1. Jeffrey D. Carlson

    40 St. George Street
    Toronto, Ontario  M5S 2E4
    Canada

    St. George office: PG 303A | UTM office: DH 3021

     

    Navigate:   Recent and Ph.D. research   |   Other research   |   CV   |   Teaching

Hi, I'm a topologist working in the symplectic geometry group at the University of Toronto. Last year I worked in the geometry group at USP-IME in São Paulo in a visiting post-doctoral position sponsored by IMPA.

My dissertation (at Tufts University) was supervised by Loring Tu and was about Borel equivariant cohomology, an algebro-topological tool for studying continuous group actions, in my case specifically as applied to Lie group actions on homogeneous spaces. Since those days I have been adopted by geometers.

 


Recent work

I recently produced this expository note on how to compute the Borel equivariant cohomology of real Grassmannians, following on their recent computation through GKM-theoretic methods by Chen He.

I am currently editing a joint preprint with Chi-Kwong Fok also dealing with equivariant formality in K-theory.

My dissertation, still being revised, is nominally about Borel equivariant cohomology, but in the writing became a book-length account of the cohomology of homogeneous spaces. My opinion, perhaps self-serving, is that it is the fastest introduction to this theory. The original portion of the dissertation is about equivariant formality of isotropy actions on homogeneous spaces. One result is a classification of for which circles S in compact, connected Lie groups G the isotropy action of S on G/S is equivariantly formal. Here is a one-page statement of this classification. A proof is contained in this preprint, which I am also updating and streamlining.

 


Other research

  • I have also done geometric topology, and a preprint on commensurability of two-multitwist pseudo-Anosovs arose. It provides the first nontrivial examples (to my knowledge) of subgroups of mapping class groups of different surfaces simultaneously covering, element-wise, a subgroup of the mapping class group of a covered surface.
  • Boris Hasselblatt once assigned, to a dynamics class I was sitting in on, the problem of fixing a broken proof of a "folklore" result on equivalence of different definitions of topological transitivity from his and Anatole Katok's Introduction to the Modern Theory of Dynamical Systems. Because it turned out that there indeed did not exist published proofs, this assignment turned into "Conceptions of Topological Transitivity," a joint paper with Ethan Akin (arXived copy at http://arxiv.org/abs/1108.4710 ), published in 2012.
  • At one point in 2011, I decided that if I did all the exercises in Atiyah & Macdonald, I would become deeply knowledgeable about commutative algebra. Though that assessment turns out to have been optimistic, I am now the proud owner of a complete solution set (only slightly longer than the book itself!). The document was originally set in a Garamond absent from my present TEX installation, resulting in some occasional tragic miskerning in this copy, and worse, my inability to recompile the source due to several inexplicable errors. Still, there's not much else I'd change about it.
    • As if sent from on high to reprimand me for my complacency, Hao Guo has informed me that there is an error to the solution to problem 4.6. He suggests the following cogent emendment:
      "[S]ince X is infinite, pick an x0 in X that is different from the xi. Then since the qi are contained uniquely in the maximal ideals, there must exist, for each qi, an fi inside that does not vanish at x0, so that the product of these fi will be a non-zero element inside the intersection of the fi."
  • In Exercise 4.10.1 of their venerable Differential Forms in Algebraic Topology, Bott & Tu ask the reader to show that the image of a proper map is closed. While a pedant might object that the claim is not actually true, it does hold of most reasonably decent spaces. In this note I show that it suffices the codomain be Hausdorff and either first countable or locally compact, but without some sort of countability criterion, there still exist counterexamples even when the codomain is T5.
  • Does the Weyl integration formula follow as a corollary of ABBV localization as applied to the conjugation action of a maximal torus T of a Lie group G on the flag manifold G/T? The statements are tantalizingly similar. This note, providing explicit equivariant extensions with respect to the natural S1-action of a maximal torus of Sp(1) on Sp(1)/S1, is geared toward a comparison of these two formulae.

 


CV and documentation

 


Teaching

 


Department of Mathematical and Computational Sciences, University of Toronto   |   Mississauga, ON   L5L 1C6