2010 AMS Levi L. Conant Prize Recipient
PATTERNS IN THE PRIMES
Abstract: In 2004, Ben Green and Terence Tao made a stunning breakthrough, showing that the primes contain arbitrarily long arithmetic progressions.
Perhaps even more impressive is the fusion of methods and results from number theory, ergodic theory, harmonic analysis, and combinatorics used in its proof. The starting point for their proof is the celebrated theorem of Endre Szemerédi from the 1970's: a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Shortly thereafter, Hillel Furstenberg gave a new proof of this theorem, uncovering beautiful connections between dynamics and additive combinatorics. More recently, Timothy Gowers gave a new proof of Szemerédi's Theorem vastly improving quantitative bounds in the finite version. Although the various proofs, Szemerédi's, Furstenberg's, and Gowers's, seem to use very different methods, they have several features in common: in each, a key idea is the dichotomy in the underlying space between randomness and structure. Green and Tao's proof draws on all of these proofs and exploits such a dichotomy. The talk will be an overview of the connections between these topics, with a focus on recent developments.
Bryna Kra earned her undergraduate degree from Harvard University in 1988 and her PhD from Stanford in 1995. Before her appointment to Northwestern University in 2004, she held postdoctoral positions at the Hebrew University of Jerusalem, the University of Michigan, the Institut des Hautes Études Scientifiques, and Ohio State University, and was an assistant professor at Pennsylvania State University. Kra works in dynamical systems and ergodic theory, with a focus on problems related to combinatorics and number theory. She was an invited speaker at the 2006 International Congress of Mathematicians, was awarded a Centennial Fellowship, also in 2006, and was awarded the Conant Prize in 2010. Kra organizes a mentoring program for women in mathematics at Northwestern, runs a math enrichment program for children at a local elementary school, and is currently chair of the Northwestern University math department.