J. L. Doob Research Assistant Professor

Department of Mathematics

University of Illinois at Urbana Champaign

Email: jsapir2 (at) illinois (dot) edu

I was a student of Maryam Mirzakhani at Stanford University. I'm interested in geometric topology and geometric group theory.

Briefly, I'm currently studying the behavior of geodesics on hyperbolic surfaces. One class of problems that interest me are counting problems. I have results on counting the number of closed geodesics with given bounds on length and self-intersection number. These counting results allow me to extend a theorem of Birman and Series. Their theorem states that the set of all complete geodesics on a finite type surface with at most *K* self-intersections lie in a nowhere dense set on the surface, of Hausdorff dimension 1. We extend their result by looking at complete geodesics with infinitely many self-intersections, where we restrict the self-intersection rate. The methods for obtaining these results have applications to other problems as well. For example, for any closed geodesic on a surface *S*, we can bound the degree of a cover of *S* to which the geodesic lifts simply.

Check out the prototype! Anyone can play: No math experience required! Feedback is very welcome. Please use the feedback tab in the game website.

We're developing an online video game, together with Cary Malkiewich and a team of graduate and undergraduate students. This game is designed to provide intuition, rather than instruction. We plan to incorporate it into our linear algebra classes as a grounding experience in linear algebra concepts. Our prototype demonstrates the concept of a linear transformation, and shows how linear transformations send lines to lines and elipses to elipses. In the future, we hope to have games that demonstrate row and column spaces, eigenvectors and eigenvalues, and so on.

This project is partitially supported by the IGL at UIUC.

- In Spring 2017, I'm teaching Math 415: Applied Linear Algebra. If you are in my class, you can find the website here.
- In Spring 2015 and 2016, I taught Math 415: Applied Linear Algebra.
- In Fall 2014, I taught Math 402: Non-Euclidean Geometry.