Scott Bailey
Research Interests

I research in the area of algebraic topology, specifically in the branch called stable homotopy theory. In the past, I have been interested in calculating central powers of $v_n$ self maps of finite p-local spectra. In doing so I have utilized the Adams and Adams-Novikov spectral sequences along with certain vanishing line arguments.

Pengelley demonstrated that H*MO<8> = A \otimes_A(2) M, i.e., that the cohomology of MO<8> is an extended A(2)-module. Currently, I am attempting to understand the A(2)-module structure of M. Furthermore, I am interested determining the homotopy type of tmf ^ tmf.

Publications
Title & Description MathRev Paper
On the spectrum bo^tmf
M. Mahowald, in his work on bo-resolutions, constructed a bo-module splitting of the spectrum bo ^ bo into a wedge of summands related to integral Brown-Gitler spectra. In this paper, a similar splitting of bo ^ tmf is constructed. This splitting is then used to understand the bo_*-algebra structure of bo_* tmf and allows for a description of bo^* tmf.
[doi:10.1016/j.jpaa.2009.06.005]
  to appear, JPAA
Inside the Levy Dragon
(Joint with Kim & Strichartz) We explore the interior of the fractal, and give a classification of its components.
MR JSTOR

Submitted Papers
Title & Description MathRev Paper
On the Tate spectrum of tmf at the prime 2
The Tate spectrum of tmf at the prime 2 is shown to split as a wedge of 8-fold suspensions of bo, proving the conjecture in my thesis.
  arXiv

Preprints & Drafts
Title & Description Stage Paper
Topological splittings of spectra related to tmf
(Ph.D. Thesis) The homotopy groups of bo ^ tmf are shown to be isomorphic to the homotopy groups of a wedge of a suspensions of spectra related to integral Brown-Gitler spectra. We will then restate Mahowald's proof of the topological splitting of bo ^ bo and subsequently apply similar techniques to construct a map realizing the algebraic splitting of \pi_* (bo ^ tmf) as a topological splitting on the level of spectra. As an application, we use our results to provide ample groundwork demonstrating the splitting of the Tate spectrum of tmf.
    
Graph Jacobians and Mackey Functors
This paper is based on my work at the Louisiana State University REU over the summer of 2001. It explores the so-called Jacobian of a graph, and shows that it forms a cohomological Mackey functor over any finite graph, X. As a result from Boltje and Perlis, for all p not dividing the genus invariant, \nu, of the Gassmann triple (G,H_1,H_2) the p-Sylow subgroups of the Jacobians of X/H_1 and X/H_2 are isomorphic. The Jacobian also satisfies Brauer relations.
draft Request

Calculations
Title & Description    
Central power of the v_1 self map of the mod-p Moore spectrum
This draft implements the splitting of M ^ M to demonstrate the p-th power of the v_1 self map lies in the center of End(M) -- the result only holds for odd primes.
    
Central power of the v_1 self map of the mod-p Moore spectrum
In this draft, I use a vanishing line argument in the Adams-Novikov spectral sequence converging to End(End(M)) to show the p-th power of the v_1 self map lies in the center of End(M) for all p.
   
The KO-splitting of a false wedge of bo's
In this draft, I show that if the cohomology of a tmf-module spectrum X is isomorphic to a wedge of (0 mod 8) suspensions of bo, then after defining an inversion spectral sequence utilizing the Yoneda composition product to invert elements of tmf, it is shown that the K(1)-localization of X is homotopy equivalent to a wedge of suspensions of KO.
    

Presentations
Title Date File
The connective real K-theory of tmf.
Penn-State Altoona Geometry/Topology Seminar. 		December 2008  
On the Tate spectrum of tmf at the prime 2.
University of Rochester Topology Seminar 		November 2008  
On the Tate spectrum of tmf at the prime 2.
1044th AMS Meeting (Southeast Section). Huntsville, AL. Special Session on Homotopy Theory and Algebraic Topology, III.

October 2008

Slides
On the Tate spectrum of tmf at the prime 2.
1043rd AMS Meeting (Central Section). Kalamazoo, MI. Special Session on Homotopy Theory, II.		 October 2008 Slides
The splitting of bo ^ tmf.
1033rd AMS Meeting (Southeastern Section). Murfreesboro, TN. Special Session on Recent Advances in Algebraic Topology.

November 2007

 Slides
Inside the Levy Dragon fractal.
969th AMS Meeting (Central Section). Columbus, OH. Special Session on Fractals.		 September 2001  

Research Links

Fun Links

  • Klein Four
    Mathematics a cappella in your face!
  • Ph.D. Comics
    "Piled Higher & Deeper comic strip.
  • Funny Math
    As a calculus instructor, I have seen some funny answers on exams. I guess so have other graders! Here are some common fun answers.
  • The Pejorative Calculus
    Shows that Alexander the Great didn't exist, and had an infinite number of limbs! In other words, one should use care with inductive arguments.
  • Math Genealogy
    My branch on the mathematics genealogy tree.
Research