Abstract: Consider a set A of either F_3^n or [N] = {1,…,N}. Remarkable work of Roth (and Meshulam) proved that if A has size at least 1% of the ambient group then A contains 3 term arithmetic progressions. We sketch a proof of this result via Fourier analytic methods. We then discuss briefly why this proof does not adapt in an trivial manner to longer term arithmetic progressions and discuss the topic of higher order Fourier analysis.

Abstract: Inkscape is a free program that many mathematicians (especially low-dimensional topologists) use to create figures for their papers. Instead of giving a research talk, I thought I'd give a tutorial on how this program works, so that you can use it when you write papers!

Abstract: The cohomology of a topological space X acquires extra structure when X is a complex variety, functorially with respect to algebraic maps. The weight filtration is part of this structure, and in practice, weight-zero compactly supported cohomology is particularly tractable and combinatorial. I will sketch how to define and calculate this invariant, touching upon recent work with collaborators.

Abstract: The global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou-Klainerman in 1993. In 2007, Bieri extended the result of Christodoulou-Klainerman under lower decay and regularity assumptions on the initial data. In 2010, Lindblad-Rodnianski provided a new proof of the global stability of Minkowski by wave coordinates. In this talk, I will first introduce the basic set-up of Minkowski stability. I will then report a recent work, which extends the results of Minkowski stability to minimal decay assumptions.

Abstract: After introducing hyperbolic geometry in all dimensions, I will discuss some of the key features that make hyperbolic manifolds behave very differently in dimension two, three, and at least four, and point out some basic open problems to highlight these differences.