Friday, March 19, 2021

2021-03-26 / 10:00 ~ 12:00

by 김경훈(고려대학교)

Many types of diffusion equations have been used to describe diverse natural phenomena. The classical heat equation describes the heat propagation in homogeneous media, and the heat equation with fractional time derivative describes anomalous diffusion, especially sub-diffusion, caused by particle sticking and trapping effects. On the other hand, space-fractional diffusion equations are related to diffusion of particles with long range jumps. In this talk, I will introduce the following: 1. Elementary notion of stochastic parabolic equations 2. Stochastic processes with jumps and their related PDEs and Stochastic PDEs 3. Some regularity results of PDEs and Stochastic PDEs with non-local operators

2021-03-26 / 14:00 ~ 15:30

학과 세미나/콜로퀴엄 - 대수기하학:

by 한강진()

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants. In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

2021-03-19 / 10:00 ~ 12:00

학과 세미나/콜로퀴엄 - SAARC 세미나: Understanding Infinite-Width Deep Neural Networks

by ()

Deep neural networks have shown amazing success in various domains of artificial intelligence (e.g. vision, speech, language, medicine and game playing). However, classical tools for analyzing these models and their learning algorithms are not sufficient to provide explanations for such success. Recently, the infinite-width limit of neural networks has become one of key breakthroughs in our understanding of deep learning. This limit is unique in giving an exact theoretical description of large scale neural networks. Because of this, we believe it will continue to play a transformative role in deep learning theory. In this talk, we will first review some of the interesting theoretical questions in the deep learning community. Then we will review recent progress in the study of the infinite-width limit of neural networks focused around Neural Network Gaussian Process (NNGP) and Neural Tangent Kernel (NTK). This correspondence allows us to understand wide neural networks as different kernel based machine learning models and provides 1) exact Bayesian inference without ever initializing or training a network and 2) closed form solution of network function under gradient descent training. We will discuss recent advances, applications and remaining challenges of the infinite-width limit of neural networks.

2021-03-25 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 수리생물학:

by ()

Abstract: Millions of individuals track their steps, heart rate, and other physiological signals through wearables. This data scale is unprecedented; I will describe several of our apps and ongoing studies, each of which collects wearable and mobile data from thousands of users, even in > 100 countries. This data is so noisy that it often seems unusable and in desperate need of new mathematical techniques to extract key signals used in the (ode) mathematical modeling typically done in mathematical biology. I will describe several techniques we have developed to analyze this data and simulate models, including gap orthogonalized least squares, a new ansatz for coupled oscillators, which is similar to the popular ansatz by Ott and Antonsen, but which gives better fits to biological data and a new level-set Kalman Filter that can be used to simulate population densities. My focus applications will be determining the phase of circadian rhythms, the scoring of sleep and the detection of COVID with wearables.

2021-03-19 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 대수기하학: Introduction to infinity-categories I

by 조창연(QSMS, Seoul National University)

Infinity-category theory is a generalization of the ordinary category theory, where we extend the categorical perspective into the homotopical one. Putting differently, we study objects of interest and "mapping spaces" between them. This theory goes back to Boardman and Vogt, and more recently, Joyal, Lurie, and many others laid its foundation. Despite its relatively short history, it has found applications in many fields of mathematics. For example, number theory, mathematical physics, algebraic K-theory, and derived/spectral algebraic geometry: more concretely, p-adic Hodge theory, Geometric Langlands, the cobordism hypothesis, topological modular forms, deformation quantization, and topological quantum field theory, just to name a few. The purpose of this series of talks on infinity-categories is to make it accessible to those researchers who are interested in the topic. We’ll start from scratch and try to avoid (sometimes inevitable) technical details in developing the theory. That said, a bit of familiarity to the ordinary category theory is more or less necessary. Overall, this series has an eye toward derived/spectral algebraic geometry, but few experience in algebraic geometry would hardly matter. Therefore, everyone is welcome to join us. This is the first in the series. We’ll catch a glimpse of infinity-category theory through some motivational examples.

Events for the 취소된 행사 포함