Jeffrey D. Carlson
40 St. George Street
Toronto, Ontario M5S 2E4
St. George office:
UTM office: DH 3021
Recent and Ph.D. research
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
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.
I recently produced this
on how to compute the Borel equivariant cohomology of real Grassmannians,
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.
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.
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
of this classification.
A proof is contained in
which I am also updating and streamlining.
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.
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
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
installation, resulting in some occasional tragic miskerning in this copy,
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,
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
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
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)
is geared toward a comparison of these two formulae.
CV and documentation
Department of Mathematical and Computational Sciences,
University of Toronto