Thursday, September 19, 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.

2024-09-20 / 14:00 ~ 16:00

IBS-KAIST 세미나 - 수리생물학:

by ()

In this talk, we discuss the paper “Achieving Occam’s razor: Deep learning for optimal model reduction” by Botond B. Antal et.al., PLOS Computational Biology, 2024.

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).

2024-09-20 / 11:30 ~ 13:00

학과 세미나/콜로퀴엄 - 대수기하학:

by 김영락(부산대)

An Ulrich bundle E on an n-dimensional projective variety (X, O(1)) is a vector bundle whose module of twisted global sections is a maximal Cohen-Macaulay module having the maximal number of generators in degree 0. It was once studied by commutative algebraists, but after Eisenbud and Schreyer introduced its geometric viewpoint, many people discovered several important applications in wide areas of mathematics. In this motivating paper, Eisenbud-Schreyer asked a question whether a given projective variety has an Ulrich bundle, and what is the minimal possible rank of an Ulrich bundle if exists. The answer is still widely open for algebraic surfaces and higher dimensional varieties. Thanks to a number of studies, the answer for the above question is now well-understood for del Pezzo threefolds. In particular, a del Pezzo threefold V_d of (degree d≥3) has an Ulrich bundle of rank r for every r at least 2. The Hartshorne-Serre correspondence translates the existence of rank-3 Ulrich bundle into the existence of an ACM curve C in V_d of genus g=2d+4 and degree 3d+3. In this talk, we first recall a construction of rank-3 Ulrich bundle on a cubic threefold by Geiss and Schreyer, by showing that a "random" curve of given genus and degree lies in a cubic threefold and satisfies the whole conditions we needed. We also discuss how this problem is related to the unirationality of the Hurwitz space H(k, 2g+2k-2) and the moduli of curves M_g. An analogous construction works for d=4, however, for d=5 a general curve of genus 14 and degree 18 does not belong to V_5. We characterize geometric conditions when does such a curve can be embedded into V_5 using the vanishing resonance. This is a joint work with Marian Aprodu.

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.

2024-09-20 / 14:00 ~ 16:00

학과 세미나/콜로퀴엄 - 위상수학 세미나:

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We give L^2-signature obstructions to embedding closed 3-manifolds with infinite cyclic first homology in closed 4-manifolds with infinite cyclic fundamental group preserving first homology. From the obstructions, we obtain lower bounds on the double slice genus of a knot, and give examples of algebraically doubly slice knots with vanishing Casson-Gordon invariants whose double slice genera are arbitrarily large. This is a joint work with Taehee Kim.

2024-09-20 / 11:05 ~ 12:00

학과 세미나/콜로퀴엄 - 응용 및 계산수학 세미나:

by 이진실()

We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to other PDEs. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the two- and three-dimensional settings.

2024-09-19 / 10:30 ~ 11:30

학과 세미나/콜로퀴엄 - 대수기하학:

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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.

2024-09-23 / 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)

Events for the 취소된 행사 포함