A systematic study of large combinatorial objects has recently led to discovering many connections between discrete mathematics and analysis. In this talk, we explore the analytic view of large permutations. We associate every sequence of permutations with a measure on a unit square and show the following: if the density of every 4-element subpermutations in a permutation p is 1/4!+o(1), then the density of every k-element subpermutation is 1/k!+o(1). This solves a question of Graham whether quasirandomness of a permutation is captured by densities of its 4-element subpermutations. At the end of the talk, we present a result related to an area of computer science called property testing. A property tester is an algorithm which determines (with a small error probability) properties of a large input object based on a small sample of it. Specifically, we prove a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio asserting that hereditary properties of permutations are testable with respect to the so-called Kendal's tau distance. The results in this talk were obtained jointly with Tereza Klimosova or Oleg Pikhurko.

The finite time blow-up problem of the 3D incompressible Euler equations is one of the most
outstanding open problems in the partial differential equations. In this lecture we introduce the problem, and discuss the current status. We also discuss the studuies on the closely related equations, namely the compressible Euler equations, quasi-geostropic equations,
 and the Boussinesq equations.

3차원 오일러방정식의 해가 유한시간에 폭발하는지에 관한 문제는 편미분방정식의 가장 중요한 난제중의 하나이다. 이 강연에서는 이 문제를 소개하고 현재까지의 연구 현황을 얘기하고자 한다. 이와 더불어 이와 관련된 유체역학의 다른 방정식들의 연구 결과들도 소개한다.

Geometric invariant theory (GIT) is a powerful tool to construct moduli spaces as quotients of the Hilbert schemes. GIT was very successful to construct the moduli space of stable curves and the moduli space of vector bundles on curves. But in general it is very hard to understand explicitly (semi-)stable objects in the sense of GIT. Along the development of the minimal model program (MMP), birational geometry including MMP is again a powerful tool to construct compactified moduli space for varieties of general type. In the talk, I will introduce several notions of stabilities and discuss their relations. Recent developments of log MMP of the moduli space of stable curves will be also discussed.

Based on the newly understood dichotomy of sparse and dense
structures, we provide the general framework for study of structural
(mostly graph) limits. Our approach uses both model theoretic and
analytic tools and uses structural theory of bounded expansion
classes.