Equivariant Stable Homotopy Theory
Together with Igor Kriz, I have developed an Isotropy Separation Spectral Sequence (ISSS) that permits the computation of the coefficients of equivariant spectra. For my Ph.D. thesis, I described the coefficients of the equivariant homotopical complex cobordism ring MU_G as the limit of a diagram of fixed point sets with respect to the various subgroups of G. This is a partial equivariant analog to Milnor's computation of MU. I give the corresponding description of the equivariant formal group law of MU_G, insofar as it gives a complex stable, complex oriented cohomology theory. I also note the isomorphism between shift desuspension and loop space of suspension spectra, filling a small gap in the foundations of equivariant stable homotopy theory. My thesis has led to three papers, two of them joint with Kriz, on the topics above.
Presently, I am studying a conjecture of Greenlees that the equivariant formal group law of MU_G is algebraically universal for G-equivariant formal group laws. This would be the equivariant analog of Quillen's Theorem.
In the near future, I hope to use the ISSS to compute the coefficients of other equivariant spectra, such as equivariant Eilenberg-MacLane spectra. These have only been computed in the case where G is a cyclic group of prime power order, in that case by Po Hu and Kriz, but I hope to extend this to the finite abelian case. I would also be interested to compute the coefficients of certain multiplicative norms of real cobordism, which were essential to the work of Mike Hill, Mike Hopkins, and Doug Ravenel on the Arf-Kervaire invariant one problem.
Symbolic Dynamical Systems
As a prolonged side interest, I have been studying the Hausdorff dimension of multiplicative translates of certain p-adic Cantor sets with Jeff Lagarias. This was motivated by a conjecture of Erdős, who proposed that the ternary expansion of a power of 2 will never omit the digit 2 for any power at least 9. Our work has thus far produced three papers. The first introduces some necessary foundations, including the new symbolic dynamics concept of a path set. A path set is a set of one-sided infinite digit sequences whose digits can be described as the edge labels of infinite paths in a directed labeled graph initiating from a fixed initial vertex. These are similar to sofic shifts, but are not shift invariant. Nonetheless, many standard symbolic dynamical results apply to path sets. Our second paper introduces the notion of a p-adic path set fractal, which is a path set which can be realized as a fractal set inside the p-adic integers. Our third paper studies the Hausdorff dimension of certain 3-adic path sets relevant to a certain generalization of the Erdős conjecture.
My current work on this project, in collaboration both with Lagarias and with the student Artem Bolshakov, is two-fold: first, to compute finally the Hausdorff dimension of certain special 3-adic Cantor sets; second, to extend this work beyond the 3-adic case to other primes.
In the near future, I hope to extend this work as far as possible to the p-adic and g-adic cases, where g is not necessarily prime, and to implement computer programs to extend the experimental base of this research.