Research Projects

We consider a twisted version of topological Hochschild homology for \( C_n \)-ring spectra. For computing counit-unit composites, we use an original choice of coset representatives \( \{e, \gamma^n, \gamma^{2n}, \dots, \gamma^{(p-1)n}\} \) for \( C_{p^kn}/C_{p^{k-1}n} \) which yields the identity after restriction, mirroring the triangle identity, making this the "categorical" choice.

For \( H \subset G \), we prove that there is an identification between the twisted cyclic bar construction for \( G \) of a norm from \( H \) and the subdivided twisted cyclic bar construction for \( H \): \( i_H^G B_\bullet^{cy,G}(N_H^G R) = sd_m B_\bullet^{cy,H}(R) \). Finally, we propose a definition of \( C_n \)-twisted \( TC_{C_n} \) and prove that this definition can be identified with a homotopy equaliser.

We give a consolidated exposition of the Hopf invariant one problem. At the prime 2 we mention Adams' secondary operation argument, showing that a map \( f \colon S^{2n-1}\to S^n\) with Hopf invariant \(\pm 1\) exists only for \( n=1,2,4,8\). For an odd prime \( p\) we follow the Adams-Atiyah translation into \(K\)-theory and prove that the mod-\( p \) Hopf invariant vanishes as soon as \( n>1 \). The unique surviving cases \( n=1 \) is realised by an explicit map \( S^{2p}\to S^3 \) that we construct in detail.

We begin by introducing Poisson integration, as well as several related constructions needed to define the Levy integral. We review the Levy-Ito decomposition and show that it further decomposes into a Brownian, a compound Poisson process and a compensated Poisson process. We conclude with the construction of Levy-type stochastic integrals and an application of it to options pricing.

In this work we define many basics from homotopy theory, proving key results such as Hurewicz's Theorem. We work through some more difficult cohomology and homotopy calculations. We finish with bringing differential forms onto the scene to study rational homotopy theory.

In this work we explain the methods of Hill, Hopkins, and Ravenel in their solution of the Kervaire Invariant One problem. We make the simplification from \(C_8\) to \(C_2 \)-equivariance, describe the computational techniques involved, and talk about what these findings mean. In particular, we use the methods to compute the homotopy groups of \(KR\)-theory.