The fundamental question of contemporary number theory, which has also been the fundamental question of number theory for the last two and a half thousand years, is: what can we say about solutions to equations in whole numbers? The PI?s research explores the relation of questions about solutions to equations (in particular, how many solutions certain kinds of equations have) with questions about geometry of certain kinds of spaces; in turn, another side of the PI?s proposed research investigates the ways geometry of high-dimensional spaces can be brought to bear on other problems in mathematics and data science which might not immediately "look like" geometry. The PI?s research is closely entwined with his work as a popularizer of mathematics in print, broadcast, and social media; he is currently working on a book about geometry which will involve some of the funded research.
The project covers a wide range of problems in number theory, algebraic geometry, and data science. A central part is the work of PI and collaborators on a theory of height for rational points on algebraic stacks. The definition was pinned down and its properties studied during the previous granting period; during the present period, we will state a general heuristic for asymptotically counting points on stacks of bounded height, and work towards proving new cases. This new conjecture will include as special cases the Malle conjectures (how many number fields are there of discriminant at most X?) and the Batyrev-Manin conjectures (how many solutions are there to a given equation in integers all of which are at most X?) but applies to many new cases besides, and sheds new light even on the classical questions. In particular, our work adds to the developing consensus that we should go beyond heights attached to line bundles and study the variation of heights attached to vector bundles of arbitrary rank, opening up whole new directions of research and revealing new connections between existing sectors of the literature. The project also includes a wide range of problems in other areas, including group theory (an attempt to use the method of ?FI-groups? to prove property T for new families of groups, following the breakthrough of Kaluba, Kielak, and Nowak for Out(F_n)), arithmetic statistics (proving new results towards the Bhargava-Kane-Lenstra-Poonen-Rains conjectures on variation of Selmer groups in the function field case), multilinear algebra (understanding the algebraic and convex geometry of the locus of low-slice-rank tensors), and data science (investigation of what popular machine learning protocols do and don?t learn about symmetry from their input).
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.