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.

Currently I am interested in determining the homotopy type of the Tate cohomology spectrum of tmf at the prime 2. I am also interested in applications of the splitting of bo ^ tmf to that of tmf ^ tmf and MO<8>.

Publications

1. 2003f:28011 (with T. Kim and R. Strichartz) Inside the Levy Dragon, American Mathematical Monthly, 109 (2002), no. 8, 689--703.

Preprints

1. Topological Splittings of Spectra Related to tmf (Ph.D. Thesis)
   

   The homotopy groups of $bo \wedge tmf$ 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 $bo \wedge bo$ and subsequently apply similar techniques to construct a map realizing the algebraic splitting of $\pi_* (bo \wedge tmf)$ 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 $tmf$.

2. 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.

Calculations

1. Central power of the $v_1$ self map of the mod-p Moore spectrum.

   In this draft I utilize the splitting of $M \wedge M$ to demonstrate that the p-th power of the $v_1$ self map lies in the center of End(M), so the result only holds for odd primes.

2. 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.

3. 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 I define a inversion spectral sequence utilizing the Yoneda composition product to invert elements of tmf, it is shown that L_{K(1)} X is homotopy equivalent to a wedge of suspensions of KO.

Presentations

1. The 1033th AMS Meeting (Southeastern Section). Murfreesboro, TN. November 3-4, 2007. Special Session: Recent Advances in Algebraic Topology.
2. The 969th AMS Meeting (Central Section). Columbus, OH. September 21-23, 2001. Special Session on Fractals.
3. Northwestern Undergraduate Math Society. Evanston, IL. May 26, 2004.
4. Northwestern Undergraduate Math Society. Evanston, IL. January 20, 2005.
5. Graduate Student Seminar. Evanston, IL. November 7, 2006.
   Title:The center of endomorphism rings of some finite spectra.