My research revolves
around
the
interaction between homotopy theory and
algebra. One
of my primary interests is motivic homotopy theory, which is a blend of homotopy
theory and algebraic geometry. The basic goal of this subject is to
solve
problems in algebraic geometry using traditional methods of algebraic
topology.
Currently, my main
interest is
a
project, in conjunction with Bob
Bruner and Dan Dugger, to make fundamental computations in motivic homotopy
theory. The
most promising current direction involves the motivic
version of the
I have recently
completed a
series of
three papers, in conjunction with Dan
Dugger, that generalize results known
about
sums-of-squares formulas over fields of characteristic zero so that
they apply
over arbitrary fields of characteristic not equal to 2. The basic idea
is to
replace techniques involving cohomology
theories for
topological spaces with techniques involving algebraic cohomology
theories. See the papers under the section Sums-of-Squares
Formulas below.
Another topic of my
recent
work involves
Cayley-Dickson algebras. These are
finite-dimensional
non-associative algebras that generalize the real numbers, the complex
numbers,
the quaternions, and the octonions.
The long-term goal was to produce some interesting geometric structures
that
possess exotic homotopical
properties. Although
there probably are some interesting geometric structures here,
unfortunately it
appears that they are too complicated to analyze by our methods. In conjunction with Dan Dugger,
Daniel Biss, and Dan
Christensen, we have written three papers that attempt to classify
the
zero-divisors in Cayley-Dickson
algebras. See
the papers under the section Cayley-Dickson
Algebras below.
I am also interested in
abstract homotopy theory (i.e., model
categories), especially with
respect to homotopy theories for
pro-objects.
If you want to know
more about
my
professional record, take a look at my curriculum
vita (i.e.,
my resume). If your computer can access to Mathscinet, the electronic database of
abstracts of
mathematics papers, you can look me
up.
Below you will find my
preprints and
publications, most of which can be downloaded. See also my page about undergraduate
research for a few papers that I have written about graph theory
and my
page about expository
and recreational mathematics.
Flasque model structures
for simplicial presheaves, K-Theory 36 (2005) 371--395.
It
is
well-known that there are two useful local model structures on simplicial presheaves,
the
projective and injective. This paper is about a third model structure
that lies
in between. It is surprising that this model structure has not been
described
before, especially considering that the proofs are so easy. [The link
leads to
the article abstract. From some computers, the full text is available
below and
to the right of the abstract.]
Motivic
cell structures, Algebr.
Geom. Topol. 5
(2005)
615--652 (with D. Dugger).
Unlike
in
ordinary homotopy theory, it is not
possible to build
every motivic space out of the motivic
spheres. We study the class of motivic
spaces that
can be built out of motivic spheres. We
give some
examples and prove some general computational results that can apply
only to
these special motivic spaces.
Etale realization on the A1-homotopy
theory of schemes, Adv. Math.
184 (2004) 37--63.
This
paper
uses results from several of the other papers listed here in order to
produce a
well-behaved realization functor from motivic homotopy
theory to the homotopy theory of pro-simplicial
sets. For several years (when I was in graduate school and just
afterwards),
this was the central question around which most of my research
revolved. [The
link leads to the article abstract. From some computers, the full text
is
available above and to the right of the abstract.]
Topological
hypercovers and A1-realizations, Math. Z. 246 (2004) 667--689 (with D. Dugger).
We
study hypercovers in the classical
situation of ordinary open
covers of topological spaces. We use this machinery to produce
well-behaved
realization functors from motivic
homotopy theory (in characteristic zero)
into
ordinary homotopy theory. [The link leads
to the
article abstract. From some computers, the full text is available below
and to
the right of the abstract.]
Weak equivalences of
simplicial presheaves, in Homotopy
theory:
relations with algebraic geometry, group cohomology,
and algebraic K-theory, 97--113, Contemp.
Math. 346, Amer. Math. Soc., 2004 (with D. Dugger).
This
technical paper provides an alternative way of viewing local weak
equivalences
of simplicial presheaves
in
terms of local lifting properties.
Hypercovers and simplicial presheaves,
Math. Proc. Cambridge Philos.
Soc. 136 (2004) 9--51 (with D. Dugger
and S. Hollander).
The
main
point of this paper is to show how to produce the local model
structures on simplicial presheaves (i.e., the
ones where the weak equivalences are detected by sheaves of homotopy
groups) as left Bousfield localizations at
a certain
set of maps. These sets of maps are defined in terms of hypercovers.
[The link leads to the article abstract. From some computers, the full
text is
available above and to the right of the abstract.]
Eigentheory of Cayley-Dickson algebras, preprint, 2007 (with D. Biss, J. D. Christensen, and D. Dugger).
We
establish
foundations for an eigentheoretic approach to Cayley-Dickson
algebras. This paper describes some basic
constructions and gives some indication about why this approach is
useful. There are many accessible and unanswered questions in the
subject.
Large
Annihilators in Cayley-Dickson algebras II,
Bol. Soc. Mat. Mexicana, to appear
(with D. Biss, J. D. Christensen, and D. Dugger).
We
establish
many previously unknown properties of zero-divisors in Cayley-Dickson
algebras. The basic approach is to use a certain splitting that
simplifies
computations surprisingly. We are able to determine the
annihilators of
large classes of zero-divisors, and we work out some concrete
conclusions for
zero-divisors in the 32-dimensional Cayley-Dickson
algebra. This paper is a sequel to the following paper.
