I am interested in geometric analysis and geometric partial differential equations. I also have research interests in discrete differential geometry.
My research statement can be viewed here, and a list of publications with abstracts can be viewed here.
- "Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes" with Daniel Champion, Andrew Marchese*, and Jacob Miller* (submitted)
- "Ricci Yang-Mills Solitons on Nilpotent Lie Groups" with Michael Jablonski (submitted)
- "Regge's Einstein-Hilbert Functional on the Double Tetrahedraon" with Daniel Champion and David Glickenstein (To Appear Diff. Geom. Apps.)
- "The Cross Curvature Flow on a Negatively Curved Solid Torus" with Dan Knopf and Jason Deblois (Algebr. Geom. Topol. 10, 343-372, 2010.)
- "Stability of the Ricci Yang-Mills flow at Einstein Yang-Mills metrics" (Comm. Anal. Geom. Vol 18, No. 1, 77-100, 2010.)
- "Asymptotic Stability of the Cross Curvature Flow at a Hyperbolic Metric" with Dan Knopf (Proc. Amer. Math. Soc. 137 (2009), 699-709)
- "Modified Ricci Flow on a Principal Bundle" my Ph.D. thesis
Research Mentoring Experience
I have mentored several undergraduate and graduate students on research problems in differential geometry. Below are links to some of their work.
- Andrew Marchese and Jacob Miller: Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes. (Arizona Summer Program, Summer 2010)
- Howard Cheng and Peter McGrath: Prescribing Curvature for Piecewise Flat Triangulated 3-manifolds. (Arizona Summer Program, Summer 2010)
- Veronica Marino: An Introduction to 3-manifolds: A constructive approach. (Research Tutorial Group, Fall 2009)
- Veronica Marino: Geodesics on Polyhedra. (Masters Thesis, Spring 2010-present)
- Cindy Phillips: The Geometry of Positively Curved Combinatorial 3-Manifolds. (Research Tutorial Group, Fall 2010)
Geometric Evolutions on Computational Abstract Manifolds (GEOCAM)
Click here for the GEOCAM homepage, where you can find information about our projects and participants.