Construction of Chern classes
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 …
read moreA week at Berlin Mathematical School
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 …
read moreSetting up GitLab to automatically generate PDFs from committed LaTeX files
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 …
read moreCohomology as a measure of local to global failure
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 …
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