Research Interests and Publications

Research Interests

My research interests lie in the areas of real and complex analysis.  Specifically, I study the ways in which topological, geometric, and measure theoretic information about the level sets of continuous (in the real case) and meromorphic (in the complex case) functions determines the analytic properties of these functions.  Some of the fields on which this work bears are: real analysis, complex analysis, shape analysis, algebraic geometry, several complex variables.  Since the level curves of a meromorphic function form graphs in the complex plane (whose vertices are critical points of the function), I also use tools from graph theory and combinatorics.

Undergraduate Research

One exciting aspect of research for me in recent years has been the opportunities I have had to engage with my students in research.  I have directed undergraduate research projects in a variety of settings: as a summer research project, as an independent study research project during the semester, in our summer SOFIA (Summer Opportunities For Intellectual Activities) program, and in our Senior Capstone course.  I have a joint paper with a student (Jimmy Yau) to appear in the Pi Mu Epsilon Journal (link below).  Click the tab titled “Undergraduate Research” in the menu for more information.

Publications

Links titled “.PDF may be found here” direct you to the appropriate page at arXiv.org.

T. J. Richards and M. Younsi. Computing polynomial conformal models for low degree Blaschke productsComputational Methods and Function Theory, 19: 173-182, 2019.
(.PDF may be found here)

T. J. Richards and S. Steinerberger. Leaky roots and stable Gauss-Lucas theoremsComplex Variables and Elliptic Equations, 64(11): 1898-1904, 2019.
(.PDF may be found here)

T. J. Richards and J. Yau. Recognizing a difference quotient,  to appear in Pi Mu Epsilon Magazine.
(.PDF may be found here)

T. J. Richards. Characterizing meromorphic pseudo-lemniscatesComputational Methods and Function Theory, 18(4):609-616, 2018.
(.PDF may be found here)

T. J. Richards and M. Younsi. Conformal models and fingerprints of pseudo-lemniscates, Constructive Approximation, 45(1):129-141, 2017.
(.PDF may be found here)

K. Beanland, P. D. Humke, and T. J. Richards. On Scottish Book Problem 157, Real Analysis Exchange 41(2):331-346, 2016.
(.PDF may be found here)

T. J. Richards. Conformal equivalence of analytic functions on compact sets, Computational Methods and Function Theory 16(4):585-608, 2016.
(.PDF may be found here)

T. J. Richards. Level curve configurations and conformal equivalence of meromorphic functions, Computational Methods and Function Theory, 15(2):323-371, 2015.
(.PDF may be found here)

Under Review

T. J. Richards. Rouché’s theorem and the geometry of rational functions.
(.PDF may be found here)

T. J. Richards. Some recent results on the geometry of complex polynomials: the Gauss–Lucas theorem, polynomial lemniscates, shape analysis, and conformal equivalence.
(.PDF may be found here)

In Process

T. J. Richards. Boundary convergence and path divergence sets for bounded analytic functions on the disk.