Research

Papers

  1. Twisted equivariant differential cohomology, with U. Schreiber and H. Sati, in preparation.

  2. The geometric cobordism hypothesis, with D. Pavlov, preprint, https://arxiv.org/abs/2111.01095

  3. Extended field theories are local and have classifying spaces, with D. Pavlov, preprint https://arxiv.org/abs/2011.01208

  4. Ramond-Ramond fields and twisted differential K-theory, with H. Sati, preprint https://arxiv.org/abs/1903.08843

  5. Twisted differential KO theory, with H. Sati, preprint https://arxiv.org/abs/1905.09085

  6. Cobordisms of global quotient orbifolds and an equivariant Pontrjagin-Thom construction, preprint, https://arxiv.org/abs/1811.08794.

  7. Parametrized geometric cobordism and smooth Thom stacks, with H. Sati, preprint, https://arxiv.org/pdf/1709.00686.

  8. Differential KO theory: Constructions, computations and applications, with H. Sati, Advances in Math, 384 (2021) 107671

  9. Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers, with H. Sati, Journal of Geometry and Physics, 165 (2021) 104203

  10. Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence, with H. Sati, Algebraic and Geometric Topology, 19-6 (2019), 2899-2960.

  11. Higher twisted Deligne cohomology, Homology, Homotopy and Applications, 21-1 (2019), 129-159.

  12. Primary operations in differential cohomology, with H. Sati, Advances in Math, 335 (2018), 519-562.

  13. Twisted smooth Deligne cohomology, with H. Sati, Annals of Global Analysis and Geometry, 53-3 (2018), 445-466.

  14. Spectral sequences in smooth generalized cohomology, with H. Sati, Algebraic and Geometric Topology, 17-4 (2017), 2357-2412.

  15. Massey products in differential cohomology via stacks, with H. Sati, Journal of Homotopy and Related Structures, 13-1 (2017), 1-55.

Some notes

Brief note on anomalies in gauge theory and the AS index theorem

Lectures on cohomology, notes (incomplete) from a 2013 course taught by Hisham Sati at Pitt

Notes on Dan Freed's Lectures on invertible QFT's, at CBMS conference 2017 (with contributions by Matt Wheeler)

An introductory text that I am writing on differential cohomology .

Lectures, talks and slides

2020

  • The homotopy type of the cobordism category (3 talks), Topology and Geometry seminar, TTU

  • Introduction to equivariant homotopy theory, Quantum homotopy seminar, TTU

  • Smooth stacks and Cech cocycles (2 talks), Quantum homotopy seminar, TTU

  • Bundle gerbes with connection, Quantum homotopy seminar, TTU

  • Lifting M-theory to cohomotopy via obstruction theory, Topology and Geometry seminar, TTU

2019

  • RR-fields and twisted differential K-theory, UAE Math day, American University of Sharjah. Slides

  • An introduction to differential cohomology, Topology and Geometry seminar, TTU

  • The Atiyah–Hirzebruch spectral sequences for differential cohomology theories (2 talks), Topology and Geometry seminar, TTU

  • Twisted differential cohomology and the twisted Atiyah–Hirzebruch spectral sequence (5-talks), Topology and Geometry seminar, TTU

2018

  • A differential Riemann-Roch theorem for KO-theory, The Regional Collaborative Seminar on Geometry, Topology, and Physics, NYUAD. Slides

  • Geometric invariants of vector bundles and the index of elliptic operators, Math colloquium, NYUAD. Slides

2017

  • Twisted Differential K-theory and a refinement of the Atiyah-Hirzebruch spectral sequence, Topology seminar, KIAS. 4 talks. Slides 1, Slides 2

  • An Atiyah-Hirzebruch spectral sequence for twisted differential K-theory, Geometry-Topology seminar, University of Pennsylvania. Slides

  • An Atiyah-Hirzebruch spectral sequence for twisted differential K-theory, GAP seminar, Penn State.

  • Lectures on the Atiyah-Singer index theorem, Postdoc seminar, NYUAD.

  • Cobordism via higher geometry, Higher structures in geometry, topology and physics workshop, NYUAD. Slides

  • Parametrized geometric cobordism and smooth Thom stacks, Cohomology theories, categories and applications, University of Pittsburgh.

2016

  • Moduli stacks and higher gerbes, UAE math day, NYUAD.

  • What is differential cohomology and why should you care? American University of Sharjah.

2015

  • Massey products in differential cohomology, Higher structures and cohomology workshop, University of Pittsburgh.

2014

  • Minimal models, geometry proseminar, University of Pittsburgh.

  • Sheaf, deRham, and Čech cohomology, GTP seminar, University of Pittsburgh.

2013

  • Topological index theorem for differential K theory, topics course on AS index theorem, University of Pittsburgh.

  • An example of a rarely seen characteristic class in ‘nature’, geometry proseminar, University of Pittsburgh.

2012

  • What is a gerbe? GTP Seminar, University of Pittsburgh .