Report for the academic year - Libraries - Institute for Advanced Study

Institute for advanced study
[ Bombieri gave a series of four lectures entitled "Some elementary remarks on the
Guinand'Weil explicit formula." His approach to the Riemann Hypothesis is to give
lower hounds for the number of zeros of the Zeta function oft of Riemann's line, while the
usual i methods il analytic number theory put upper bounds on the number of zeros off the
line.
On the research front, Shahidi and Kim, experts on the Langlands-Shahidi method, felic-
itously combined for the special year with Cogdell and Piatetski-Shapiro, experts on con-
verse theorems, proved some spectacular new cases of functoriality, with applications to
,h
the Ramanujan conjectures. Cogdell, Piatctski-Shapiro, and Sarnak solved Hilbert's 1 l
problem over a number field.
Vladimir Voevodsky, who is at the Institute on a multi-year appointment, ran a special
year entitled "A' homotopy theory." In the beautiful subject, a theory is developed in
algebraic geometry that mimics usual homotopy theory from topology, with the complex
line replacing the unit interval, except that there are two different kinds of homotopy cir-
cles: the complex line minus a point and the complex line with two points identified.
This program will continue in the year 2000-2001. The program will present the mate-
rial needed to understand Voevodsky's celebrated proof of the Milnor Conjecture.
The A' homotopy theory program consisted of two courses and a weekly seminar. The
first course, entitled "Lectures on Motivic Cohomology," was given by Voevodsky him-
self. This course concerned the construction and properties of the triangulated category
of motives. The second course, "Lectures on Norm Varieties," was given by Markus Rost
of The Ohio State University, who was a Member for the year. This course concerned
various constructions related to algebraa K-theory, including Steenrod operations and
cobordism. The seminar associated with this special program was on Homotopy Theory
(the ordinary topological kind). This was the first time in many years that homotopy
theory was a serious pan ol the program at the Institute. Speakers were D. Christensen,
C. Re:k, F. Morel, G. Carlsson, M. Bendersky, A. Neeman, R Hu, and H. Miller.
This academic year was the first year that our new Professor, Avi Wigderson, was in resi-
dence. His specialty is theoretical computer science and discrete mathematics. A weekly
seminar was held on Combinatorics and Complexity Theory, run by Wigderson. This was
a continuation of our now long-standing weekly seminar on discrete mathematics. Speak-
ers were M. Saks, R. Ra:, E. Fischer, A. Razborov, H. Ben-Sasson, J. Kahn, L. Tfevisan, L.
valiant, E. Friedgut, D. Aharonov.J. Beck, A. Samorodnitsky, Y. Peres, N. Alon, M. Sudan,
1. Hast ad, R. Shaltiel, P. Winkler, B. Sudakov, A. Yao, and R. Impaglia::o. In the year
2000-2001, there will be a special year on ( 'omplexity Theory, run In Wigderson
Additional weekly seminars included the Analysis Seminal organized by 1
Bourgain and
n< er, the joint Princeton/IAS/Rutgers Nonlinear Analysis Seminar organized by S.
Klainerman, II Bezis, and |. Bourgain, and as usual, the Members Seminar, whose field
of mathematics < hanges every week. The Princeton/IAS/Rutgers Number Theory Sem-
inar was in abeyance tor the year, because all of its organizers were participating in the
mi on Modular Forms.
The Marston Morse Lectures were given by ML Hopkins of MIT They were entitled
"Modular tonus, theta functions, and algebraic topology." It was a tale ot a remarkable