Jane is currently a third year student at the ANU, where she is completing a Bachelor of Philosophy. After spending a semester studying in Hungary, she has been particularly interested in graph theory and combinatorics, but is also keen on gathering ideas and tools from a broad range of areas and being exposed to as many different fields as possible. She looks forward to developing a new-found interest in algebraic topology over the summer, as well as starting an honours project in graph theory in 2018
Spectral sequences in algebraic topology
In algebraic topology, homotopy groups are a generalisation of the fundamental group to higher dimensions, and are of great theoretical importance. Unfortunately, they are very difficult to compute. We consider instead the related but slightly more tractable problem of computing stable homotopy groups. These were basically founded on Freudenthal’s work and rose to significance after further development by Adams through the introduction of his eponymous spectral sequence. In this project, we aim to develop the tools needed to detail the process of computing stable homotopy groups using spectral sequences, and apply this method following Adams’ work to concretely compute the stable homotopy groups for certain spaces and spectra. In particular, this will entail giving constructions of and applying the Serre and Adams spectral sequences.