Homotopy theory, functor calculus, and related fields.
My personal webpage with more information may be found here: https://sites.google.com/view/nialltaggartmath
My research interests lie in algebraic topology, where one uses tools from algebra to study properties (invariants) of topological spaces. I am particularly interested in stable homotopy theory with an equivariant flavour, and the applications of such to functor calculus.
Functor calculus may be thought of as a version of Taylor's Theorem from differential calculus applied to functors satisfying particular properties. Current projects include constructing a variant of functor calculus from functors from the category of complex vector spaces to the category of topological spaces and analysing the homotopy theory of such a calculus. I am also interested in how the various versions of functor calculi relate to each other.
I am also interested in more general questions. For example, the Blakers-Massey Theorem for cubes is a well-established result relating the connectivity of homotopy limits and colimits. Together with collaborators, I am trying to extend this result to diagrams indexed on the poset of subspaces of the n-plane. This project is interesting in its own right but should also give valuable insight into analytic functors in orthogonal calculus, a rather unexplored class of functors, despite their important role in Goodwillie calculus.
I am currently a teaching assistant on PMA3017 Metric and Normed Spaces.
In previous years I have been a teaching assistant on PMA1020 Numbers, Vector and Matrices, PMA2002 Analysis, and PMA2008 Group Theory.