My primary research field has been the mathematics inspired by the quantum theory of nature—known to some as noncommutative geometry (á la Connes) or geometric functional analysis—which involves operator algebras, K-theory, and Lie theory. More specifically, a substantial portion of my research has been in index theory. And recently, I began exploring quantum information theory.
Index theory—as a study of generalizations of the Atiyah-Singer index theorem—seeks to present global properties of a geometric object (such as manifolds) in terms of the local data, and analytic quantities in terms of the topological. A similar yet different flavor is found in Lie theory, where we find a tight connection between the global and local aspects of the theory (namely, Lie groups and Lie algebras). There are beautiful examples of the interplay between index theory and Lie theory that are known to us, but no doubt there are more to be revealed out there.
Quantum information theory attempts to describe the information processing and computations based on quantum mechanics. According to our best knowledge of physics, our universe is quantum mechanical, but for most of our daily experience the approximation of classical (Newtonian) mechanics is sufficient and adequate, and thus, the current theory of computation and information is classical in its foundation. Thanks to the continued advancement in engineering, however, computation relying wholly on quantum nature is becoming more feasible. Hence, the mathematical problems in quantum information theory are likewise becoming more practical.
I count myself one of the number of those who write as they learn and learn as they write.
Papers
Refereed Journal Articles:
| • | Volume of the set of locally diagonalizable bipartite states, Journal of Physics A: Mathematical and Theoretical, 51 (2018), DOI 10.1088/1751-8121/aad3de. Available at arXiv:1611.09586 [quant-ph] |
| • | A Lie-algebraic approach to the local index theorem on compact homogeneous spaces, Advances in Mathematics, 296 (2016), 127–153, DOI 10.1016/j.aim.2016.03.038. Available at arXiv:1210.5855 [math.FA] |
| • | Harish-Chandra’s Volume Formula via Weyl’s Law and the Euler-MacLaurin Formula, Differential Geometry and Its Applications, 38 (2015), 58–68, DOI 10.1016/j.difgeo.2014.11.006. Available at arXiv:1111.2749 [math.RT] |
Unpublished Manuscripts:
| • | Borel-Weil-Bott theorem via the equivariant McKean-Singer formula. Available at arXiv:1412.3879 [math.RT]. |
| • | Duflo isomorphism and Chern-Weil theory. Available at arXiv:1502.07342 [math.DG] |
Notes
- Proof of the Atiyah-Singer index theorem using heat kernels and rescaling (pdf)