# Mentoring and Outreach

## Laboratory of Geometry

The Laboratory of geometry (LOG) is a program aimed at groups of motivated undergraduate students. They work together under the guidance of mentors to understand a mathematical concept deeply enough so that they can tell a computer to produce meaningful pictures. Then they use these pictures to get further insights and conjectures. Mentors can consist of graduate students or faculty members.

I am the co-founder of Laboratory of Geometry at IU. Here is the website:

### Current projects

I am not organizing any Geometry Lab currently, but I am mentoring an undergraduate student on a Geometry Lab-like project. The project we are working on is on relating geodesics on the once-punctured torus to rational functions in the Farey parameterization through Markov numbers and continued fractions. If you are interested in a similar research experience, please reach out!

### Past projects

At UC Davis, I mentored undergraduates on the following projects:

Hyperbolic geometry and the once-punctured torus (Spring 2023)

Gauss product for quadratic forms (Spring 2023)

Hyperbolic geometry (Fall 2022)

In the past, I've co-mentored the following two LOG projects at Indiana University:

Laboratory of Geometry at IU: Volume of knot complements, main organizer of the program and co-mentor of students: Arthur Hertz and Robert Iannuzzo.

Abstract: The students learned about hyperbolic geometry, geodesics, unit tangent bundle of surfaces, canonical lifts, and volumes of 3-manifolds. Then they studied the unit tangent bundle of the once punctured torus, canonical lifts of pairs of simple geodesics and did some computations of volume triangulating the complement of the canonical lifts and using Snappy. See a problem list here and their final slides here

Laboratory of Geometry at IU: 3-D printing fiber bundles, co-founder of the program and co-mentor of students: Jeffery Coulter, Tristan Britt and Phuong Dong Le, Spring 2019.

Abstract: We study concrete examples of fiber bundles, such as mapping tori of the torus, or the Hopf fibration. Then, we set on producing some computer code that allows us to explore the Hopf fibration via 3d-imagery. Finally, we produced a 3d model of the Hopf fibration. See a problem list here and the final slides here.

In the past, I've co-mentored the following two LOG projects at University of Michigan (https://sites.lsa.umich.edu/logm/):

A missing entry in Sullivan's dictionary (see poster), students: Colby Kelln, Sean Kelly, Jung Suk Lee (U Michigan), Winter 2018. Abstract: initially, students were introduced to complex dynamics of polynomials on the Riemann sphere, and the concept of Julia set. Then we covered the concept of Fuchsian and quasi-fuchsian group and its limit set, and they produced a Python code that draws limit sets for quasi-fuchsian groups of the once-punctured torus. Later on, they compared these pictures to the pictures obtained for Julia sets. Here are some notes that we wrote for the students.

Visualizing the Schwarzian derivative (see poster), students: Noah Luntzlara, Pleum Piriyatamwong, Mengxi Wang, Sharon Ye (U Michigan), Fall 2017. Abstract: we covered Moebius transformations, complex analytic maps, and studied the concept Schwarzian derivative. We then related the Schwarzian derivative to the curvature of images of curves, to the eccentricity of small circles. The project contains an original proof that the Schwarzian derivative is related to the rate of change of the osculating Moebius transformation. Here are some notes we wrote for the students.

## Research Experience for Undergraduates

In Summer 2019 I co-mentored the student Max Newman on a REU project on extremal length of non-simple closed curves, with Dylan Thurston as the principal faculty. Max studied families of non-simple closed curves on the thrice punctured sphere whose extremal metric is realized by the flat metric. These show up when looking at regular tessellations of the plane. Further, he implemented a computer algorithm by Zwiebach and Haedrick to compute extremal lengths via convex optimization and used it to perform an experimental study of the extremal length of the torus with one hexagonal boundary component.

## Directed Reading Program

The Indiana University Math Directed Reading Program math majors the opportunity to learn advanced math topics. Participants spend a semester on an independent math project under the supervision of a graduate student mentor. More info: http://www.indiana.edu/~mathdrp/

In the past, I have mentored one DRP project:

The Hex game via the fixed point theorem (see presentation), student: Nanjie Chen (IUB), Spring 2017. Abstract: We studied the equivalence between existence of a winner in Hex game and the existence of a fixed point theorem for a continuous function between convex compact set (Brower fixed point theorem) following a result by David Gale. She also related this to Jordan's curve theorem in topology, and finally to Sperner's lemma in graph theory.

## Outreach

The Set game: exploring combinatorics, probability and geometry, RISE STEM summer scholar program, Summer 2017 and Summer 2019. See slides.

A topological game, McCormick Creek's Elementary Math/Science night volunteer (IUB) Spring 2017

Volunteer in Mathematical New Games Workshop, Hanabi (see handout), Indiana University Science Fest, (IUB), Fall 2015

Judge in Science Fair - Ivy Tech Community College (IUB), Spring 2015

Volunteer in Mathematical Games Workshop. Indiana University Science Fest, (IUB), Fall 2014

Introduction to Math life at university of Young Mathematicians Award team challenge competition, Nrich activity, Cambridge Millenium Math Project, Michaelmas term 2013

Mathematics Outreach session for children and young people between 5-18 age. Cambridge Maths Circle, Cambridge Millenium Math Project, Michaelmas term 2013