Counting number fields and its applications

Counting number fields and its applications

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https://www.youtube.com/embed/N6uQwl_gxtA  + 733
강연자 조재현
소속 UNIST

It is a fascinating and challenging problem to count number fields with bounded discriminant. It has so many applications in number theory. We give two examples. First, we compute the average of the smallest primes belonging to a conjugacy class. Second, we compute the average residue of the Dedekind zeta functions over the family of non-Galois cubic fields.

 

프린트

Vlasov-Maxwell equations and the Dynamics of Plasmas

강연자 : 장진우 | 소속 : 포항공과대학교

 

In this colloquium talk, we study the Vlasov-Maxwell equations, a collisionless model in the field of kinetic theory. The model is a fundamental model for the dynamics of plasmas and was introduced in 1938 by Vlasov. Due to the hyperbolic nature of the system, there are still many open problems left for mathematical analysis including the global existence of classical solutions in 3D. The first part of the colloquium talk will include a brief introduction to the kinetic theory and a brief history of the mathematical analysis of the model. Then, in the rest of the talk, we introduce one kind of stability mechanism of the system, the magnetic confinement of plasmas. In the 2D annulus domain, we introduce proof for the global wellposedness of the Vlasov-Maxwell system if a huge (but finite-in-time) external magnetic potential is imposed near the boundary. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. This is a joint-work with Robert M. Strain and Tak Kwong Wong.

2023-1 Stochastic PDE(이재윤)

강연자 : 이재윤 | 소속 : KIAS

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2023-1 Probabilistic Potential Theroy (강재훈)

강연자 : 강재훈 | 소속 : BK21

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Study stochastic biochemical systems via their underlying network structures

강연자 : 김진수 | 소속 : 포항공과대학교

 

When a biological system is modeled using a mathematical process, the following step is normally to estimate the system parameters. Despite the numerous computational and statistical techniques, estimating parameters for complex systems can be a difficult task. As a result, one can think of revealing parameter-independent dynamical properties of a system. More precisely, rather than estimating parameters, one can focus on the underlying structure of a biochemical system to derive the qualitative behavior of the associated mathematical process. In this talk, we will discuss study of qualitative behaviors of stochastic biochemical systems using reaction networks. A reaction network is a graphical configuration of a biochemical system. One of the key problems in this field is to relate dynamical properties and the underlying reaction network structure. The goal of this talk is to 1. walk you through the basic modeling aspect of the stochastically modeled reaction networks, and 2. to show how to derive stability (ergodicity) of the associated stochastic process solely based on the underlying network structure.

<학부생을 위한 ɛ 강연> 복잡한 생명현상을 위한 21세기 현미경, 수학!

강연자 : 김재경 | 소속 : 카이스트

지난 50년간 진행된 분자생물학의 혁명으로 인해 생명 시스템이 수많은 분자들의 상호작용으로 구성되어 있음을 알게 되었습니다. 이러한 복잡한 시스템을 이해하기 위해서는 현미경과 같이 생명현상을 관찰하는 도구와 함께 수학은 생명과학 분야에서 필수적인 도구로 자리 잡았습니다. 수학적 언어를 사용하여 복잡한 시스템을 가상 공간에 구현하고, 이를 통해 직관적으로 이해하기 어려운 다양한 생명현상을 이해할 수 있습니다. 이러한 수리모델링을 통해 노벨상 수상자들이 놓친 생체시계의 핵심 작동 원리와 우리가 가지고 있는 좋은 수면에 대한 인식의 오류를 어떻게 발견하였는지 소개하겠습니다. 또한, 이러한 생명현상을 수학적 언어로 번역하는 과정에서 새로운 수학 문제들이 등장하여 수학 분야를 더욱 풍성하게 만들어줍니다. 특히, 수리모델에 관한 본 노이만의 유명한 이야기인 “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”이 수학적으로 과연 타당한지 논의하겠습니다.

 

Birational Geometry of varieties with effective anti-canonical divisors

강연자 : 최성락 | 소속 : 연세대학교

Fano varieties are fundamental objects in algebraic geometry. These can be considered as the unique output of the -K -minimal model program on the varieties with effective anticanonical divisors. Thus the initial models should encode the information of their resulting Fano varieties. In this talk, I will present some results related to the -K-MMP.

 

Contact instantons and entanglement of Legendrian links

강연자 : 오용근 | 소속 : IBS-CGP /POSTECH

We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Hamiltonian-perturbed equation and establish the Gromov-Floer-Hofer type convergence result for (Hamiltonian-perturbed) contact instantons of finite energy and construct its compactification of the moduli space. Then we explain how we can apply this analytical machinery in the study of contact topology and contact Hamiltonian dynamics. As an example, we explain our proof of a conjecture of Sandon and Shelukhin on the translated points of contactomorphisms on compact contact manifold. This is the contact analog to the Arnold's conjecture type in symplectic geometry.

 

<학부생을 위한 ɛ 강연> Self-Supervised Learning in Computer Vision

강연자 : 현윤석 | 소속 : 인하대학교

In recent years, artificial intelligence has made remarkable progress in developing algorithms that can learn from vast amounts of carefully labeled data. This paradigm of supervised learning has made great success in training specialist models that perform extremely well on the task they were trained to do, but there’s a limit to building more intelligent, generalized models without a massive amount of labeled data.
Self-supervised learning is an unsupervised learning method where the supervised learning task is created out of the unlabeled input data. It enables machines to understand new concepts quickly after seeing only a few examples that are labeled.
In this seminar, we will briefly review the basic concepts of deep learning and computer vision, and explain the recent development of self-supervised learning methods, focusing on representation learning in the field of computer vision.

  

<학부생을 위한 강연> 수학과 보험산업

강연자 : 유신옥 | 소속 : 라이나생명

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Now

현재 Counting number fields and its applications

강연자 : 조재현 | 소속 : UNIST

It is a fascinating and challenging problem to count number fields with bounded discriminant. It has so many applications in number theory. We give two examples. First, we compute the average of the smallest primes belonging to a conjugacy class. Second, we compute the average residue of the Dedekind zeta functions over the family of non-Galois cubic fields.