Large
Annihilators in Cayley-Dickson algebras, Comm. Algebra, to appear (with D. Biss
and D. Dugger).
We
show that
the 2n-dimensional Cayley-Dickson
algebra
possesses zero-divisors whose annihilators have dimension 2n
- 4n +
4. In fact, any integer between 0 and this upper bound occurs as the
dimension
of an annihilator, provided that the integer is a multiple of 4. The
paper also
contains various other miscellaneous results about Cayley-Dickson
algebras.
The Hopf condition for
bilinear forms
over arbitrary fields, Ann. Math 165
(2007) 943--964 (with D. Dugger).
We
make a
computation in
motivic cohomology.
This
gives us a proof of the classical Stiefel-Hopf
condition for sums-of-squares formulas over arbitrary fields of
characteristic
not equal to 2.
Algebraic K-theory and sums-of-squares formulas, Doc. Math. 10
(2005) 357--366
(with D. Dugger).
We
compute
the algebraic K-theory of a certain open subvariety
of projective space. This gives us a proof of the Atiyah
condition for sums-of-squares formulas over arbitrary fields of
characteristic
not equal to 2.
Etale homotopy
and sums-of-squares formulas, Math.
Proc. Cambridge Philos. Soc., to appear (with
D. Dugger).
Like
the two
papers listed above, this paper generalizes some results about
sums-of-squares
formulas from characteristic zero to characteristic p > 2.
Instead of ordinary motivic cohomology
or algebraic K-theory, this paper uses a relative of BP-cohomology called etale
BP2.
Model
structures on pro-categories,
Homology Homotopy Appl.
9 (2007)
367--398 (with H. Fausk).
The
purpose
of this paper is to find a common framework that unifies the various homotopy theories of pro-objects. I consider
this project
to have been only partially successful because the framework described
in the
paper is very complicated and unintuitive. There really ought to be a
better
viewpoint.
t-model
structures, Homology Homotopy Appl. 9 (2007) 399--438 (with H. Fausk).
This
paper is
a generalization of the preprint Generalized cohomology
of pro-spectra. Rather than studying homotopy
theories of pro-spectra, we work with pro-objects in any stable model
category
equipped with a t-structure. Independently of the application
to
pro-categories, this paper contains some ideas about the interactions
between t-structures
and model categories that deserve further exploration.
Generalized
cohomology of pro-spectra, preprint, 2004.
This
paper is
a stable analogue of my paper A
model structure on the category of pro-simplicial
sets. I define weak equivalences of pro-spectra in terms
of pro-homotopy groups. I show how
generalized cohomology
of pro-spectra is representable in the
associated homotopy category, and I
construct an Atiyah-Hirzebruch
spectral sequence.
Completions
of pro-spaces, Math. Z. 250
(2005) 113--143.
This
paper
constructs model structures on pro-spaces in which the weak
equivalences are
detected by cohomology isomorphisms.
The interesting thing is that fibrant
replacement is
closely linked to the Bousfield-Kan
R-tower. [The
link leads to the article abstract. From some computers, the full text
is
available below and to the right of the abstract.]
Duality and pro-spectra, Algebr. Geom. Topol. 4 (2004)
781--812 (with J.
D. Christensen).
We
construct
a model structure on the category of pro-spectra that uses cohomotopy
groups to detect weak equivalences. This model structure turns out to
be
equivalent to the opposite of ordinary stable homotopy
theory. This is similar to what happens for vector spaces, where the
opposite
of the category of k-vector spaces is equivalent to the
category of
pro-(finite-dimensional k-vector spaces).
Strict
model
structures for pro-categories,
in Categorical decomposition techniques in algebraic topology (
Starting
with
an arbitrary proper model category C, I construct a model
structure on
pro-C. This is almost always the first step in establishing more
interesting model structures on pro-categories.
Calculating limits
and colimits in pro-categories, Fund. Math.
175 (2002) 175--194.
This
paper is
a collection of results that are useful for making constructions in
pro-categories. Some of these constructions are not as intuitive as one
might
expect. In particular, the nature of colimits
in pro-categories (and dually limits in ind-categories)
are subtle.
A
model structure on the category of pro-simplicial
sets, Trans. Amer. Math. Soc.
353 (2001) 2805--2841.
This
is the
published version of my Ph.D. thesis. I constructed a model structure
on the
category of pro-simplicial sets such that
the weak
equivalences are detected by pro-isomorphisms
of pro-homotopy groups. This is precisely
the right context for
studying the etale homotopy
type. [The link leads to the article abstract. From some computers, the
full text
is available above the abstract.]
Obstruction
theory in model categories, Adv. Math. 181
(2004) 396--416 (with J. D.
Christensen
and W. G. Dwyer).
Working
in an
arbitrary model category, we classify the kinds of maps that have an
obstruction theory in the classical sense. Not surprisingly, some kind
of
principal fibration condition is
necessary. The point
is to shed light on traditional obstruction theory by understanding its
abstraction. [The link leads to the article abstract. From some
computers, the
full text is available above and to the right of the abstract.]
Email:
isaksen at math.wayne.edu