15 – 17 May 2019

Organised in collaboration with the School of Mathematics, University of Bristol, UK

Jordan Ellenberg, University of Wisconsin, Madison, USA

Jordan Ellenberg is a highly distinguished arithmetic algebraic geometer whose broad interests include the study of rational points on varieties, the enumeration of number fields and other arithmetic objects, and incidence problems and algebraic methods in combinatorial geometry. Major results include his ground breaking work on the so-called cap set problem in additive combinatorics (Annals of Mathematics, 2017).

Jordan is a Fellow of the American Mathematical Society (the Inaugural Class of 2013), a Guggenheim Fellow in 2015 and a plenary speaker at the 2013 Joint Mathematics Meetings of the AMS. In addition, he is a prolific expositor of mathematics, via his well known blog “Quomodocumque”, public lectures and regular appearances in the media. He is also the author of the bestselling book, “How Not To Be Wrong: The Hidden Maths of Everyday Life”.

Caps, Sets, Lines, Ranks, Polynomials, and (the absence of) Arithmetic Progressions

Here is an innocent-looking problem. Suppose you wish to construct a subset of the numbers from 1 to 1,000,000 — or, more generally, from 1 to some large number N — with the property that no three of the numbers ever form an arithmetic progression. How big can your subset be?

It’s not clear that this problem is hard and it’s not clear that it’s important.  In fact it is both! I’ll talk about the long history of this problem and its variants, including the “cap set” problem, which is related to the card game Set: how many cards can be on the table if there is no legal play? This problem sounds different but is in many ways the same.  

I’ll talk about a sudden burst of progress on the cap set problem that took place in 2016, and explain what it all has to do with polynomials over finite fields, spinning needles (they’re also over finite fields), notions of rank for NxNxN “matrices”, and the data science of embedding points in space.

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