This is a series of talks I gave in late 2020 about my Honours project.
The first talk was a 50-minute lecture given to the VUW Logic Seminar, on Monday 14th September 2020:
Abstract: Cousin’s lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. Using reverse mathematics, we analyse the strength of Cousin’s lemma for various classes of functions. Joint work with Downey and Greenberg.
The second was a 20-minute talk given to the VUW Honours seminar and the staff of VUW, on Thursday 15th October 2020. This talk was part of the research project requirement of my Honours degree. The slides are below.
Abstract: Cousin’s lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We attempt to find the optimal proof of Cousin’s lemma for various functions, using the toolkit of reverse mathematics. Joint work with Downey and Greenberg.
honstalk_v4_handoutThe third was a 5-minute conference talk, given as part of the NZ Mathematics and Statistics Postgraduate Conference 2020. The conference was held on Thursday 26th November 2020, over Zoom.
Title: The best proof of Cousin’s lemma
Abstract: Kurzweil and Henstock’s gauge integral generalises the usual Riemann and Lebesgue integrals, allowing a wider class of functions to be integrated. Cousin’s lemma is a compactness principle that arises naturally when studying the gauge integral. Our aim is to find the “best” proof of Cousin’s lemma, using the toolkit provided by reverse mathematics. We have done this for continuous functions; however, the problem remains open for functions of Baire class n \geq 1. Joint work with Downey and Greenberg.
Here is the recording: