Photo credit: Cristina Taho Izuno
Jeffrey D. Carlson
180 Queen's Gate
Hi! I'm a topologist presently working as a postdoc at Imperial College London.
In previous lives I've worked in a joint appointment between the Fields Institute and Western University,
symplectic geometry group at the University of Toronto,
the geometry group
at the Universidade de São Paulo
in a position sponsored by
My dissertation, supervised by
Loring Tu at Tufts University,
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. It is in the long process of being reshaped into a textbook.
My work since has continued to mostly be connected with the equivariant topology
of manifolds in a broad sense, often in collaboration with geometers.
Fixed points and semifree bordism.
We show the ABBV localization identities completely determine
the possible discrete fixed-point sets of a semifree circle action on a stably complex manifold.
This approach turns out to recover the cobordism ring of such actions (it's Z[S2])
through extremely naive (19th-century–style) manipulations of combinatorial identities in about a page.
A generalization of this approach will—God willing—enable computation of torus-equivariant
complex cobordism in other cases as well.
Realization of abstract GKM isotropy data,
with Elisheva Adina Gamse and Yael Karshon.
The ABBV localization identities for a torus action put strong constraints on isotropy data and we show the converse,
that such data must arise from a torus action on stably complex manifold.
This comes with a concrete, visually appealing construction
in dimensions up to four, and the proof otherwise relies
on considerations connected with equivariant complex cobordism.
My dissertation (now a book).
This work was nominally about
Borel equivariant cohomology, but in the writing became predominantly an
account of the cohomology of homogeneous spaces.
My opinion (perhaps self-serving) is that it is the fastest introduction
to this theory.
The exposition has some things that don't appear elsewhere.
This is not the final version.
This book is shortly to be the subject of a major revision at
the behest of Springer and its polite but insistent referees.
- The equivariant K-theory of an isotropy action.
The main result of this paper establishes as corollaries the equivariant K-theory rings in basically all the known equivariantly formal examples.
We also show that under the same conditions, surjectivity of the forgetful map in K-theory
is equivalent to weak equivariant formality in the sense of Harada–Landweber.
The main tool is a map of spectral sequences from Hodgkin's Künneth spectral sequence to the
analogous spectral sequence in Borel cohomology.
- The equivariant K-theory of a cohomogeneity-one action.
An outgrowth of the joint paper below. The appendices include lemmas providing a general framework for such computations in an arbitrary equivariant cohomology theory vanishing in certain degrees and a curious additional structure
on the Mayer–Vietoris sequence no one seems to have discussed.
- The equivariant cohomology ring of a cohomogeneity-one action, with Oliver Goertsches, Chen He, and Liviu Mare.
This one computes the object in the title, extending results of the first and third author.
Cohomogeneity-one actions are an important and very well-studied class
of examples in differential geometry whose topology is determined entirely by
an inclusion diamond of four Lie groups, so this was a natural question to ask.
Published in Geometriae Dedicata.
The Borel equivariant cohomology of real Grassmannians.
had recently computed this
through his development of GKM-theory and I noticed older methods gave a clean proof as well.
This also gave me the chance to publish a proof of a structure result from a late, unpubished version of my thesis which was in danger of becoming folkore.
To be published in the Proceedings of the American Mathematical Society.
Equivariant formality of isotropy actions, with Chi-Kwong Fok.
Also deals with equivariant formality in K-theory,
and proves the underlying homogeneous space is formal and a
regularity criterion for equivariant formality of an isotropy action.
Published in the Journal of the London Mathematical Society.
Equivariant formality of isotropic torus actions.
A development of my thesis research. Reduces (nearly) equivariant formality of an isotropy action to the case of a torus in a semisimple group, and provides a complete classification for circles.
Appendices handle (partially) the case of a compact Hausdorff group and a disconnected group.
Published in the Journal of Homotopy and Related Structures.
Commensurability of two-multitwist pseudo-Anosovs.
This provides the first nontrivial examples of subgroups of mapping class
groups of different surfaces simultaneously covering, element-wise,
a subgroup of the mapping class group of a covered surface, and a lot of examples of invariants distinguishing noncommensurable elements.
Conceptions of topological transitivity,
Ethan Akin (arXived).
once assigned a dynamics class I was sitting in on
the problem of fixing a broken proof
of a "folklore" result on
from his and Katok's
Introduction to the Modern Theory of Dynamical Systems,
so I did.
Boris suggested I verify
with "a hard-core topological dynamicist"
there were no published proofs of the desired result,
and proposed Akin.
Published in Topology and its Applications.
At one point in grad school, 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!).
- In the literature on cohomology of fibrations,
there has long been in use a term coexact,
which J. C. Moore insisted upon,
by all available accounts, owing to categorical
considerations that show the situation in question is
dual to the familiar situation of an exact sequence.
This notion doesn't seem to get a whole lot of press,
so I wrote a brief
expounding on the notion and the difference
from exactness, as well as conditions under which the two agree.
It was originally a letter to Larry Smith, in case a
"you" still appears anywhere in the revised note.
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.
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