- Mathematics Graduate Program Director, University of Nevada, Reno, 2021-present
- Professor, University of Nevada, Reno, 2009-present
- Core Mathematics Director, University of Nevada, Reno, 2008- 2014
- Associate Professor, University of Nevada, Reno, 2003- 2009
- Assistant Professor, University of Nevada, Reno, 1999-2003
- Visiting Assistant Professor, Swarthmore College, 1996-1999
- Britton Postdoctoral Fellow, McMaster University, Hamilton, Ontario, 1994-1996
- Visiting Researcher, Max Planck Institut für Mathematik, Bonn, Germany, January-June 1995
- Visiting Assistant Professor, University of California, Irvine, 1992-1994
My current research interests include low-dimensional topology, knot theory, symplectic geometry, and gauge theory.
My current projects are related to traceless SU(2) character varieties of knots and links in 3-manifolds. Consider a surface splitting the knot into tangles. The traceless character varieties of the tangles produce Lagrangian submanifolds in the traceless character variety of the splitting surface, which is a symplectic manifold. The Fukaya category of this symplectic manifold provides a very useful algebraic framework to study these situations.
These character varieties discussed above are related to singular instanton knot Floer homology, Khovanov homology and Casson-Lin invariants.
- University of California, Berkeley B.A., Mathematics 1985
- University of California, Berkeley Ph.D., Mathematics 1992
Select publications and preprints
- The correspondence induced on the pillowcase by the earring tangle (with G. Cazassus, P. Kirk, and A. Kotelskiy). Submitted. Preprint at arXiv:2010.04320.
- Tangles, relative character varieties, and holonomy perturbed traceless moduli spaces (with G. Cazassus and P. Kirk), to appear in Open Book Series. Preprint at arXiv:2101.00901.
- The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology (with M. Hedden, M. Hogancamp, and P. Kirk), Trans. Amer. Math. Soc. 373 (2020), 8391-8437. Preprint at arXiv:1808.06957.
- A note on orderability and Dehn filling (with X. Zhang), Proc. Amer. Math. Soc. 147 (2019), 2815-2819. Preprint at arXiv:1807.00742.
- Holonomy perturbations in a cylinder, and regularity for traceless SU(2) character varieties of tangles (with P. Kirk), Quantum Topology , 9 (2018), no. 2, 349-418. Preprint at arXiv:15111.00308.
- The pillowcase and traceless representations of knot groups, II: A Lagrangian-Floer theory in the pillowcase (with M. Hedden and P. Kirk), J. Symplect. Geom. 16 (2018), no. 3, 721-815. Preprint at arXiv:1501.00028.
- The SU(2) Casson-Lin invariant of the Hopf link (with H. Boden), Pacific J. Math. 285 (2016), no. 2, 283-288. Preprint at arXiv:1601.06017.