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Summary

• Foundation Professor, 2023-present
• Mathematics Graduate Program Director, University of ÁùºÏ±¦µä, Reno, 2021-present
• Professor, University of ÁùºÏ±¦µä, Reno, 2009-present
• Core Mathematics Director, University of ÁùºÏ±¦µä, Reno, 2008- 2014
• Associate Professor, University of ÁùºÏ±¦µä, Reno, 2003- 2009
• Assistant Professor, University of ÁùºÏ±¦µä, 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

Research Interests

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.

Education

• University of California, Berkeley B.A., Mathematics 1985
• University of California, Berkeley Ph.D., Mathematics 1992

Select publications and preprints

• Examples of homology 3-spheres whose unperturbed Chern-Simons function is not Morse-Bott, (with H. Boden and P. Kirk), "Frontiers in Geometry and Topology," AMS Proceedings of Symposia in Pure Mathematics (to appear). Preprint at arXiv:2301.03676.
• The correspondence induced on the pillowcase by the earring tangle (with G. Cazassus, P. Kirk, and A. Kotelskiy). J. Topol. 15 (2022), Issue 4, 2472-2543. Preprint at arXiv:2010.04320
• Tangles, relative character varieties, and holonomy perturbed traceless moduli spaces (with G. Cazassus and P. Kirk), Open Book Ser. 5 (2022), No. 1, Math. Sci. Publ., 95-121. 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.