Analysis

As a computer scientist by trade, it is worth noting that I never was expected to go into analysis. In particular during undergrad it was spectacular if a CS major (exclusively) had taken a variation of three dimensional calculus. I find it peculiar as I have found that mathematics only becomes fun after you stop focusing on calculations. Returning to pure mathematics during my thesis and studying proof theory was incredibly entertaining. Thus, since my thesis I have been working my way through analysis in the hope of creating a personal intuition for an introduction to topology. What at first was a means to an end, has created a new respect for calculus in me. During calculus one is taught about limits in such restrictive and brief a format that its not common (at least in me) to think past that brief introduction. That is however, a great mistake. It is not obvious to me that before learning these $\epsilon$-$\delta$ definitions that one can quite comprehend a limit at all. In particular it is of interest to me due to the usefulness of Complex-analysis and Topology in computer science theory. Specifically in the realms of complexity and computability which I plan to write on more.

I have studied real analysis with Cummings and Baby Rudin. This gave me a good understanding of proof based analysis and $\epsilon-\Delta$ definitions. Now I am moving my attention to Complex analysis using a combination of Conway: Functions of One complex variable & Lang: Complex analysis & Needham: Visual complex analysis. My main source of information is Conway, using the others as supplemental material and at times a fresh perspective. I chose Conway because I can understand his wording better on average.

I have now written a blog post on various constructions of the real numbers. In particular focusing on Cantor’s and Dedekind’s constructions. I mentioned Weierstrass’s construction only due to its overwhelming difference in approach to either of the previous constructions. You can find the blog post here The Completeness Axiom.