Math

Here you can find a summary of my time as an academic mathematician -- my technical writings and past experiences in the classroom. 

Teaching

Teaching and mentorship was truly a joy for me. At the time I experimented with how to center my students' autonomy and build a culture of mathematical self-efficacy. That exploration led me to receive the 2022 Graduate Teaching Assistant of the Year award, the Class of 1934 Excellence in Teaching award, and an Outstanding Evaluations award. Despite the institutional recognition, I would say my efforts were met with varying degrees of success. My teaching philosophy from that moment in time can be found here. Below are a few comments I received anonymously from my students. 

• The instructor does an exceptional job not only getting the main concepts across, but also creating an extremely valuable support network built on collaboration and appreciating the subject matter.

• They created a healthy learning environment for students of all levels.

• Santana highly respected every student and was a very understanding and kind professor. I did not feel overlooked or not cared about and that is really influential in success in a class.

• I absolutely appreciated Santana. They are truly there to help students understand and not feel the pressure of learning math. Math is tough, but Santana was amazing at explaining the concepts and answering not just the questions we asked but also the questions we meant to ask. They were great at anticipating difficulties and addressed hard concepts very well.

• I really loved their enthusiasm about teaching us things about math in the real world and how to help us be better humans in the world.

Mentorship

Through a directed reading program, I worked together with a number of students to explore some math they might not see in their coursework. Generally, our goals were simple: have fun, learn something new, and see some beautiful mathematics. I am extraordinarily proud of everyone I have had a chance to serve as a mentor; you can find the students I worked with below along with the topics we discussed.

• Fall 2022

  • Proof writing and social justice with Sid Farron

• Spring 2020

  • • Category theory via group theory with Noah Caplinger and Caleb Partin

    • Combinatorial graph theory with Nina Moorman

    • Metric spaces and hyperbolic geometry with Stanley Pritchard

    • Geometric Group Theory with Esther Gallmeier

• Fall 2019

  • • Group theory and Cayley graphs with Stanley Pritchard

    • Algebraic topology with Noah Caplinger

• Spring 2019

  • • Group theory in algebraic topology with Sidhanth Raman

    • Topics in geometric group theory with Alvin Chiu and Eric Zhu

• Fall 2018

  • • Algebraic topology with Dorian Kidonakis

• Summer 2018 (mentored an REU jointly with Justin Lanier) 

  • • Twists of the Rabbit Polynomial with Xian Li and Abigail Saladin

Technicalities:

As a research mathematician, I was drawn to groups naturally associated to various geometries and understanding the relationship between geometric structure and group structure. I ended up doing case studies in three different geometries: discrete, topological, and algebraic.

• Constructing a random model for the action of Frobenius on fundamental groups of curves (Thesis, to be released).

• Heegaard Floer homology and the word metric on the Torelli group (joint with Miriam Kuzbary and Tye Lidman)

• Nielsen realization for infinite-type surfaces (joint with Danny Calegari, Joe Chen, and Rylee Lyman).

  • • Proceedings of the AMS, Volume 149 (2021), no. 4, pp. 1791-1799

    • Here are slides for the talk below.

• On prime order automorphisms of generalized quadrangles (joint with Eric Swartz)

  • • Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 143-160