Broadly my research focuses on analyzing discrete sets arising from geometry. My inspiration often comes from the study of translation surfaces. This Numberphile video is a great introduction to the types of problems I enjoy working on. I'm also working with people in the field of applied algebraic geometry to better understand explicit computations moving between Riemann surfaces and algebraic curves. You can see some of my work on crossing the transcendental divide for an idea. I'm also pursuing new projects which connect geometric and algebraic techniques to real-world problems in collaboration with local industries.
Below are my projects, with links to papers and sometimes recorded talks.
Here's some other talks:
"Why would a dynamicist care about combinatorial algebraic geometry?"
C. Burrin, S. Fairchild with an appendix by J. Chaika
ArXiv, to appear in Journal of the European Mathematics Society
J. Chaika and S. Fairchild
ArXiv, published in Geometriae Dedicata
Video of talk at UCSD Group actions seminar
P. Breiding, S. Fairchild, P. Santarsiero, E. Shehu
J.S. Athreya, S. Fairchild, H. Masur
T. Ö. Çelik, S. Fairchild, Y. Mandelshtam