Pierre Deligne wins Abel Prize

Viscount Pierre Deligne of Belgium won the $1 million Abel Prize, a major award given to mathematicians, today. It's a well-deserved honor, I think.

He is perhaps most famous for the proof of several Weil conjectures, (now) theorems about various Riemann zeta functions and related facts generalized to the case of finite fields. This is enough to see that Deligne is primarily an algebraic geometer, in the tradition of Alexander Grothendieck.

However, the depth of a mathematician may arguably be measured by the extent to which his or her insights start to influence cutting-edge theoretical physics in the present and, if possible, in the future. For decades, this criterion was equivalent to the role that the mathematical insights play in string theory. Deligne gets an A in this subject, too.




First of all, he has made quite some work on the Hodge theory which is concerned with differential forms on manifolds and its role in string theory is self-evident. However, this is a sort of trivially stringy part of Deligne's research. There is a more abstract example that shows that some research by Deligne is very close to some principles that certain string theorists may consider central yet heavily underappreciated.




What am I talking about?

Well, I mean the algebraic stacks – Deligne-Mumford stacks and their generalizations, Artin stacks. They were originally introduced to describe the fine moduli space of genus \(g\) curves (Riemann surfaces) and as algebraic counterparts of orbifolds. The links to string theory world sheets and stringy orbifolds is obvious.

However, there's much more stuff beyond these structures that isn't obvious but may be important for string theory. Imagine that this portion of the blog entry contains an insightful lecture on stacks, sheaves, derived categories, and their ability to classify everything that may be included into the most generalized notion of D-branes. Apologies: I will omit it because it wouldn't satisfy the quality standards I demand here.

Shamefully enough, Deligne's textbook of field theory and string theory for mathematicians has never been mentioned on this blog. So at least, I am trying to undo the grievance right now. To a large extent, it is a rather standard physics textbook. Nevertheless, it may be interesting for a physicist to listen to the "accent" that a top mathematician displays when he talks about similar issues.