In many different areas of mathematics (such as number theory, discrete geometry, and combinatorics), one is often presented with a large "unstructured" object, and asked to find a smaller "structured" object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and present some recent progress on one of the central problems in the area: bounding the so-called "diagonal" Ramsey numbers. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

We first survey on nodal solutions for coupled elliptic equations, using results from nonlinear scalar field equations as motivations. Then we discuss work for constructing multiple nodal solutions using various variational methods. In particular we discuss in some details the results about solutions having componentwisely-shared nodal numbers of coupled elliptic systems. These works are done by further developing minimax type critical point theory with built-in flow invariance of the associated gradient or parabolic flows, which has been a useful tool to give locations of critical points via minimum methods, also revealing complex dynamic behavior of the flow.

We will first introduce Homogeneous dynamics, especially the mixing property of flows on spaces of hyperbolic nature. We will then survey applications of homogeneous dynamics to various problems in Number theory. (Part of the talk is based on joint work with Keivan Mallahi-Karai and Jiyoung Han.)

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The date has been postponed from March 16 to March 30.

In this lecture we introduce some challenging problems in the mathematical fluid mechanics. Although fluid mechanics is one of the most important physical phenomena we experience in everyday life, and has been studied for long time in history by top class mathematicians, still there are many problems which are open even at the fundamental level. We explain these problems and briefly review some of the recent progress.