


Jeffrey D. Carlson
40 St. George Street
Toronto, Ontario M5S 2E4
Canada
St. George office:
PG 303A

UTM office: DH 3021
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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 USPIME in São Paulo
in a visiting postdoctoral position sponsored by
IMPA.
My dissertation (at Tufts University) was supervised by
Loring Tu and
was about Borel equivariant cohomology,
an algebrotopological 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 GKMtheoretic methods by Chen He.
I am currently editing a joint preprint with
ChiKwong Fok also dealing with equivariant formality in Ktheory.
My dissertation,
still being revised, is nominally about
Borel equivariant cohomology, but in the writing became a booklength
account of the cohomology of homogeneous spaces.
My opinion, perhaps selfserving, 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
onepage 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 twomultitwist pseudoAnosovs arose.
It provides the first nontrivial examples (to my knowledge) of subgroups of mapping class
groups of different surfaces simultaneously covering, elementwise,
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
T_{E}X
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 x_{0} in X
that is different from the x_{i}.
Then since the q_{i}
are contained uniquely in the maximal ideals, there must exist,
for each q_{i},
an f_{i}
inside that does not vanish at x_{0},
so that the product of these f_{i}
will be a nonzero element inside the intersection of the f_{i}."

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 T_{5}.
 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
S^{1}action of a maximal torus of Sp(1)
on Sp(1)/S^{1},
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
