Studying the dynamics under iteration yields beautiful Julia set fractals with intricate nested structure. How can that structure be best understood? One approach is combinatorial or topological, giving concrete models for the Julia set and tools for cataloguing the possibilities.
Host: 백형렬     영어     2024-02-22 11:14:37
Optimal Transport (OT) problem investigates a transport map that bridges two distributions while minimizing a specified cost function. OT theory has been widely utilized in generative modeling. Initially, the OT-based Wasserstein metric served as a measure for assessing the distance between data and generated distributions. More recently, the OT transport map, connecting data and prior distributions, has emerged as a new approach for generative models. In this talk, we will introduce generative models based on Optimal Transport. Specifically, we will present our work on a generative model utilizing Unbalanced Optimal Transport. We will also discuss our subsequent efforts to address the challenges associated with this approach.
This is part of an informal seminar series to be given by Mr. Jaehong Kim, who has been studying the book "Hodge theory and Complex Algebraic Geometry Vol 1 by Claire Voisin" for a few months. There will be 6-8 seminars during Spring 2024, and it will summarize about 70-80% of the book.
Host: 2734 박진현     Contact: 2734 박진현 (2734 박진현)     미정     2024-01-26 13:26:53
List flow is a geometric flow for a pair $(g,u)$, where $g$ is a Riemannian metric and $u$ a smooth function. This extended Ricci flow system has applications to static vacuum solutions of the Einstein equations and to Ricci flow on warped products. The coupling induces additional difficulties compared to Ricci flow, which we overcome by proving an improved bound on the Hessian of the function u. This allows us to prove a convergence result, a singularity classification result and a surgery result in three dimensions.
Host: 박지원     영어     2024-02-21 21:29:46