# Undergrad Research

**University of Michigan REU:**

- Natalia M. Pacheco-Tallaj (Harvard University).
*Thurston norm of 2-generator, 1-relator groups**,*Summer 2017. - (Co-advised with Kevin Schreve)
- Alex Pieloch (Duke University).
*Curve Complexes of non-orientable surfaces**,*Summer 2016. - (Co-advised with Matthew Durham)
- Ben Lowe (University of Chicago).
*Rigidity of representations of free groups in PSL(2,*Summer 2016.**C**), - (Co-advised with Richard Canary)

**Laboratory of Geometry at Michigan, LOG(M):**

In the Winter 2017 semester, the University of Michigan started its own geometry lab, named the Laboratory of Geometry at Michigan, or LOG(M) for short. The lab was started by Anton Lukyanenko, Caleb Ashley, and myself in the spirit of the Experimental Geometry Lab run by Bill Goldman. Many such labs have popped up at universities throughout the country; they are organized into a collective unit called Geometry Labs United, or GLU for short.

Briefly, LOG(M) is meant to provide a vertically integrated research experience for undergraduates by incorporating both a faculty and graduate student mentor. The faculty member is responsible for designing a project and the mentoring of the undergraduates is split between the faculty member and the graduate student. The projects tend to be motivated by visualization in geometry. The visualization aspect requires students to familiarize themselves with some standard computer programs and languages, which is an added benefit.

**LOG(M) Project:**

Title: *Visualizing the Birman-Series set on punctured tori**.*

Faculty Mentor: Nicholas Vlamis (me)

Graduate Mentor: Mark Greenfield

Students: Connor Davis, Ben Gould, Luke Kiernan

Summary: Birman-Series showed that the union of all simple closed geodesics on a hyperbolic surface has Hausdorff dimension 1; in particular, they take up very little room on the surface. This was quantified by the work of Buser-Parlier, where they showed that given a hyperbolic surface there exists an embedded hyperbolic disk that no simple closed geodesic enters, where the radius depends only on the topology of the surface (not the geometry!). The goal of this project is to visualize the union of the simple closed geodesics on hyperbolic punctured tori at different points in moduli space.

Pictured above on the left is the union of the first 600 closed geodesics (up to word length) followed on the right by the union of first 100 simple closed geodesics (up to word length) in a fundamental domain in the hyperbolic plane for a once-punctured torus. These images were drawn by the students using a program they wrote using the language Sage. The code can be downloaded here. The program includes an interactive GUI allowing the user to set the number of geodesics pictured as well as to move through a 1-parameter family of hyperbolic structures.

For more info on the students work, see the poster they made for a presentation at a conference run by Geometry Labs United.