There’s a remarkable new paper out from Shing-Tung Yau, entitled Perspectives on Geometric Analysis. Yau is probably the dominant figure in the field of Geometric Analysis in recent years, and in this paper he gives his personal perspective on the field, including many comments on its recent history, where it is now, and where he thinks it is going.
The paper begins with a dedication to Chern, and some personal history of Yau’s interactions with him. It includes an outline of the distant and recent history of geometric analysis, mainly by giving names of the mathematicians involved. There are 755 references in the reference section, listing pretty much all the papers that Yau sees as important for one reason or another. This has to be some kind of record for number of references in a paper, especially a paper whose main text is only about 50 pages long.
Yau covers an immense amount of ground, commenting on a very wide variety of topics. This is a paper aimed at those who already know quite a bit about the subject, or who are beginning to learn it and would appreciate recommendations of what they should be reading. It includes very little in the way of expository material aimed at the beginner. There is a long section on “Ricci flow” techniques, which are the topic of a lot of current research and that Yau considers to be “the most spectacular development in the last thirty years.” He also has quite a bit to say about “Calabi-Yau” manifolds and their use in physics, commenting that they provide “a good testing ground for analysis, geometry, physics, algebraic geometry, automorphic forms and number theory.”
Another expository paper also appeared on the arXiv last night, but one of a very different nature. It’s by Ravi Vakil, a young algebraic geometer at Stanford, and it is aimed at explaining how Gromov-Witten theory has been used in recent years to study the moduli space of curves. It includes a lot of expository material about the moduli space of curves, and is designed to be understandable by the non-expert.