Wednesday, March 19, 2025

2025-03-21 / 10:00 ~ 11:30

by 이호민()

In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role to understand group actions. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures. In this talk, we will study non-linear actions on surfaces using random dynamics. We will discuss about absolutely continuity and exact dimensionality of stationary measures as well as classification of orbit closures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

2025-03-20 / 11:50 ~ 12:30

대학원생 세미나 - 대학원생 세미나:

by 서해송(카이스트 수리과학과)

Hyperbolicity is a fundamental concept that connects differential geometry and algebraic geometry. It is in general very hard to determine whether a given manifold or variety is hyperbolic or not. A key tool for verifying hyperbolicity is symmetric differentials; more precisely, the positivity of the cotangent bundle. In this talk, I will introduce various notions of hyperbolicity and explore their geometric properties. I will also discuss how the cotangent bundle, or more generally the syzygy bundle, plays a crucial role in this context.

2025-03-25 / 16:00 ~ 17:00

SAARC 세미나 - SAARC 세미나:

by 김동한(KAIST 수리과학과)

H. Föllmer introduced in 1981 a version of Itô's formula without any probabilistic assumptions. It has been generalized in several aspects, including pathwise Tanaka's formula, high-order, and functional change-of-variable formula. Its drawbacks and a brief application to mathematical finance will also be presented.

2025-03-20 / 14:00 ~ 15:30

학과 세미나/콜로퀴엄 - 정수론:

by 김연수(전남대학교)

We establish the generic local Langlands correspondence by showing the equality of Langlands-Shahidi L-functions and Artin L-functions in the case of even unitary similitude groups. As an application, with one assumption on L-function, we prove both weak and strong versions of the generic Arthur packet conjectures in the cases of even unitary similitude groups and even unitary groups. Furthermore, we describe and define generic L-packets and therefore we were able to remove the above assumption. With our definition of L-packets, we recently prove its expected properties such as Shahidi's conjecture and finiteness of L-packets. This is in preparation and joint work with Muthu Krishnamurthy and Freydoon Shahidi.

2025-03-26 / 10:00 ~ 11:00

학과 세미나/콜로퀴엄 - 위상수학 세미나:

by 서동균(서울대학교)

A surface can be decomposed into a union of pairs of pants, a construction known as a pants decomposition. This fundamental observation reveals many important properties of surfaces. By forming a simplicial graph whose vertices represent pants decompositions, connecting two vertices with an edge whenever the corresponding decompositions differ by a simple move, we obtain a graph that is quasi-isometric to the Weil–Petersson metric on Teichmüller space. Meanwhile, topologists often study a structure called a rose, formed by attaching multiple circles at a single point. A rose is homotopy equivalent to a compact surface with boundary. Consequently, we can define a pants decomposition of a rose as the pants decomposition of a surface homotopy equivalent to it. In this talk, we will explore the concept of pants decompositions specifically in the context of roses.

2025-03-20 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

by 김연수(전남대학교 수학교육과)

The Langlands program, introduced by Robert Langlands, is a set of conjectures that attempt to build bridges between two different areas: Number Theory and Representation Theory (Automorphic forms). The program is also known as a generalization of a well-known theorem called Fermat’s Last Theorem. More precisely, when Andrew Wiles proved Fermat’s Last Theorem, he proved a special case of so-called Taniyama-Shimura-Weil Conjecture, which states that every elliptic curve is modular. And as a corollary, he was able to prove Fermat’s Last Theorem since Taniyama-Shimura-Weil Conjecture implies that certain elliptic curves associated with Fermat-type equations must be modular, leading to a contradiction. Note that the Langlands program is a generalization of the Taniyama-Shimura-Weil conjecture. In the first part of the colloquium, we briefly go over the following subjects: (1) Fermat’s Last Theorem (2) Taniyama-Shimura-Weil conuecture And then, in the remaining of the talk, we start to explain a bit of the Langlands program (3) Langlands program and L-functions (4) (If time permits) Recent progress This colloquium will be accessible to graduate students in other fields of mathematics (and undergraduate students who are interested in Number theory) at least in the first part.

2025-03-21 / 11:00 ~ 12:00

IBS-KAIST 세미나 - IBS-KAIST 세미나:

by ()

Our present healthcare system focuses on treating people when they are ill rather than keeping them healthy. We have been using big data and remote monitoring approaches to monitor people while they are healthy to keep them that way and detect disease at its earliest moment presymptomatically. We use advanced multiomics technologies (genomics, immunomics, transcriptomics, proteomics, metabolomics, microbiomics) as well as wearables and microsampling for actively monitoring health. Following a group of 109 individuals for over 13 years revealed numerous major health discoveries covering cardiovascular disease, oncology, metabolic health and infectious disease. We have also found that individuals have distinct aging patterns that can be measured in an actionable period of time. Finally, we have used wearable devices for early detection of infectious disease, including COVID-19 as well as microsampling for monitoring and improving lifestyle. We believe that advanced technologies have the potential to transform healthcare and keep people healthy.

2025-03-24 / 16:00 ~ 17:30

편미분방정식 통합연구실 세미나 - 편미분방정식:

by 김태규()

In this note, we investigate threshold conditions for global well-posedness and finite-time blow-up of solutions to the focusing cubic nonlinear Klein–Gordon equation (NLKG) on $\bbR^{1+3}$ and the focusing cubic nonlinear Schrödinger equation (NLS) on $\bbR$. Our approach is based on the Payne–Sattinger theory, which identifies invariant sets through energy functionals and conserved quantities. For NLKG, we review the Payne–Sattinger theory to establish a sharp dichotomy between global existence and blow-up. For NLS, we apply this theory with a scaling argument to construct scale-invariant thresholds, replacing the standard mass-energy conditions with a $\dot{H}^{\frac12}$-critical functional. This unified framework provides a natural derivation of global behavior thresholds for both equations.

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