Geometry of Arithmetic Statistics and Related Topics

A fundamental theme of contemporary number theory is that questions about numbers (for instance: how likely is it that two randomly chosen large numbers have no prime factors) have analogues in geometry — in this case — how can a set of red dots and a set of blue dots move around the surface of a sphere if no red dot is ever allowed to collide with a blue dot? One relevant geometric fact in this setting is that the red dots (and equally so the blue dots) can be rearranged in whatever order you like without ever breaking the no-collisions rule; this would not, for instance, be true if the red and blue dots were located on a line instead of on a sphere. In a non-technical setting it isn’t easy to say why this has anything to do with the likelihood of two numbers having a prime factor in common; suffice it to say that this passage between contexts has consistently generated new ideas in both number theory and geometry, and is a central theme of the PI’s proposed research. Beyond that, the PI has several projects at the interface of geometry and machine learning -- for example, can we use the kind of techniques that enable machines to play very strong chess and Go to find large configurations of points in in a grid such that no three form an isosceles triangle? This is a toy problem but the techniques we develop will tell us a lot about the prospects for accelerating progress in pure mathematics using machine learning techniques. The PI’s research is closely entwined with his work in outreach to the community outside academic mathematics, which includes a best-selling book on geometry published in 2021; during the funding period he will continue developing programs to train early-career scientists in writing for the public. This award will also support graduate student research. The proposed research covers a range of problems at the interface of number theory, algebraic topology, and algebraic topology. One main goal will be exploiting the techniques developed in PI’s collaboration with Tran and Westerland to prove upper bounds (and in some cases lower bounds) for Malle’s conjecture over function fields, an old problem which in the number field case remains almost entirely inaccessible. The new techniques suggest an interesting role for perverse sheaves in arithmetic statistics, will the PI will explore over the granting period. The PI also proposes a range of projects in arithmetic geometry, including: new directions in “twisted” arithmetic statistics (for instance: how many cubic extensions of F_q(t) are there with prime conductor?) and computational and theoretical work on the variation of the Ceresa class in families of algebraic curves, The PI will also continue collaborative work with researchers in industry on the development of machine learning techniques adapted to enable progress in pure mathematics, especially extremal combinatorics. 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.