Analysis, Spectra, and Number Theory

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The grant will support a conference "Analysis, Spectra and Number theory", to be held to be held at Princeton University and the Institute for Advanced Study, December 15 - 19, 2014. There will be approximately 20 speakers and an anticipated 200 participants. The conference will focus on number theory, with emphasis on its many relationships with analysis and spectral theory. Both (Fourier) analysis and spectral theory have their origins in understanding oscillating systems, interpreted broadly. It is very surprising, then, that they have been found to play a basic role in number theory. In his 1859 study of primes, Riemann observed that the number of primes up to a given integer could be expressed simply as a sum of oscillating components. It was later observed that that the "frequencies" occurring in Riemann's analysis show many regularities, and behave as if they were the eigenvalues of a large unitary matrix - i.e., an abstraction of the frequency spectrum of a drum. A further link was the discovery, beginning in the work of Maass and Selberg, of highly symmetric geometries (locally symmetric spaces) whose frequency spectrum appears to control many problems in number theory. These themes have expanded in many directions since, encompassing the field of analytic number theory as well as much of automorphic forms. The conference will examine the latest developments in these areas.Topics to be highlighted include arithmetic quantum chaos, analysis of families of L-functions, arithmetic statistics, and connections with ergodic theory. These areas have seen a flurry of activity in recent years, including: the resolution by Lindenstrauss of quantum unique ergodicity for arithmetic surfaces; spectacular breakthroughs by Bhargava and his colleagues concerning the statistics of number fields and elliptic curves; detailed models and predictions for the zero and value distribution of L-functions that were inspired via connections with random matrix theory; construction and the analysis of highly efficient expander graphs; the development of additive combinatorics based on the work of Green and Tao. The conference will include a problem session to suggest future directions for the field. The conference website can be found at https://sites.google.com/site/asnt2014/.