My primary research focus is low-dimensional manifolds and knot theory. I am especially
interested in invariants
of 3-manifolds and knots that are constructed using gauge theory, including Floer
homology and the Casson invariant, as well as many related invariants such
as knot polynomials, signature, Khovanov homology.
Much of my recent work involves traceless character varieties of knots and links. The
traceless character variety is the set of homomorphisms from the knot or link group to
SU(2). There are also higher rank analogs, character varieties with a condition fixing
the meridional conjugacy classes.
I also have been involved in several interdisciplinary projects
related to engineering and climate science.
Below are my publications, some of which can be downloaded
in pdf form.
Preprints and Publications
The SU(2) Casson-Lin invariant of the Hopf link, with H. Boden,
Pacific J. Math. 285 (2016), no. 2, 283-288. Preprint (2016)
The pillowcase and traceless representations of knot groups, II:
A Lagrangian-Floer theory in the pillowcase, with M. Hedden and P. Kirk,
to appear in J. Symplect. Geom. Preprint (2015), arXiv:1501.00028.
The pillowcase and perturbations of traceless representations of knot groups,
with P. Kirk and M. Hedden, Geom. Topol. 18 (2014), 211-287.
Simple Climate Models to Illustrate How Bifurcations Can Alter
Equilibria and Stability, with S. Kurita and A. Telyakovskiy,
J. Contemp. Water Res. Ed., 152 (2013), 14-21.
UNR Mathematics Dodges the Budget AxeÑFor Now, Notices Amer. Math. Soc.,
59 (2012), no. 02, 261.
Simulation of aggregates with point-contacting monomers in the clusterÐdilute
regime. Part 2: Comparison of two and three-dimensional structural properties
as function of fractal dimension, with R. K. Chakrabarty, M. A. Garro, B.A. Garro,
S. Chancellor, H. Moosmüller, Aerosol Sci. Technol. 45 (2011), no. 8, 893-898
Simulation of aggregates with point-contacting monomers in the cluster-dilute
regime. Part 1: Determining the most reliable technique for obtaining
three-dimensional fractal dimension from two-dimensional images dimension,
with R. K. Chakrabarty, M. A. Garro, B.A. Garro, S. Chancellor, H. Moosmüller,
Aerosol Sci. Technol. 45 (2011), no. 1, 75-80.
Extending the SU(3) Casson invariant to rational homology 3-spheres,
AMS/IP Studies in Advanced Mathematics, Proceedings of Chern-Simons Theory:
20 Years After, edited by Andersen, Boden, Hahn, and Himpel 50 (2011), 115-130.
Metabelian representations, twisted Alexander polynomials, knot slicing,
and mutation, with P. Kirk and C. Livingston), Math. Z. 24 (2009) 925-949.
FracMAP: A User-Interactive Package for Performing Simulation and
Orientation-Specific Morphology Analysis of Fractal-like Solid Agglomerates,
with R. Chakrabarty, M. Garro, S. Chancellor, and H. MoosmŸller, Comput. Phys. Commun.
180 (2009), 1376-1381.
Thermal/Fluid Characteristics of Elliptic Cross Section Filament Box Lattice
Matrices As Heat Exchanger Surfaces, with D. Sarde and R.A. Wirtz,
Paper #3810, AIAA Thermophysics and Heat Transfer Conference,
San Francisco, CA, June 6-8, 2006.
The SU(3) Casson Invariant for Brieskorn Homology 3-Spheres, with H. Boden
and P. Kirk, J. Differential Geom. 71 (2005), no. 1, 23-83.
Porosity, Specific Surface Area and Effective Thermal Conductivity of
Anisotropic Open Cell Lattice Structures, with K. Balantrapu, D. Sarde, and R.A. Wirtz,
paper IPACK2005-73191, in Proceedings of ASME InterPack Conference 2005, July 17-22.
On the integer valued SU(3) Casson invariant, with H. Boden and P. Kirk,
Topology and geometry of manifolds (Athens, GA, 2001), 209--236, Proc. Sympos.
Pure Math., 71 , Amer. Math. Soc., Providence, RI, 2003.
SU(3) generalizations of the Casson invariant from gauge theory, Geometry,
integrability, and quantization (Varna, 2001), 278--289, Coral Press Sci.
Publ., Sofia, 2002.
Chern-Simons gauge theory on 3-manifolds, The Proceedings of the Conference
on Geometric Structures on Manifolds (Seoul 1997),
Lecture Notes Ser. No. 46, Seoul Nat. Univ., Seoul 1999.
Existence of irreducible representations for knot
complements with nonconstant equivariant
signature, Math. Ann. 309 (1997) 21—35.
Flat connections, the Alexander invariant, and Casson's invariant, Comm.
Anal. Geom., 5 (1997) 93—120.
Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary,
Comm. Anal. Geom., 2 (1994) 337—413.
This webpage is under construction. Check back for updates. (In the meantime, many of
the recent papers listed above are listed on the mathematics preprint arxiv.)
last updated 1/31/17
Math Dept or UNR