Saturday, July 22, 2023

2023-07-23 / 16:00 ~ 17:00

by ()

For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, it is an open question if it is possible to construct solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative $L^2$ stability theory of Vasseur. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE. In this talk, I will discuss recent work which shows the impossibility, for a large class of 2x2 systems, of doing convex integration via the use of $T_4$ configurations. Our work applies to every well-known 2x2 hyperbolic system of conservation laws which verifies the Liu entropy condition. This talk is based on joint work with László Székelyhidi.

2023-07-28 / 15:00 ~ 16:00

학과 세미나/콜로퀴엄 - 대수기하학: Hodge theory and essential dimension

by Patrick Brosnan(University of Maryland)

The essential dimension of an algebraic object E over a field L is heuristically the number of parameters it takes to define it. This notion was formalized and developed by Buhler and Reichstein in the late 90s, who noticed at the time, that several classical results could be interpreted as theorems about essential dimension. Since the paper of Buhler and Reichstein, most of the progress on essential dimension has had to do with essential dimension of versal G-torsors for an algebraic group G. But recently Farb, Kisin and Wolfson showed that interesting theorems can be proved for certain (usually) non-versal torsors arising from congruence covers of Shimura varieties. I'll explain this work, some extensions of it proved by me and Fakhruddin, and a conjecture on period maps which generalizes the picture.

2023-07-25 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Hamilton cycles and optimal matchings in a random subgraph of uniform Dirac hypergraphs

by 강동엽(IBS 극단조합및확률그룹)

A loose cycle is a cyclic ordering of edges such that every two consecutive edges share exactly one vertex. A cycle is Hamilton if it spans all vertices. A codegree of a $k$-uniform hypergraph is the minimum nonnegative integer $t$ such that every subset of vertices of size $k-1$ is contained in $t$ distinct edges. We prove "robust" versions of Dirac-type theorems for Hamilton cycles and optimal matchings. Let $\mathcal{H}$ be a $k$-uniform hypergraph on $n$ vertices with $n \in (k-1)\mathbb{N}$ and codegree at least $n/(2k-2)$, and let $\mathcal{H}_p$ be a spanning subgraph of $\mathcal{H}$ such that each edge of $\mathcal{H}$ is included in $\mathcal{H}_p$ with probability $p$ independently at random. We prove that a.a.s. $\mathcal{H}_p$ contains a loose Hamilton cycle if $p = \Omega(n^{-k+1} \log n)$, which is asymptotically best possible. We also present similar results for Hamilton $\ell$-cycles for $\ell \geq 2$. Furthermore, we prove that if $\mathcal{H}$ is a $k$-uniform hypergraph on $n$ vertices with $n \notin k \mathbb{N}$ and codegree at least $\lfloor n/k \rfloor$, then a.a.s. $\mathcal{H}_p$ contains a matching of size $\lfloor n/k \rfloor$ if $p = \Omega(n^{-k+1} \log n)$. This is also asymptotically best possible. This is joint work with Michael Anastos, Debsoumya Chakraborti, Abhishek Methuku, and Vincent Pfenninger.

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