Last Wednesday, the conversation in my office veered towards the words we hated the most in math. Not surprisingly, the list included the usual suspects like normal, simple, and regular. It's probably the same reason that these words also make it to the top five of this MathOverflow post. These …

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 …

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 …

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 …