Thomas Treloar

Thomas I. Treloar

Chairman and Professor of Mathematics
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.
— Johannes Kepler

Faculty Information

Additional Faculty Information for Thomas I. Treloar


Ph.D., University of Maryland – College Park, 2001

B.S., Buena Vista College, 1995

VIGRE Postdoctoral Fellow, University of Arizona, 2001-2004

Leitzel Fellow, Project NExT/Mathematical Association of America, 2004-present


American Mathematical Society

Mathematical Association of America

Intercollegiate Studies Institute

National Association of Scholars

Society for Classical Learning


“An iterative Markov rating method,” Journal of Quantitative Analysis in Sports.

A Network Diffusion Ranking Family that Includes the Methods of Markov, Massey, and Colley, Journal of Quantitative Analysis in Sports.

Evolution of Cooperation through the Heterogeneity of Random Networks, Physical Review E.

Cooperation in an Evolutionary Prisoner’s Dilemma on Networks with Degree-Degree Correlations, Physical Review E.

Network-based Criterion for Success of Cooperation in an Evolutionary Prisoner’s Dilemma, Physical Review E.

The Symplectic Structure of Polygons in the 3-sphere, Canadian Journal of Mathematics.

The Symplectic Structure of Polygons in Hyperbolic 3-space (with Kapovich and Millson), Asian Journal of Mathematics.

Arithmetic: Leading the Mind Toward Truth, The Journal of the Society of Classical Learning.

The Purpose of Mathematics in a Classical Education, The Imaginative Conservative.

Courses Taught

MTH 105: Mathematics and Deductive Reasoning

MTH 112/113: Integrated Calculus A/B

MTH 120: Calculus I

MTH 220: Calculus II

MTH 370: Probability

MTH 375: Game Theory

MTH 405: Complex Analysis

MTH 420: Mathematical Statistics

MTH 425: Statistical Learning

Student research supervised for Departmental Honors Thesis

Samuel Cassels, Probabilistic Interpretations in Markov Ranking Systems, 2019.

Kirk Williams, Markov Methods for Ranking and Prediction in NCAA Basketball, 2018.

Sarah Onken, The Mathematical Philosophy of the Common Core: How the Standards Answer the Question: ‘What is Mathematics?’, 2015.

Nathan English, The Effect of Hubs in Populations with Evolutionary Prisoner’s Dilemma, 2011.

Heidi Schweizer, Non-Manipulability in Social Choice Theory, 2011.

Micah Seppanen, Cooperation with Evolutionary Prisoner’s Dilemma Games, 2008.

Angela Moore, Fourier Analysis, Wavelets, and Multi-Resolution Analysis, 2008.

Christin Alford, A Study of Game Theory and Evolving Strategies, 2006.

Jasmine Spady, Knots in Chemistry, 2006.


Current research interests: Complex Systems, Sports Analytics, and Markov Processes

Much of my current research is in complex systems. The field of complex systems includes the study of dynamical systems in large populations modeled by networks with a view towards how topological properties (the structure of the networks) influence behavior. Within this field, much of my current work has been in evolutionary game theory, the study of how strategies played by members of a large structured population change over time. Future work will include adapting strategies in an attempt to understand how population structures influence the spread of diseases.

I am also interested in the field of Sports Analytics, which involves the use of mathematical tools in analyzing processes within sports competitions. In particular, I am looking at the use of alternative ranking systems within sports, and other areas of competition, using Markov processes and random walks on networks. This work includes both iterative and global methods with interest in understanding how ratings might provide a probabilistic interpretation to me used in predictive tools.

Past research: Symplectic Geometry and Integrable Systems

Symplectic geometry is an area of differential geometry which arose out of the Hamiltonian reformulation of classical mechanics in which the space is equipped with a closed, nondegenerate two-form. In this research, I studied the symplectic structure on the moduli space of polygonal linkages (that is, polygons with fixed side lengths) in three-dimensional spaces of constant curvature.

Other interests:

I am interested in K-12 mathematics education, in particular, the study of mathematics in a classical education. I have advised the Barney Charter School Initiative through Hillsdale College by reviewing the K-12 mathematics curriculum, and conducting many presentations and teacher training sessions at the affiliated classical charter schools.