Craig Westerland

Twisted Morava K-theory and E-theory

For a class H ∈ H n+2 (X; Z), we define twisted Morava K-theory K(n) ∗ (X;H )a t the prime 2, as well as an integral analogue. We explore properties of this twisted cohomology theory, studying a twisted Atiyah–Hirzebruch spectral sequence, a universal coefficient theorem (in the spirit of Khorami). We extend the construction to define twisted Morava E-theory, and provide applications to string theory and M-theory.

A higher chromatic analogue of the image of J

We prove a higher chromatic analogue of Snaiths theorem which identifies the K-theory spectrum as the localisation of the suspension spectrum of CP^\infty away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP^\infty = K(Z,2). Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K(n)-local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross-Hopkins dual of the sphere. Our main technical tool is a K(n)-local notion generalising complex orientation to higher Eilenberg-MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.

The symplectic Verlinde algebras and string K-theory

We construct string topology operations in twisted K-theory. We study the examples given by symplectic Grassmannians, computing K τ ∗(LHP � ) in detail. Via the work of Freed–Hopkins– Teleman, these computations are related to completions of the Verlinde algebras of Sp(n). We compute these completions, and other relevant information about the Verlinde algebras. We also identify the completions with the twisted K-theory of the Gruher–Salvatore pro-spectra. Further comments on the field-theoretic nature of these constructions are made.

Operads of moduli spaces of points in C d

We compute the structure of the homology of an operad built from the spaces THd,n of configurations of points in Cd, modulo translation and homothety. We find that it is a mild generalization of Getzler’s gravity operad, which occurs in dimension d = 1.

Operads of moduli spaces of points in Cd

We compute the structure of the homology of an operad built from the spaces THd,n of configurations of points in Cd, modulo translation and homothety. We find that it is a mild generalization of Getzler’s gravity operad, which occurs in dimension d = 1.