Abstract
We construct new versions of orthogonal calculus, a unitary version which considers complex vector spaces, and a calculus with reality, an extension of the unitary calculus which takes into account the complex conjugation action on the complex vector spaces. These calculi produce Taylor towers approximating a functor, and we show through a zigzag of Quillen equivalences (in both versions of the calculi) that the layers of these towers are classified by spectra with an action of either U(n) in the unitary case, or the semidirect product of C_2 with U(n) in the calculus with reality, where C_2 acts on U(n) by termwise complex conjugation of the matrices.From the complexificationrealification adjunction between real and complex vector spaces we construct functors between the orthogonal and unitary calculi, allowing for movement between these two versions of calculus, and direct comparisons of the Taylor towers. We introduce a class of functors, which we call ``weakly polynomial'' and we show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.
Date of Award  Dec 2020 

Original language  English 
Awarding Institution 

Sponsors  Northern Ireland Department for the Economy 
Supervisor  Ivan Todorov (Supervisor), David Barnes (Supervisor) & Lisa McFetridge (Supervisor) 