Photo credit: Cristina Taho Izuno
Jeffrey D. Carlson
180 Queen's Gate
Hi! I'm a topologist presently working as a postdoctoral researcher at Imperial College London.
In lay terms, I am working in a temporary job where I
think about properties of shapes that don't change under stretching, and also symmetries.
In more mathematical language,
my work has started off in dynamics on surfaces,
is mostly connected with the equivariant algebraic topology of manifolds,
often in collaboration with geometers,
and has expanded to have a bit of overlap with sympectic geometry
and A-infinity algebras,
with some connections with Galois cohomology on the horizon.
In previous lives I've worked in a
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.
Not all of this is up (or up-to-date) on the arXiv
and this needs to be fixed shortly.
A ring structure on Tor
Cartan computed the cohomology ring of a homogeneous space over the reals in 1949,
setting off a search for cohomology groups over a more general field which more or less concluded
All these approaches looked at the Eilenberg–Moore spectral sequence of a fiber bundle
where the base and total space had polynomial cohomology; Hans Munkholm's approach,
however, considered the EMSS of a pullback square with polynomial cohomology more generally,
and thus applies to Borel cohomology of a homogeneous space,
cohomology of a biquotient,
or cohomology of the free free loop space of a space with polynomial cohomology
over any principal ideal domain.
The answer is that it is Tor of the cohomology rings of the span of input spaces.
All these results were additive, but
we show Munkholm's isomorphism is multiplicative,
resolving a seventy-year old question.
To be published
in Forum of Mathematics, Sigma.
Products on Tor
We compare the product on Tor we construct to show Munkholm's isomorphism is additive
with a product suggested by Munkholm and a product on the two-sided bar construction
defined by Franz figuring in the biquotient paper.
Collapse results for Eilenberg–Moore spectral
This is a historical survey of Eilenberg–Moore collapse results
establishing who did what when, which seemed appropriate to compile
given the seeingly final closure of this family of questions.
Not yet arXived.
Ask for these.
The topology of Gelfand–Zeitlin fibers,
with Jeremy Lane.
Gelfand–Zeitlin systems are examples of classical interest in symplectic geometry
satisfying many of the conclusions of standard theorems about toric integrable systems
without satisfying their hypotheses, and so giving an interesting class of edge cases.
In this work we find an closed-form description of the diffeomorphism
type of any fiber of a GZ system and compute the cohomology ring and first three homotopy groups
of a given fiber.
Each of these expressions turns out to admit a transparent description in terms of the combinatorics
of a certain graph associated to the fiber.
The cohomology of biquotients via a product on the two-sided bar construction, with an appendix joint with Matthias Franz.
We compute the Borel equivariant cohomology ring of the left K-action on a homogeneous space G/H,
where G and its subgroups H and K are compact, connected Lie groups and 2 and the torsion primes of the groups are units of the coefficient ring.
Particularly, this computes the cohomology rings of biquotients H\G/K.
(This had been done for H = 1 by Franz.)
The extension depends on a version of the Eilenberg–Moore theorem developed in the appendix,
where we show mod-2 formulas proposed by Franz give a sort of weak product structure
on a two-sided bar construction B(A', A, A'')
whenever A' ← A → A'' is a pair of maps of homotopy Gerstenhaber algebras.
To be published in Algebraic & Geometric Topology.
The proof of the main result relies
among other things on a lemma proven in Joel Wolf's 1973 Brown dissertation
but never published,
which I wound up reproving for lack of access.
(The one copy of this thesis in the world
is not available for loan.)
On a recent trip to Providence,
I reserved time in the rare books room at
John Hay Library to digitize it,
and it is now
available to the public
Fixed points and bordism of semifree actions.
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.
Following some correspondence we have included generous background on the problem.
Revisions have been requested and material will be added to this in the coming months.
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.
Equivariant formality of corank-one isotropy actions and products of rational spheres, with Chen He.
We characterize the pairs of connected Lie groups G > K such that the difference of their ranks is one and the left action of K on G/K is equivariantly formal. The analysis requires us to correct and extend an existing partial classification of homogeneous quotients G/K with the rational homotopy type of a product of an odd- and an even-dimensional sphere.
To be published in Mathematische Zeitschrift.
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, diligent referees.
- The 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.
This work is under revision following referee comments at Advances and will
only be updated on the arXiv when finalized.
The finished version will differ substantially from the arXiv version.
- K-theory and formality.
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.
in International Mathematics Research Notices.
The published version differs substantially from the arXiv version,
which will be updated in early November.
Grassmannians and the equivariant cohomology of isotropy actions.
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 folklore.
Published in the Proceedings of the American Mathematical Society.
This needs an arXiv update.
The K-theory of the conjugation action.
This is a 1999 result of Brylinski and Zhang; my observation was simply that with some planning, much of the proof could be sidestepped, resulting in a proof of less than a page.
Published in Comptes Rendus.
Not yet arXived.
- 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.
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 hardcore 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.
I taught commutative algebra at Imperial in Fall 2021 and
was compelled to transcribe complete lecture notes,
reproduced here for your approval.
And here are the problem sets and solutions.
I also once overwrote a number of word problems
which were received with some mixture of enthusiasm and confusion
in a vector calculus course at the University of Toronto at Mississauga.
CV and documentation