The natural way to talk about two spaces having the same shape is by asking if they are homeomorphic. Homotopy theory provides a weaker version of this equivalence. Since homotopy theory plays such a vital role in algebraic topology, a natural question arises: Is there a general setting in which we can define homotopy theory? The answer to this is given by model categories as introduced by Quillen.
In this project we aim to define and understand what it means for a category to be a model category. This allows for the construction of the homotopy category – the tool for defining homotopy theory on the model category. We will then use this to examine the homotopy theory of various model categories, such as the category of topological spaces. With this
framework we can proceed to define spectra which will allow us to introduce the notion of stable homotopy theory through the stable homotopy category. Time permitting, it would also be insightful to explore recent developments which define model categories through the use of weak factorization systems and also explore G-equivariant spectra with the hope of introducing stable equivariant homotopy theory.
The Australian National University
James is about to enter honours in mathematics at the Australian National University, after completing his Bachelor of Science (mathematics major) in 2019. In his undergraduate studies, James explored areas of physics, computer science, economics, nance, and statistics alongside his mathematics studies. He was working towards doing honours in stochastic analysis before being introduced to the exciting world of Topological Data Analysis. This naturally pushed him to undertake an introductory course in algebraic topology and is not looking back.
During the summer of 2018/19 James completed an Australian National University summer research scholarship with Dr Benjamin Andrews. Here he examined numerical solvers for various curvature equations, such as curve shortening flow.
James is excited to deepen his interest in algebraic topology over the 2019/20 summer and explore some areas which will not be too relevant to his honours project in topological data analysis in 2020.