Sunday, September 8, 2024

2024-09-12 / 10:30 ~ 11:30

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The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

2024-09-10 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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2024-09-09 / 16:00 ~ 17:30

편미분방정식 통합연구실 세미나 - 편미분방정식: Singularities in the ion dynamics: Formation, Structure, and Propagation

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We consider the Euler-Poisson system, which describes the ion dynamics in electrostatic plasmas. In plasma physics, the pressureless model is often employed to simplify analysis. However, the behavior of solutions to the pressureless model generally differs from that of the isothermal model, both qualitatively and quantitatively - for instance, in the case of blow-up solutions. In previous work, we investigated a class of initial data leads to finite-time C^1 blow-up solutions. In order to understand more precise blow-up profiles, we construct blow-up solutions converging to the stable self-similar blow-up profile of the Burgers equation. For the isothermal model, the density and velocity exhibit C^{1/3} regularity at the blow-up time. For the pressureless model, we provide the exact blow-up profile of the density function, showing that the density is not a Dirac measure at the moment of blow-up. We also consider the peaked traveling solitary waves, which are not differentiable at a point. Our findings show that the singularities of these peaked solitary waves have nothing to do with the Burgers blow-up singularity. We study numerical solutions to the Euler-Poisson system to provide evidence of whether there are solutions whose blow-up nature is not shock-like. This talk is based on collaborative work with Junho Choi (KAIST), Yunjoo Kim, Bongsuk Kwon, Sang-Hyuck Moon, and Kwan Woo (UNIST)

2024-09-10 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: The Helly number of Hamming balls and related problems

by Zhihan Jin(ETH Zürich)

We prove the following variant of Helly’s classical theorem for Hamming balls with a bounded radius. For $n > t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X| > 1$ and all $n > t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory. This is joint work with N. Alon and B. Sudakov.

Events for the 취소된 행사 포함