Motivic Homotopy Theory and Applications to Enumerative Geometry

This project studies the number of solutions to certain equations, using the shape of spaces associated to these equations, as well as this shape itself. There is a new invariance property of the number of such solutions. This invariance not only applies to the number of solutions in the complex numbers, but also solutions which are ordinary fractions, such as one half.

It is obtained by using A1-homotopy theory due to Morel and Voevodsky. Mathematics education is furthermore supported by continuing a program of week-long summer math jobs for gifted high school students from diverse backgrounds. During each of the summers of the project period, approximately eight high school students will work on an important mathematical problem, learning the background material as necessary, and solving it as a group.

They will be accompanied by two high school teachers. A Research Experience for Undergraduates aimed at the graduates of the program is provided to continue mathematical training and provide research mentorship.

A1-homotopy theory was introduced by Morel and Voevodksy in the late 1990's and allows the successful import of tools from algebraic topology into the study of solutions to polynomial equations. The PI and collaborators are studying the interaction between A1-homotopy theory and classical questions from enumerative geometry such as "How many lines meet four lines in space?" A1-homotopy theory functions well over very general base schemes and in particular over any field, resulting in enumerative results over non-algebraically closed fields valued in bilinear forms.

This project searches for such results connected with characteristic classes, Gromov--Witten theory, and zeta functions and develops tools in motivic homotopy theory suggested by these applications.

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.