Topology and differential geometry for singular spaces (foliations, stratified orbifolds and moduli spaces from Gromov-Witten theory), including local index theories and higher index theories. Mathematical physics including, topological and differential quantum field theory, gerbes and D-branes.
Potential project opportunities
- Higher bundle gerbes and cohomology classes in gauge theories, (with A. Carey, M. Murray), Journal of Geometry and Physics, 21 (1997), no. 2, 183-197.
- Seiberg-Witten Monopoles in Three Dimensions , (with A. Carey, J. McCarthy, R. Zhang), Letters in Mathematical Physics, 39 (1997), no. 3, 213-228.
- Equivariant Seiberg-Witten-Floer theory, (with M. Marcolli), Communications in Analysis and Geometry, Vol.9 N.3 (2001) 451-640.
- Featured Review in MathSciNet. Seiberg-Witten invariant and Casson-Walker invariant for rational homology 3-spheres , (with M. Marcolli), Geometriae Dedicata. Vol.91 (1): 45-58, 2002.
- Weak unique contiuation principal and perturbed monopole equations, (with B. Booss-Bavnbek, M. Marcolli), International Journal of Mathematics, Vol. 13, No. 9 (2002), 987-1008.
- The geometric triangle for 3-dimensional Seiberg-Witten monopoles, (with A. Carey, M. Marcolli), Communications in Contemporary Mathematics, Vol. 5, no. 2, 2003, 197-250.
- Seiberg-Witten-Floer homology and gluing formulae, (with A. Carey), Acta Mathematica Sinica, Vol 19, No.2, 245-296, 2003.
- Variants of Equivariant Seiberg-Witten-Floer Homology, (with M. Marcolli), Contemporary Mathematics, Vol. 366, 225-238, 2005.
- Seiberg-Witten-Floer homology of a surface times a circle, (with V. Munoz), Mathematische Nachrichten, Vol. 278, No. 7-8, 844-863, 2005.
- Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, (with A. Carey, S. Johnson, M. Murray, D. Stevenson), Communications in Mathematical Physics, Vol. 259 No. 3 577-613, 2005.
- On the relationship of gerbes to the odd families index theorem, (with A. Carey), J. Geom. Phys. 57 (2006), no. 1, 23-38.
- Fusion of Symmetric D-branes and Verlinde ring. (with A. Carey), Communications in Mathematical Physics, Vol. 277, No. 3, February, 2008, 577-625.
- Thom isomorphism and Push-forward map in twisted K-theory. (with A. Carey), Journal of K-theory, (2008), 1 : 357-393.
- Geometric cycles, Index theory and twisted K-homology . Journal of Noncommutative Geometry. 2 (2008), no. 4, 497-552.
- Differential Twisted K-theory and its Applications (with A. Carey and J. Mickelsson), J. Geom. Phys. 59 (2009), no. 5, 632-653.
- Riemann-Roch and index formulae in twisted K-theory. (with A. Carey), "Topology, C*-algebras, and String Duality", Proceedings of Symposia in Pure Mathematics Vol. 81, 95-131, 2009.
- Twisted longitudinal index theorem for foliations and wrong way functoriality. (with Paulo Carrillo Rouse). Advances in Mathematics, Vol. 226, No. 6, 4933-4986, 2011.
- Delocalized Chern character for stringy orbifold K-theory (arχiv:1110.0953). (with Jianxun Hu), Accepted in for publication in Transactions of the American Mathematical Society.