My curriculum vita (i.e., my resume).
Wayne State University hosts an active topology seminar. Older seminar schedules are archived here.
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 classical 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. This research is supported by a grant from the National Science Foundation.
One promising current direction involves the motivic version of the Adams spectral sequence. Over C, this spectral sequence converges to the 2-completed motivic stable homotopy groups. Unfortunately, convergence is uncertain over other fields. In particular, we are interested in working over R. Nevertheless, the spectral sequence does converge to something interesting. Computer calculations play a major role in this project.
Another promising direction concerns geometric constructions and composition methods, i.e., Toda brackets. Using this approach, we can produce many interesting generators and relations in the motivic stable homotopy groups.
I 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.
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.
Below are some charts that display the results of recent calculations. Most of these charts make use of full color. I highly recommend the color versions because the colors are useful for making sense of the information.
Motivic Adams spectral sequence over C
Motivic Adams-Novikov spectral sequence over C
Speculative motivic modular forms over C
Cohomology of A(1) over C, and other related computations
Motivic connective K-theories and the cohomology of A(1), 2010, preprint (with A. Shkembi).
In classical homotopy theory, there is a connection between connective real K-theory and the subalgebra A(1) of the Steenrod algebra. Working over C, this article establishes similar results for motivic homotopy theory. The paper contains some explicit computations.
The motivic Adams spectral sequence, Geometry and Topology, to appear (with D. Dugger).
Working over C, this paper shows how to compute with the motivic version of the Adams spectral sequence. The paper gives explicit computations through the 34-stem. The methods can certainly be extended to higher stems with more effort. The May spectral sequence is a key computational tool.
The cohomology of motivic A(2), Homology Homotopy and Applications 11 (2009) 251--274.
Working over C, this paper computes the cohomology of motivic A(2). The main tools are the May spectral sequence and Massey products. The paper also provides some information about the structure of a speculative "motivic modular forms" spectrum.
Motivic cell structures, Algebraic and Geometric Topology 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.
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.]
Etale realization on the A1-homotopy theory of schemes, Advances in Mathematics 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, Mathematische Zeitschrift 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, Contemporary Mathematics 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, Mathematical Proceedings of the Cambridge Philosophical Society 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.]
The Hopf condition for bilinear forms over arbitrary fields, Annals of Mathematics 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, Documenta Mathematica 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, Mathematical Proceedings of the Cambridge Philosophical Society 145 (2008) 1--25 (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.
Eigentheory of Cayley-Dickson algebras, Forum Mathematicum 21 (2009) 833--851 (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, Boletin de la Sociedad Matematica Mexicana 13 (2007) 269--292 (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, Communications in Algebra 36 (2008) 632--664 (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.
Model structures on pro-categories, Homology Homotopy and Applications 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 and Applications 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, Mathematische Zeitschrift 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, Algebraic and Geometric Topology 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 (Isle of Skye, 2001), 179--198, Progress in Mathematics 215, Birkhauser, 2004.
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, Fundamenta Mathematicae 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, Transactions of the American Mathematical Society 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, Advances in Mathematics 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