Contact instantons and entanglement of Legendrian links

Contact instantons and entanglement of Legendrian links

3800
https://www.youtube.com/embed/ne87FwWXGQU  + 900
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.

 

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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.

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

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Birational Geometry of varieties with effective anti-canonical divisors

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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.

 
Now

현재 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

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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

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