Current Research


Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, in other words, with the symmetries of a space. A localization theorem computes a global invariant in terms of local data, say at the zeros of a vector field or at the fixed points of the group action. The localization theorem of Atiyah--Bott and Berline--Vergne often reduces an integral over a manifold with a group action to a finite sum over the fixed point sets of the action. Currently I am interested in localization theorems and their applications to more classical problems in geometry and topology. An expository article I wrote about this subject appeared in the March 2011 issue of the Notices of the American Mathematical Society. In 2020 I published a book Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies vol. 204, Princeton University.



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Selected Articles Since 2000

  1. Equivariant characteristic classes in the Cartan model (with Raoul Bott), in Geometry, Analysis, and Applications (Varanasi, 2000), World Scientific Publishing, River Edge, NJ, pp. 3--20. preprint

    In this article we construct an equivariant Chern--Weil homomorphism and prove that the topological and differential-geometric definitions of equivariant characteristic classes coincide.

  2. The life and works of Raoul Bott, in The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, edited by S.-T. Yau, International Press, Somerville, MA, 2003, pp. 85--112. An updated version appeared in the Notices of the American Mathematical Society 53 (2006), pp. 554--570. reprint

    In a career spanning five decades, Raoul Bott had wrought profound changes on the landscape of geometry and topology. This is an authorized biography of him, which he proofread and approved. In the article I recount the salient incidents in his life and discuss a selection of nineteen articles that he considered to be his favorites.

  3. Reminiscences of working with Raoul Bott, in The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, edited by S.-T. Yau, International Press, Somerville, MA, 2003, pp. 121--124. preprint

  4. Une courte démonstration de la formule de Campbell-Hausdorff (A short proof of the Campbell-Hausdorff formula), Journal of Lie Theory 14 (2004), pp. 501--508. reprint

  5. A partial order on partitions and the generalized Vandermonde determinant, Journal of Algebra 278 (2004), pp. 127--133. reprint

  6. On the localization formula in equivariant cohomology (with Andrés Pedroza), Topology and Its Applications 154 (2007), pp. 1493--1501. reprint

  7. Computing characteristic numbers using fixed points, in A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings and Lecture Notes, vol. 50, American Mathematical Society, Providence, RI, 2010, pp. 185--206. preprint

  8. What is equivariant cohomology, Notices of the American Mathematical Society 58 (2011), pp. 423--426. reprint

  9. The Abdus Salam School of Mathematical Sciences in Pakistan, Notices of the American Mathematical Society 58 (2011), pp. 938--943. reprint

  10. From sheaf cohomology to the algebraic de Rham theorem, Chapter 2 in Hodge Theory, Mathematical Notes 49, edited by E. Cattani, F. El Zein, P. A. Griffiths, L. D. Trang, Princeton University Press, 2014, pp. 69--121. preprint

  11. On the genesis of the Woods Hole fixed point theorem, Notices of the American Mathematical Society 62 (2015), pp. 1200--1206. reprint

  12. Computing the Gysin map using fixed points, in Algebraic Geometry and Number Theory, Proceedings of the CIMPA Summer School on Algebraic Geometry and Number Theory (Istanbul, June 2--11, 2014), edited by H. Mourtada, C. C. Sarioglu, C. Soulé, and A. Zeytin, Birkhäuser, 2017. preprint

  13. Computing topological invariants using fixed points, in Proceedings of the Sixth International Congress of Chinese Mathematicians, vol. II, Adv. Lect. Math. (ALM) 37, pp. 285--298, International Press, Somerville, MA, 2017. preprint

  14. Equivariant characteristic classes, in Raoul Bott: Collected Papers , vol. 5, Birkhäuser, 2017, pp. 103--105. preprint

  15. Lefschetz fixed point theorem for correspondences, in Mathematics Going Forward , Lecture Notes in Mathematics vol.~2313, Springer, to appear in April 2023. preprint