My research is oriented by chromatic homotopy theory. Roughly speaking, here is the story. As a homotopy theorist, the one thing I really want to understand more than anything else are the homotopy groups of spheres. There are some reductions/refinements one can make: 

• Freudenthal tells us that these groups stabilize, so let's focus on computing the stable homotopy groups of spheres; 

• Serre tells us that the non-negative stable stems are finite abelian groups, so let's work one prime at a time;

• Morava (or maybe God) tells us that at a fixed prime, the stable stems are organized into vn-periodic families, so let's focus on computing one family at a time.

There is a lot of fun one can have now in vn-periodic homotopy theory. For example, there are two localization functors which allow one to access these periodic families, one more algebraic and one more topological, and the telescope conjecture equates these two localizations. Famously, the telescope is false at all primes and all heights 2 and above. However, at the height 1, and say now at the prime 2, Mahowald (or maybe God) used like a zillion spectral sequences and computed the v1-periodic stable stems entirely and showed that the two aforementioned localizations actually agree at height 1, proving the height 1 telescope conjecture. 

This is the motivation behind most of my current research: to try to understand periodicity in other contexts by means of a zillion spectral sequences, and more generally to understand the telescope conjecture and other cool chromatic things in other contexts.

Most of my research has been housed in motivic homotopy theory, a variant of stable homotopy theory which incorporates algebraic-geometry over a fixed base scheme. One can ask for a hands on approach to v1-periodicity in motivic homotopy theory, and my current research studies this at all primes over the complex numbers, the real numbers, and finite fields. However, the motivic stable stems are much more interesting than the classical stable stems. I am interested in studying the following:

• determining exotic periodic elements in the motivic stable stems and, more generally, better understanding exotic periodicity;

• understanding the correct motivic analogue of the telescope conjecture, one which captures classical and exotic periodicity;

• moving towards a more integral class of base scheme.

Most of my current work has used the Adams spectral sequence as a guiding principle (although recently, I have had an eye towards the slice spectral sequence...). I am particularly fond of comparing and contrasting these results as the base scheme varies and with classical stable homotopy theory, and I am eager to connect my work over the real numbers with C2-equivariant homotopy theory.

We construct spectrum level splittings of BPGL<1> ^ BPGL<1> and BPGL<0> ^ BPGL<0> in terms of motivic Adams covers. These calculations are performed at all primes p and over R, C, and all finite fields Fq where char(Fq) ≠ p. This lifts classical results in spite of the current lack of motivic Brown--Gitler spectra, and opens the door to the study of the BPGL<1>-motivic Adams spectral sequence. 

We compute the ring of cooperations for the very effective Hermitian K-theory spectrum kq over finite fields Fq of where char(Fq)≠2. To do this, we use the motivic Adams spectral sequence and show that the differentials are determined by the motivic cohomology of Fq. We extend our results to compute the E1-page of the kq-resolution.

We compute the ring of cooperations for the the very effective Hermitian K-theory spectrum kq over the real numbers. This result involves a delicate series of spectral sequences and is a first step towards the kq-resolution. Additionally, we prove a splitting result for the very effective symplectic K-theory spectrum ksp using a motivic integral Brown-Gitler spectrum.

We examine Li’s double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double determinantal varieties are smooth. We use this framework to give a straighforward formula for their dimension. Finally, we use the smallest nontrivial toric double determinantal variety to provide some empirical evidence concerning an open problem in local algebra.