| Speaker: Julia Wolf Affiliation: University of Cambridge Time: 2:00 pm Monday, 9 March, 2020 Location: 303-101 |
| A long-standing open problem in additive number theory is the following: how large does a set of integers have to be before it is guaranteed to contain a non-trivial arithmetic progression of length 3? In the first half of this talk we shall survey recent progress on this problem, and the techniques used to solve it and related questions about additive structures in finite abelian groups. In particular, we shall explain the idea behind the so-called "arithmetic regularity lemma" pioneered by Green, which is a group-theoretic analogue of Szemeredi's celebrated regularity lemma for graphs. In the second half of the talk we shall describe a host of recent results showing that under natural model-theoretic assumptions the conclusions of such regularity lemmas can be significantly strengthened. Most of the relevant model-theoretic notions were conceived several decades ago and have been studied extensively in an abstract context since, but it is only now that their significance in the finitary setting is becoming apparent. This talk aims to be accessible to postgraduate students across all areas of mathematics. Julia Wolf is a University Lecturer in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, and a Fellow of her alma mater, Clare College. After obtaining a PhD in pure mathematics under the supervision of Fields medalist Timothy Gowers, she held a sequence of postdoctoral appointments in the United States between 2007 and 2010. From 2010 to 2013, she was a Hadamard Associate Professor at Ecole Polytechnique in Paris from 2010 to 2013, and from 2013 until her return to Cambridge in 2018 a Heilbronn Reader at the University of Bristol, where she also served as the Associate Chair of the Heilbronn Institute for Mathematical Research for three years. In 2016 Julia received the Anne Bennett Prize of the London Mathematical Society in recognition of her "contributions to additive number theory, combinatorics and harmonic analysis, and to the mathematical community". |