School of Mathematics

Fundamental questions in basic and applied ecology alike involve complex adaptive systems, in which localized interactions among individual agents give rise to emergent patterns that feed back to affect individual behavior. In such systems, a central challenge is to scale from the “microscopic” to the “macroscopic,” in order to understand the emergence of collective phenomena, the potential for critical transitions, and the ecological and evolutionary conflicts between levels of organization.

The trace formula has been the most powerful and mainstream tool in automorphic forms for proving instances of Langlands functoriality, including character relations. Its generalization, the relative trace formula, has also been used to prove functoriality between "spaces" (more precisely: spherical homogeneous varieties), which is a generalization of functoriality for groups, including relations between periods of automorphic forms.

In 1985 Misha Gromov proved his Nonsqueezing Theorem, and hence constructed the first symplectic 1-capacity. In 1989 Helmut Hofer asked whether symplectic d-capacities exist if 1 < d < n. I will discuss the answer to this question and its relevance in symplectic geometry. This is joint work with San Vu Ngoc.

In 2007, Zeev Dvir shocked experts by giving a one-page proof of the finite field Kakeya problem. The new idea in the proof was to introduce high degree polynomials into a problem about points and lines. This idea has led to progress on several problems of combinatorial geometry. The most famous of these is the distinct distance problem in the plane. In 1946, Erdos raised the problem how many distinct distances can be determined by n points in the plane, and he observed that a square grid gives roughly n/ (log n)^{1/2} distances.