Mathematics abounds with examples of cross-fertilisation between traditionally seperate disciplines. One of the most striking is surely the profound interaction between number theory and representation theory.In its modern incarnation, this interaction has led to notable advances in the arithmetic of elliptic curves (such as the proofs of the Shimura-Taniyama conjecture, the Sato-Tate conjecture, and the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank at most one) and has been instrumental in fleshing out a unifying vision connecting arithmetic objects (Galois representations, and motives) to analytic ones (automorphic forms and representations). The time is appropriate to reflect on the flurry of recent developments at the interface between number theory and representation theory, particularly those related to the Gross-Zagier formula.
The conference should help greatly in introducing graduate students and post-doctoral researchers to these important areas of mathematics, thereby enhancing collective expertise in a branch of mathematical science which is of great importance for a modern information-based economy. It should also help established researchers more fully appreciate the connections between work in different parts of this domain. The conference will hopefully produce a list of interesting open problems in the area which may inspire future research.
This grant supported a conference on the exciting and diverse interactions between two rather different areas of pure mathematics: number theory and representation theory. There were 19 one hour talks given during the conference by. The standard of the talks was very high, with most speakers making a substantial effort to be widely comprehensible. The program was more intense than we had been hoping, but it was hard to keep the number of talks this low. Many celebrated participants did not have a chance to speak. There was also a 90 minute panel discussion of open problems, which should prove a useful guide to graduate students and post-docs. Notes of this are posted at the conference, so they can be more widely useful. There were at least 188 participants at the conference, though we suspect a few more attended without registering. Of these 28 were women (15%); 52 were graduate students (28%); and at least 16 were post-docs (9%). Of the graduate student attendees 17 (33%) were women. The NSF funds were used to provide travel/lodging for 29 grad students (including 8 women), 3 post-docs (including 1 woman), 11 junior faculty (including 1 woman) and one senior faculty, who likely otherwise could not have attended.
