## Nir Elber L&S Math & Physical Sciences

### Frobenius Symmetries in Motivic Galois Groups

One goal of arithmetic geometry is to enumerate the points on geometric surfaces with rational coordinates. Over the past century, it has been profitable to study the geometry of the surface directly. For example, a “cohomology theory” is a way to assign a sequence of geometric invariants to the surface; it turns out that one can use cohomology in order to count points. Given a surface, there tend to be many reasonable cohomology theories. This project is interested in the symmetries of a cohomology theory. Given a cohomology theory, it turns out that there is a canonical symmetry attached to the theory. However, it is a conjecture of Serre that these canonical symmetries (on the different cohomology theories) are essentially the same! The goal of the project is to further what is known about this conjecture. Explicitly, it is known that these symmetries are the same when they are small in some sense, so it remains to understand what occurs without this smallness assumption.

#### Message To Sponsor

The project I am starting this summer marks the beginning of a big transition in my academic career towards thinking questions much closer to my true mathematical interests. Previous research I have done has largely been focused on furthering an existing program of a couple professors, but this summer I'm excited to be able to answer questions which have come more naturally to me. Thank you for so much for this opportunity!**Major:**Mathematics

**Mentor:**Yunqing Tang

**Sponsor:**Kwatinetz Fund