I'm almost a week into the algebraic geometry workshop now, and I've learnt a lot. I've learnt a few things about varieties, and also a bit of commutative algebra, but the most important takeaway for me from the first week was the sheaf theoretic way of looking at smooth manifolds …

I'm getting lazy. I thought I would be posting more often once the summer holidays started but May came and went with nary a post. In my defence, I was fairly busy, dealing with the usual bureaucratic nonsense that comes with leaving your institution for good and moving to another …

Characteristic classes

Given a manifold \(M\), one way to study vector bundles over \(M\) is to use the theory of characteristic classes. A characteristic class is a way of assigning to each vector bundle over \(M\) an element of the cohomology ring \(H^{\ast}(M, G)\). This assignment is not …

I spent the last week (18th to 24th February) at Berlin, courtesy Berlin Mathematical School, who invited me over for the BMS Days (where I had an interview for a PhD position), as well as the BMS Student Conference which immediately followed the BMS Days. I heard a lot of …

I had been meaning to get started with GitLab's continuous integration to generate PDFs of my assignments and notes, rather then generating the PDFs offline and committing them to the repository as well, but I always kept delaying the migration because of the lack of sufficient documentation on the matter …

Motivation for cohomology

In most introductory algebraic topology courses, cohomology is rather poorly motivated. It's most commonly seen form in an algebraic topology course is singular cohomology, which arises as a the homology of the dual of the singular chain complex, but that doesn't really tell you why it's of …