Mark Greenfield

About Me

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I am a Ph.D. student in pure mathematics in the Department of Mathematics at the University of Michigan in Ann Arbor. My advisor is Lizhen Ji. I started here in September 2015. I obtained my B.S. in mathematics from Caltech in June 2015. Before that, I lived in Cary, NC, and before that, I lived in Rochester, NY.

Here is my CV.

Research Interests

I am interested in geometry and topology. My favorite types of questions are those in which both geometry and topology play a major role. Specifically, I have been studying Teichmüller spaces and moduli spaces of Riemann surfaces and connections to other geometric objects, such as symmetric spaces, and generalizations. I am also interested in differential geometry, hyperbolic geometry, mapping class groups, and other related objects. Occasionally, I am interested in applications to other fields (not necessarily pure mathematics).


Greenfield, M. A lower bound for Torelli-K-quasiconformal homogeneity. Geometriae Dedicata 177 (2014), Issue 1, 61-70. Available at arXiv: 1309.2666.

Greenfield, M., Marcolli, M. and Teh, K., Twisted spectral triples and quantum statistical mechanical systems, p-Adic Numbers Ultrametric Anal. Appl. 6 (2014), no. 2, 81-104. Available at arXiv: 1305.5492.

Greenfield, M., Zhang, J. Resolving the topological social choice paradox via the addition of a null preference. In Preparation.

Greenfield, M., Ji, L. A new modular characterization of the hyperbolic plane. In Preparation.


I am not teaching during the 2016-2017 school year. However, I am available to tutor students in many of the undergraduate mathematics courses at U of M, as well as younger students who can come to U of M's central campus. Feel free to email me if interested.

I am serving as the manager and a grad student mentor for the Laboratory of Geometry at Michigan, LOG(M). This is a new program here at U of M in which small teams of motivated undergraduates are set up with a grad student and a postdoc in order to learn more about some interesting topics in geometry, and create some nice visualizations of the concepts. To learn more, visit the LOG(M) website!

Past Teaching at U of M

Fall 2015 Math 115: Calculus I (Primary Instructor)
Winter 2016 Math 215: Calculus III (Mathematica Lab Instructor)

Past Teaching at Caltech

Summer 2013, 2014 Math 0: Transition to Mathematical Proofs (Lead TA, 2014)
Spring 2014, 2015 IST 4: Information and Logic

Older Files

Several years ago I wrote a few brief expositions for Math 0, aimed at early college students interested in mathematics who have not seen much of it yet. References to courses or forums are based on Caltech classes. Additionally, here is my (imperfect) Senior Thesis about Riemann surfaces, which also includes my undergraduate research projects. Here are some old slides about my project on quasiconformal homogeneity, and here are some old slides about my project on Riemann surfaces and Quantum Statistical Mechanical systems.

Other Interests

Apart from my research interests, logic and foundations of mathematics are of interest to me. Outside of mathematics, I am most interested in science, both the basics and current advancements. I have a casual interest in philosophy (particularly the philosophy of science). Some broader issues important to me are equality and inclusiveness in academia and the STEM community, and mathematics education, especially with regards to showing wider audiences what (non-applied) mathematics is really about. On a lighter note, I absolutely love dogs.

Last Updated: April 2017.

I am indebted directly to Angus Chung and transitively to Corey Everlove for assistance in constructing this webpage and for the template code.