I work in noncommutative geometry and index theory. I am particularly interested in applications of noncommutative techniques (C*-algebras, K-theory, cyclic (co)homology, 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. A note on traces for the Heisenberg calculus. Preprint. arXiv:2410.11982
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