Research interests of Don Davis

I work in homotopy theory, which is a branch of algebraic topology. Most of my work from 1988 to 2003 was devoted to computing v1-periodic homotopy groups of compact simple Lie groups. v1-periodic homotopy groups are a localization of the actual homotopy groups, which describe roughly the portion of the p-local homotopy groups detected by K-theory. Each v1-periodic homotopy group is a direct summand of some actual homotopy group.

The compact simple Lie groups G consist of the classical groups and five exceptional Lie groups. In 2003, I completed the determination of the p-local v1-periodic homotopy groups for all compact simple Lie groups and all primes p. This project had been suggested to me by Mimura in 1989. One important advance was a 50-page paper, published in Memoirs of the AMS in 2002 detailing the calculation of the v1-periodic homotopy groups of (E8,5) and (E8,3) by a new method pioneered by Bousfield. The method here is representation theory, while ealier papers utilized the Unstable Novikov Spectral Sequence. One consequence of my work is nice new exponent results for SU(n). In 2004, Bendersky and I published a 90-page paper in the Memoirs of the AMS computing the v1-periodic homotopy groups of (SO(n),2). A paper with Bendersky and Mahowald published in 2006 applies this computation to obtain new results about geometric dimension of vector bundles. In 2003, I published a paper on the case (E7,2) and (E8,2), which completed Mimura's challenge. This depended on new work of Bousfield, which was in part motivated by my needs. In 2007, I computed v1-periodic homotopy groups of p-compact groups, which are the homotopy-theoretic generalization of compact Lie groups.

Throughout my career, but especially between 1970 and 1984, I have done much work on immersions of projective spaces. With Mahowald, I developed many methods for establishing immersions and nonimmersions. I proved many specific results and one celebrated general result, which was published in the Annals of Mathematics. In a paper with Bruner and Mahowald published in 2002, I applied the new spectrum tmf to obtain new nonimmersion results. I maintain a table of all known results on these questions for real projective spaces, and in a paper published in 2008, I explained the current status of the problem for complex projective spaces.

Other areas in which I have done much work are the Steenrod algebra, cohomology of the Steenrod algebra, and stable homotopy groups of spheres. In an early paper, I proved a formula for the antiautomorphism of the Steenrod algebra which has been frequently applied and generalized. In another early paper with Anderson, I proved a vanishing theorem for Ext groups over the Steenrod algebra, which has been extremely important to the subsequent development of the subject. My Topology paper with Mahowald on the Image of the J homomorphism in the stable homotopy groups of spheres is considered by many to be the definitive work on that topic, which served as a focal point for much of algebraic topology in the 1960s.

Since about 2007, much of my work has been in combinatorial number theory, especially focused on topics arising from studying exponents of primes in Stirling numbers of the second kind, as these are closely related to homotoy groups of SU(n). I have become interested in the way that p-adic integers arise in this study. Since about 2012, I have been interested in topological complexity, which has possible applications to robotics. This is a measure of how many rules are required to move between any two points of a topological space. The application is when the space is all configurations of a robot.