All the singularities*

Karen Smith.

Karen Smith (a little blurry).

Yesterday, I had the great pleasure of attending the AWM-AMS Noether Lecture, delivered by Karen Smith. Smith took us on a tour of modern algebraic geometry, and showed us how many contributions Emmy Noether made to this field of mathematics.

Smith introduced the audience to algebraic varieties (essentially sets of common zeros of polynomials), and the fact that they are everywhere in math. But her main goal was to show us the problems algebraic geometers are interested in, and in particular the question of deciding whether a variety is smooth, and if not how bad the singularities are. Her explanations of resolutions of singularities were great (and I appreciated the many pictures), and she has a level of energy and excitement that is really contagious.

The main technique for analyzing the “badness” of singularities is, instead of studying the variety itself, to study the ring of functions on the variety and reduce this to prime characteristic. This method of reducing this geometric problem to an algebraic one really goes back to Noether and the first isomorphism theorem (which Smith attributed to Noether even though the literature does not). Smith got a chuckle from the audience when she mentioned the “freshman’s dream”, in which reducing to characteristic p really allows you to say that (f+g)p=fp+gp. The upshot of this is that the p-th power map (whose fancy name is the Frobenius map) is actually a ring homomorphism (behaves nicely with addition and multiplication). By a Theorem of Kunz, a variety is smooth if the ring of functions decomposes in a nice way according to the Frobenius, so we really have reduced the problem of finding singularities to a simple algebraic problem! Finally, she mentioned some generalizations and other results by her and her collaborators.

As a fan of algebraic geometry, of course I liked this talk, but I think she did a great job for the general audience too. The link between algebra and geometry was clear, and Noether’s influence was adequately honored throughout. Really great talk indeed.

*To be sung to the tune of “Single Ladies”, by Beyonce.

All the singularities, all the singularities… If you liked it then you should have put a ring (of functions) on it!

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