I am a Postdoctoral Fellow at the Instituto de Matemáticas, Unidad Cuernavaca of the Universidad Autómina Nacional de México, working under Alberto Verjovsky. We are working on applications of quaternion algebras to geometry and topology. In particular, we are working on a usage of quaternion orders to generalize Hilbert-Blumenthal surfaces to (higher real-dimensional) surfaces over Hamilton’s quaternions. This generalizes a former paper of my collaborator in which Kleinian groups with entries in quaternion orders give rise to hyperbolic 4- and 5-manifolds.

I hold a PhD from the Pure Mathematics program of The Graduate Center, City University of New York, granted in April 2016. My adviser was Abhijit Champanerkar. My thesis was about uniting classical and contemporary theories on quaternion algebras for applications to a class of hyperbolic 3-manifolds. I reinterpreted a classical theorem of Macfarlane in modern context and used it to develop an algorithm for computing a Dirichlet domain embedded within the quaternion algebra generated by the manifold’s fundamental group.

Teaching is also an important part of my work, after all if we cannot explain our findings to other people then what good are they? I worked as a Quantitative Reasoning Fellow at Bronx Community College, where I designed lessons and worked with interdisciplinary faculty on math remediation. Prior to that I taught math for 5 years as an Adjunct Lecturer at Hunter College, and spent many years tutoring, both privately and at the Dolciani Math Learning Center. In the summer I am an Instructor for the Johns Hopkins Center for Talented Youth, where I design and teach sections of their Paradoxes and Infinities course. I have also been known to give math presentations amidst art and music lineups.

models for hyperbolic space embedded in quaternion algebras

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Research Statement

Teaching Statement

josephanthonyquinn [at] gmail.com