강연영상 11 페이지 > 서울대학교 수리과학부

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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.

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.

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.

 

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.

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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.

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