Tuesday, February 15, 2022

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2022-02-16 / 10:00 ~ 12:00
학과 세미나/콜로퀴엄 - PDE 세미나: 인쇄
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.
2022-02-21 / 16:30 ~ 17:30
IBS-KAIST 세미나 - 이산수학: A stronger version of Tutte’s wheel theorem for vertex-minors 인쇄
by 김동규(IBS 이산수학그룹 / KAIST)
Tutte (1961) proved that every simple $3$-connected graph $G$ has an edge $e$ such that $G \setminus e$ or $G / e$ is simple $3$-connected, unless $G$ is isomorphic to a wheel. We call such an edge non-essential. Oxley and Wu (2000) proved that every simple $3$-connected graph has at least $2$ non-essential edges unless it is isomorphic to a wheel. Moreover, they proved that every simple $3$-connected graph has at least $3$ non-essential edges if and only if it is isomorphic to neither a twisted wheel nor a $k$-dimensional wheel with $k\geq2$. We prove analogous results for graphs with vertex-minors. For a vertex $v$ of a graph $G$, let $G*v$ be the graph obtained from $G$ by deleting all edges joining two neighbors of $v$ and adding edges joining non-adjacent pairs of two neighbors of $v$. This operation is called the local complementation at $v$, and we say two graphs are locally equivalent if one can be obtained from the other by applying a sequence of local complementations. A graph $H$ is a vertex-minor of a graph $G$ if $H$ is an induced subgraph of a graph locally equivalent to $G$. A split of a graph is a partition $(A,B)$ of its vertex set such that $|A|,|B| \geq 2$ and for some $A'\subseteq A$ and $B'\subseteq B$, two vertices $x\in A$ and $y\in B$ are adjacent if and only if $x\in A'$ and $y\in B’$. A graph is prime if it has no split. A vertex $v$ of a graph is non-essential if at least two of three kinds of vertex-minor reductions at $v$ result in prime graphs. We prove that every prime graph with at least $5$ vertices has at least two non-essential vertices unless it is locally equivalent to a cycle. It is stronger than a theorem proved by Allys (1994), which states that every prime graph with at least $5$ vertices has a non-essential vertex unless it is locally equivalent to a cycle. As a corollary of our result, one can obtain the first result of Oxley and Wu. Furthermore, we show that every prime graph with at least $5$ vertices has at least $3$ non-essential vertices if and only if it is not locally equivalent to a graph with two specified vertices $x$ and $y$ consisting of at least two internally-disjoint paths from $x$ to $y$ in which $x$ and $y$ have no common neighbor. This is joint work with Sang-il Oum.
2022-02-15 / 13:00 ~ 15:00
학과 세미나/콜로퀴엄 - PDE 세미나: 인쇄
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.
2022-02-17 / 11:00 ~ 12:00
학과 세미나/콜로퀴엄 - 대수기하학: 인쇄
by ()

2022-02-16 / 15:00 ~ 16:00
학과 세미나/콜로퀴엄 - 대수기하학: 인쇄
by ()

2022-02-15 / 16:30 ~ 17:30
IBS-KAIST 세미나 - 이산수학: Independent domination of graphs with bounded maximum degree 인쇄
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.
2022-02-18 / 10:30 ~ 11:45
학과 세미나/콜로퀴엄 - 대수기하학: An introductory guide to mixed Hodge modules #2 인쇄
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.
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