Monday, 6 June 2016

Never Mind the Proof, Why Is The Riemann-Roch Theorem True?

A very long time ago, I began a project to understand the modern theory of algebraic geometry, and specifically the proof of the Riemann-Roch theorem for projective curves. It's finally over. The completed paper, Never Mind the Proof, Why Is The Riemann-Roch Theorem True? is available here.

Why should you read it? Because it will actually explain why the theorem is a) difficult in the first place, and b) true. You won’t drown in endless algebra, rather swim in a sea of geometric intuitions. You will see the Zariski topology being used to provide geometric insight and understand why flatness is at once difficult and yet easy. You will thoroughly understand the difference between a vector, a co-vector and a one-form (a lot of people who write textbooks don’t) and so why a global holomorphic one-form doesn’t give rise to a global holomorphic function. There’s a simple geometric way of thinking about spectrums and sheaves, and an explanation of where twisting sheaves really come from. It’s all about the informal illustrations, arguments and analogies.

It's inspired by Imre Lakatos' championing of informal mathematics, especially in his essay Proofs and Refutations. The aim is to show that informal argument and exposition can lead to greater understanding of abstract ideas and complicated proofs. I used the Riemann-Roch theorem for Riemann surfaces and projective curves because it provides an example of the informal approach in action on a deep but accessible theorem, rather than a toy example.

It has my real name on it and I did think about that. W S Gosset, aka “Student”, was the only serious mathematician to rock a pseudonym (Nicholas Bourbaki might be someone’s real name, oh wait…) and it’s just pretentious for me. If I was someone whose name appears in the Financial Times, maybe a pseudonym might be appropriate, but I’m just another hack in an open-plan office. Nobody I work with or know is ever going to Google “Riemann-Roch” and find this page by chance. Everyone else will be a stranger - but welcome - now and in the future, so my real name is much the same as a pseudonym.

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