Eigentheory of Cayley-Dickson algebras (2009)
Biss, Daniel K., Christensen, J. Daniel, Dugger, Daniel, Isaksen, Daniel C.
We show how eigentheory clarifies many algebraic properties of Cayley-Dickson algebras. These notes are intended as background material for those who are studying this eigentheory more closely.
The cohomology of motivic A(2) (2009)
Working over an algebraically closed field of characteristic zero, we compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq^1, Sq^2, and Sq^4. The...
ON K∗-ULTRAHOMOGENEOUS GRAPHS (2009)
Daniel C. Isaksen, Chris Jankowski, Stephanie Proctor
Abstract. Let C be any class of graphs. A graph G is C-ultrahomogeneous if every isomorphism between induced subgraphs belonging to C extends to an automorphism of G. We study graphs that are...
Stephen G. Hartke, Daniel C. Isaksen, Philip Matchett Wood
address the question: How can graduate student mentors improve Research Experiences for Undergraduates (REU) programs? A typical graduate student mentor is in the early stages of graduate study and...
The motivic Adams spectral sequence (2009)
Dugger, Daniel, Isaksen, Daniel C.
We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field. We discuss several features of the associated Adams spectral sequence, including the basic...
Homology, Homotopy and Applications, vol. 9(1), 2007, pp.399–438 t-MODEL STRUCTURES (2008)
Halvard Fausk, Daniel C. Isaksen
For every stable model category M with a certain extra structure, we produce an associated model structure on the pro-category pro-M and a spectral sequence, analogous to the Atiyah-Hirzebruch...
acknowledged as a valuable part of the total mathematics education and research community. In order to secure long-term funding for such projects and to increase even further the support for such...
Halvard Fausk, Daniel C. Isaksen
We introduce a notion of a filtered model structure and use this notion to produce various model structures on procategories. We give several examples, including a homotopy theory for G-spaces, where...
We compare Friedlander’s definition of the e ´ tale topological type for simplicial schemes to another definition involving realizations of pro-simplicial sets. This can be expressed as a notion...
ON K∗-ULTRAHOMOGENEOUS GRAPHS (2008)
Daniel C. Isaksen, Chris Jankowski, Stephanie Proctor
Abstract. Let C be any class of graphs. A graph G is C-ultrahomogeneous if every isomorphism between induced subgraphs belonging to C extends to an automorphism of G. We study graphs that are...
Large annihilators in Cayley-Dickson algebras II (2007)
Biss, Daniel K., Christensen, J. Daniel, Dugger, Daniel, Isaksen, Daniel C.
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.
A Cohomological Viewpoint on Elementary School Arithmetic (2007)
Daniel C. Isaksen, In Part, Daniel C. Isaksen
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Etale homotopy and sums-of-squares formulas (2006)
Dugger, Daniel, Isaksen, Daniel C.
This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over...
DOI 10.1007/s10977-006-7113-z Flasque Model Structures for Simplicial (2006)
Abstract. It is well known that there are two useful families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes,...
Large annihilators in Cayley-Dickson algebras (2005)
Biss, Daniel K., Dugger, Daniel, Isaksen, Daniel C.
Cayley-Dickson algebras are an infinite sequence of non-associative algebras starting with the reals, complexes, quaternions, and octonions. We study the zero-divisors in the higher Cayley-Dickson...
Fausk, Halvard, Isaksen, Daniel C.
For every stable model category $\mathcal{M}$ with a certain extra structure, we produce an associated model structure on the pro-category pro-$\mathcal{M}$ and a spectral sequence, analogous to the...
Model structures on pro-categories (2005)
Fausk, Halvard, Isaksen, Daniel C.
We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several...
C.: Motivic cell structures (2005)
Daniel Dugger, Daniel C. Isaksen
Abstract An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of...
Documenta Math. 357 Algebraic K-Theory and Sums-of-Squares Formulas (2005)
Daniel Dugger, Daniel C. Isaksen, Communicated Stefan Schwede
Abstract. We prove a result about the existence of certain ‘sums-ofsquares’ formulas over a field F. A classical theorem uses topological K-theory to show that if such a formula exists over R,...
Generalized cohomology of pro-spectra (2004)
We present a closed model structure for the category of pro-spectra in which the weak equivalences are detected by stable homotopy pro-groups. With some bounded-below assumptions, weak equivalences...
Algebraic K-theory and sums-of-squares formulas (2004)
Dugger, Daniel, Isaksen, Daniel C.
We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to obtain obstruction conditions when the...
Completions of pro-spaces (2004)
For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak...
Duality and Pro-Spectra (2004)
Christensen, J. Daniel, Isaksen, Daniel C.
Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a...
Flasque model structures for simplicial presheaves (2004)
By now it is well known that there are two useful (objectwise or local) families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For...
Duality and Pro-Spectra (2004)
J. Daniel Christensen, Daniel C. Isaksen
Abstract Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically...
GENERALIZED COHOMOLOGY OF PRO-SPECTRA (2004)
Abstract. We present a closed model structure for the category of pro-spectra in which the weak equivalences are detected by stable homotopy pro-groups. With some bounded-below assumptions, weak...
DOI: 10.1007/s00209-003-0607-y Topological hypercovers and A 1-realizations (2004)
Daniel Dugger, Daniel C. Isaksen
Abstract. We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors...
Motivic cell structures (2003)
Dugger, Daniel, Isaksen, Daniel C.
An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We...
The Hopf condition for bilinear forms over arbitrary fields (2003)
Dugger, Daniel, Isaksen, Daniel C.
We settle an old question about the existence of certain "sums-of-squares" formulas over a field F (which are the simplest examples of composition formulas for quadratic forms). A classical theorem...
Weak equivalences of simplicial presheaves (2002)
Dugger, Daniel, Isaksen, Daniel C.
The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete...
Hypercovers and simplicial presheaves (2002)
Dugger, Daniel, Hollander, Sharon, Isaksen, Daniel C.
We prove that Jardine's model category of simplicial presheaves can be obtained by localizing the `discrete' version at the collection of all hypercovers. One consequence is that the fibrant objects...
Daniel Dugger, Sharon Hollander, Daniel C. Isaksen
We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by...
Hypercovers in topology (2001)
Dugger, Daniel, Isaksen, Daniel C.
We show that if U is a hypercover of a topological space X then the natural map from hocolim U to X is a weak equivalence. This fact is used to construct topological realization functors for the...
Obstruction Theory in Model Categories (2001)
Christensen, J. Daniel, Dwyer, William G., Isaksen, Daniel C.
Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a...
Strict model structures for pro-categories (2001)
We show that C if is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. The strict model structure...
Etale realization on the A^1-homotopy theory of schemes (2001)
We compare Friedlander's definition of the etale topological type for simplicial schemes to another definition involving realizations of pro-simplicial sets. This can be expressed as a notion of...
A model structure on the category of pro-simplicial sets (2001)
We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is...
Calculating limits and colimits in pro-categories (2001)
We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are...
A model structure on the category of pro-simplicial sets / (1999)
Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999.
A model structure on the category of pro-simplicial sets / (1999)
Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999.
Graphs That Are Randomly Traceable from a Vertex (1993)
A graph G is randomly traceable from one of its vertices v if every path in G starting at v can be extended to a hamiltonian path of G that starts at v. A complete classification of these graphs will...