I work in noncommutative geometry and index theory. I am particularly interested in applications of noncommutative techniques (C*-algebras, K-theory, groupoids) to problems in "classical" analysis and topology. Much of my research is motivated by explorations of the index theory of hypoelliptic operators.
With Alexander Gorokhovsky. Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrose. In the book: Cyclic Cohomology at 40: Achievements and Future Prospects, Proceedings of Symposia in Pure Mathematics, Vol 105 (2023), 185-196. arXiv:2010.02884
With Alexander Gorokhovsky. The Heisenberg calculus, index theory and cyclic (co)homology. Advances in Mathematics, Vol. 399 (2022), 62 pages. arXiv:2010.02884
With Paul Baum. The index theorem for Toeplitz operators as a corollary of Bott periodicity. The Quarterly Journal of Mathematics, Vol 72, Issue 1-2, June 2021, Atiyah Memorial Issue, 547-569. arXiv:2010.02906
With Robert Yuncken. A groupoid approach to pseudodifferential operators. Journal für die reine und angewandte Mathematik ("Crelle's Journal"). Vol 2019, Issue 756 (2019), 151-182. arXiv:1511.01041
With Alexander Gorokhovsky. Index theory and noncommutative geometry: A survey. In the book: Advances in Noncommutative Geometry - On the occasion of Alain Connes' 70th birthday, 2019, 421-462. arXiv:2010.02900
With Robert Yuncken. On the tangent groupoid of a filtered manifold. Bulletin of the London Mathematical Society 49 (2017), no. 6, 1000-1012. arXiv:1611.01081
With Paul Baum. K-homology and Fredholm operators I: Dirac operators. Journal of Geometry and Physics, Vol 134 (2018), 101-118. arXiv:1604.03502
With Paul Baum. K-homology and Fredholm operators II: Elliptic operators. Pure and Applied Mathematics Quarterly, Vol. 12, No. 3 (2016), 1-17. arXiv:1604.03535
With Pierre Julg. The Geometry of the Osculating Nilpotent Group Structures of the Heisenberg Calculus. Journal of Lie Theory, Vol. 27 (2017), 107-138. arXiv:1007.4759
With Paul Baum. K-homology and index theory on contact manifolds. Acta Mathematica 213:1 (2014), 1-48. arXiv:1107.1741
With Dana Williams. Groupoid Crossed Products of Continuous-Trace C*-Algebras. Journal of Operator Theory, Vol. 72, Issue 2 (2014), 557-576. arXiv:1309.1115
With Calder Daenzer. T-duality for Langlands dual groups. Advances in Theoretical and Mathematical Physics, Vol. 8, No. 6 (2014), 1267-1285. arXiv:1211.0763
Contact structures in higher codimensions and idempotents in the Heisenberg calculus. Preprint. arXiv:1001.5426
The index of hypoelliptic operators on foliated manifolds. Journal of Noncommutative Geometry, Vol. 5, Issue 1 (2011) 107-124. arXiv:0811.1969
The Atiyah-Singer index formula for subelliptic operators on contact manifolds, part I. Annals of Mathematics 171 (2010), 1647-1681. arXiv:0804.2490
The Atiyah-Singer index formula for subelliptic operators on contact manifolds, part II. Annals of Mathematics 171 (2010), 1683-1706. arXiv:0804.2492
Noncommutative topology and the world's simplest index theorem. Proceedings of the National Academy of Sciences 107 (2010), No. 19, 8549-8556 . arXiv:1001.5299
An index formula for the extended Heisenberg algebra of Epstein, Melrose and Mendoza. Journal of Topology and Analysis, Vol. 2, No. 4 (2010) 527-542. arXiv:1007.3532