## 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

Kwatinetz, Thank you for sponsoring my project! I've had a lot of fun this summer, working towards doing the sort of mathematics that I hope to do during graduate school. I've found from the past two REUs I did that there is quite a bit of gap between the sort of work (in number theory) which can reasonably be considered accessible to undergraduates and the sort of work which is popular (and interesting to me) in today's landscape. Thus, I am very grateful to have had the time and opportunity to read what I needed to and begin thinking in the directions towards the more modern number theory. I plan on applying to graduate school in the fall, and I am now much more optimistic of my plans and outcomes. Thank you, Nir**Major:**Mathematics

**Mentor:**Yunqing Tang

**Sponsor:**Kwatinetz