My primary research area is 3-dimensional topology and knot theory. I am verry interested in invariants of 3-manifolds and knots that are constructed using gauge theory, such as Floer homology, the Casson invariant, and SU(n) generalizations of these invariants. I am also interested in the topological and smooth knot concordance groups and Casson-Gordon invariants, untwisted and twisted Alexander polynomials, Cochran-Teichner-Orr invariants and invariants arising from Heegaard Floer theory which shed light on knot concordance and sliceness.
My work in higher rank Casson invariants has also given rise to an interest in equivariant transversality problems. Many standard foundational results about manifolds and maps between them become either much trickier, not known, or just plain untrue if one insists that the functions preserve some symmetries of the manifolds. In other words, there are many good problems left to work on in this area.
I have also been involved in several interdisciplinary projects recently.
Below are my publications, some of which can be downloaded in pdf form.
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, with P. Kirk and C. Livingston. (submitted)
FracMAP: A User-Interactive Package for Performing Simulation and Morphological Analysis of Fractal-Like Solid Agglomerates, with R. Chakrabarty, M. Garro, and H. Moosmuller. (submitted)
Thermal/Fluid Characteristics of Elliptic Cross Section Filament Box Lattice Matrices as Heat Exchanger Surfaces, with D. Sarde, and R. Wirtz,Paper #3810, AIAA Thermophysics and Heat Transfer Conference, San Francisco, CA, June 6-8, 2006.
Porosity, specific surface area and effective thermal conductivity of anisotropic open cell lattice structures, with K. Balantrapu, D. Sarde, and R. Wirtz, paper IPACK2005-73191, Proceedings of ASME InterPack Conference 2005, July 17-22.
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