Arithmetic Topology Conference

This award supports US participants to attend the workshop 'Arithmetic Topology' at the Pacific Institute of Mathematical Sciences (PIMS) from June 10-14, 2019 in Vancouver. Recent years have seen spectacular advances at the intersection of number theory, specifically problems asking how many solutions an equations has, and topology which studies mathematical properties of shapes and spaces. While mathematicians have understood for several decades that individual equations have an associated space whose properties reflect the equation, many natural equations (and many natural spaces) come in infinite sequences. The last decade has seen spectacular advances (leading in part to Venkatesh's 2018 Fields Medal, highest honor in mathematics) in our understanding of how the asymptotic behavior in natural sequences of equations ('arithmetic statistics') governs and is governed by the asymptotic behavior in natural sequences of spaces ('homological stability'). This workshop aims to bring together a diverse group of leading and emerging researchers working in number theory, algebraic geometry and topology to obtain a global view of a fast emerging and multidisciplinary area, to train participants in the range of methods available, and to generate a robust problem list that can guide activity in the area for the next 5-10 years.

The last 10 years have brought a burst of activity at the intersection of algebraic topology, number theory and algebraic geometry. This has led to a wealth of:

1) new theorems, such as Ellenberg-Venkatesh-Westerland's proof of the

Cohen-Lenstra heuristics for function fields;

2) new sources of heuristics in topology, such as Vakil-Wood's predictions from the Grothendieck ring, or the notion and coincidences of homological densities as in Farb-Wolfson-Wood;

3) refinements of classical enumerative theorems using modern topological tools, such as Kass-Wickelgren's arithmetic count of the 27 lines on a cubic surface; and

4) a renewed focus on unstable homology, as in Galatius-Kupers-Randal-Williams and Miller-Wilson.

The organizers of the workshop believe that these results are just the beginning of the emerging area of arithmetic topology. They are organizing a 5 day workshop to bring together a diverse group of junior and senior researchers from across these areas with the goal of:

1) giving participants a global view of a fast emerging and multidisciplinary area,

2) giving participants a detailed awareness on the range of methods available, and

3) emerging with a robust problem list which can help guide activity in the area for the next 5-10 years.

More information is available at the conference website:

https://www.pims.math.ca/scientific-event/190610-pwat.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.