Research

Groups, Geometry, and Dynamical Systems

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 currently working with people in the field of nonlinear algebra to apply translation surfaces as a way of constructing algebraic curves with certain properties. This article gives an overview of nonlinear algebra. Finally, I have a budding interest in the interactions between dynamics and machine learning. If you don't know what machine learning is, see for example this nice AMS Notices article.

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?"

Crossing the transcendental divide: from Schotty groups to algebraic curves

S. Fairchild, Á.D. Ríos Ortiz

Mean value theorems for the S-arithmetic primitive Siegel transforms

S. Fairchild and J. Han

Pairs in discrete lattice orbits with applications to Veech surfaces

C. Burrin, S. Fairchild with an appendix by J. Chaika

ArXiv, to appear in Journal of the European Mathematics Society

Shrinking rates of horizontal gaps for generic translation surfaces

J. Chaika and S. Fairchild

ArXiv, published in Geometriae Dedicata

Video of talk at UCSD Group actions seminar

Average degree of the essential variety

P. Breiding, S. Fairchild, P. Santarsiero, E. Shehu

Arxiv, Published in La Matematica

Video recording of talk at Banff

Counting pairs of saddle connections

J.S. Athreya, S. Fairchild, H. Masur

ArXiv, Published in Advances in Mathematics

Crossing the transcendental divide: from translation surfaces to algebraic curves

T. Ö. Çelik, S. Fairchild, Y. Mandelshtam

Arxiv, Published in Experimental Mathematics

Van der Corput Sequence

Families of well approximable measures

S. Fairchild, M. Goering, C. Weiß

Arxiv or Published in Uniform Distribution Theory

H7 applied to e1

Higher Moments for the Siegel--Veech transform over quotients by Hecke triangle groups

S. Fairchild

Arxiv or Published in Groups, Geometry, and Dynamics

Undergraduate Research

The abelian sandpile model on fractal graphs

S.Fairchild, I. Haim, R.G. Setra, R. Strichartz, T. Westura. From the Cornell 2014 REU Arxiv

Rigid Tilings of quadrants by L-shaped n-ominoes and notched rectangles

A. Calderon, S.Fairchild, M.C. Muir, V. Nitica, S. Simon. From the Penn State 2013 REU

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