Saturday, September 21, 2024

2024-09-26 / 14:30 ~ 15:30

by 조재현()

Let J={a,b} be an unordered pair of F_q, and E_J the associated elliptic curve of the form y^3=(x-a)(x-b) over \F_q. We show that there are "only three possible values" for the trace of Frobenius of E_J. Furthermore, these three values can be computed via a certain Jacobi sum. As applications, we first compute the average analytic rank of a certain family of elliptic curves. Next, we generate elliptic curves with designated extremal primes. After computing a variant of the n-th moment of Traces of Frobenius, we give explicit values and average values on class numbers of every constant field extension of K_J=F_q(\sqrt[3]{(T-a)(T-b)}). Finally, we compute the exact values and the average values on Euler-Kronecker constants of K_J. This is a joint work with Jinjoo Yoo.

2024-09-24 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Extremal theory of 0-1 matrices

by Gábor Tardos(Alfréd Rényi Institute of Mathematics)

We say that a 0-1 matrix A contains another such matrix (pattern) P if P can be obtained from a submatrix of A by possibly changing a few 1 entries to 0. The main question of this theory is to estimate the maximal number of 1 entries in an n by n 0-1 matrix NOT containing a given pattern P. This question has very close connections to Turan type extremal graph theory and also to the Devenport-Schinzel theory of sequences. Results in the extremal theory of 0-1 matrices proved useful in attacking problems in far away fields as combinatorial geometry and the analysis of algorithms. This talk will concentrate on acyclic patterns and survey some old and recent results in the area and will also contain several open problems.

2024-09-26 / 11:50 ~ 12:40

대학원생 세미나 - 대학원생 세미나:

by 은남현()

In this talk, we will discuss the uniqueness and stability of a Riemann shock solution to the compressible Euler system, which is a self-similar entropy solution connecting two different constant states, in a physical vanishing viscosity limits. We focus on the one dimensional compressible full Euler system and consider the Brenner-Navier-Stokes-Fourier system, which is an amendment of the Navier-Stokes-Fourier system, to describe the physical perturbation class. (This is a joint work with Moon-Jin Kang (KAIST) and Saehoon Eo (Stanford University).

2024-09-27 / 13:30 ~ 14:30

학과 세미나/콜로퀴엄 - Topology, Geometry, and Data Analysis: Introduction to Graph Neural Networks (Part 2)

by 서동엽(KAIST)

2024-09-25 / 13:30 ~ 14:30

학과 세미나/콜로퀴엄 - Topology, Geometry, and Data Analysis: Introduction to Graph Neural Networks (Part 1)

by 서동엽(KAIST)

2024-09-26 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

by 조재현(UNIST 수리과학과)

First, we briefly introduce arithmetic statistics. After that we give some AS problems related to L-functions such as class numbers and analytic rank of elliptic curves.

2024-09-23 / 16:00 ~ 17:00

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

by ()

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)

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