Tuesday, February 8, 2022

2022-02-15 / 13:00 ~ 15:00

by 김찬우()

We talk about a convergence of kinetic vorticity of Boltzmann toward the vorticity of incompressible Euler in 2D. The talk would be self-contained (1) covering necessary background in basic Boltzmann theory, asymptotic expansion (Hilbert expansion). (2) When the Euler vorticity is below Yudovich, we prove a weak convergence toward Lagrangian solutions, (3) while for the Yudovich class we have a strong convergence toward a unique solution with a rate.

2022-02-14 / 13:00 ~ 15:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 김찬우()

2022-02-15 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Independent domination of graphs with bounded maximum degree

by 김진하(IBS 이산수학그룹)

An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or $\Delta\ge 6$, every connected $n$-vertex graph of maximum degree at most $\Delta$ has an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta })(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta })n$. This is joint work with Eun-Kyung Cho, Minki Kim, and Sang-il Oum.

2022-02-08 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: A unified Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups

by Pascal Gollin(IBS 이산수학그룹)

Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. We therefore say that cycles satisfy the Erdős-Pósa property. However, while odd cycles do not satisfy the Erdős-Pósa property, Reed proved in 1999 an analogue by relaxing packing to half-integral packing, where each vertex is allowed to be contained in at most two such cycles. Moreover, he gave a structural characterisation for when the Erdős-Pósa property for odd cycles fails. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then the cycles whose values avoid a fixed finite set for each abelian group satisfy the half-integral Erdős-Pósa property, and we similarly give a structural characterisation for the failure of the Erdős-Pósa property. A multitude of natural properties of cycles can be encoded in this setting. For example, we show that the cycles of length $\ell$ modulo $m$ satisfy the half-integral Erdős-Pósa property, and we characterise for which values of $\ell$ and $m$ these cycles satisfy the Erdős-Pósa property. This is joint work with Kevin Hendrey, Ken-ichi Kawarabayashi, O-joung Kwon, Sang-il Oum, and Youngho Yoo.

2022-02-11 / 10:30 ~ 11:45

학과 세미나/콜로퀴엄 - 대수기하학: An introductory guide to mixed Hodge modules #1

by 정승조(전북대학교)

Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.

Events for the 취소된 행사 포함