Saturday, November 28, 2020

2020-12-02 / 17:00 ~ 18:00

by 이준경()

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erd\H{o}s, conjectured that every graph is common. The conjectures by Erd\H{o}s and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Despite its importance, the full classification of common graphs is still a wide open problem and has not seen much progress since the early 1990s. In this lecture, I will present some old and new techniques to prove whether a graph is common or not.

2020-11-30 / 17:00 ~ 18:00

학과 세미나/콜로퀴엄 - Discrete Math:

by 이준경()

Abstract: Ramsey's theorem states that, for a fixed graph $H$, every 2-edge-colouring of $K_n$ contains a monochromatic copy of $H$ whenever $n$ is large enough. Perhaps one of the most natural questions after Ramsey's theorem is then how many copies of monochromatic $H$ can be guaranteed to exist. To formalise this question, let the \emph{Ramsey multiplicity} $M(H;n)$ be the minimum number of labelled copies of monochromatic $H$ over all 2-edge-colouring of $K_n$. We define the \emph{Ramsey multiplicity constant} $C(H)$ is defined by $C(H):=\lim_{n\rightarrow\infty}\frac{M(H,n)}{n(n-1)\cdots(n-v+1)}$. I will discuss various bounds for C(H) that are known so far.

2020-12-04 / 09:30 ~ 11:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 권현주()

In the theory of turbulence, a famous conjecture of Onsager asserts that the threshold Hölder regularity for the total kinetic energy conservation of (spatially periodic) Euler flows is 1/3. In particular, there are Hölder continuous Euler flows with Hölder exponent less than 1/3 exhibiting strict energy dissipation, as proved recently by Isett. In light of these developments, I'll discuss Hölder continuous Euler flows which not only have energy dissipation but also satisfy a local energy inequality.

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