This collaborative proposal is concerned with developing a new higher rank version of a fundamental identity known classically as the Kuznetsov trace formula, which relates the spectrum of a certain differential operator to the geometry of the space on which the operator acts. The aim is to establish either asymptotics or strong bounds for all the different terms appearing in the formula. A first application is to obtain the symmetry types of certain thin families of L-functions in various higher rank situations. A broad range of further applications are expected. Additionally, the following research problems will be investigated: a search for a new class of Multiple Dirichlet Series will be executed; supercuspidal representations in higher rank will be studied; the Affine Linear Sieve will be combined with bilinear forms methods to exhibit thin orbits containing an infinitude of primes; and finally, effective infinite-volume counting problems will be attacked.
The theory of automorphic forms, representations, and L-functions is a central theme in modern number theory, and has provided links between such diverse areas of mathematics as algebraic geometry, representation theory, probability, combinatorics, and mathematical physics. Thus progress in the understanding of the aforementioned objects often has a significant impact in other fields. For example, cryptographic algorithms which secure wireless communication for the internet and cellular phones often rely heavily on deep properties of prime numbers. The proposal also includes a significant educational and dissemination component in the mentoring of undergraduate, graduate students, and postdocs working in these evolving parts of mathematics, with the hope of bringing traditionally under-represented goups into the field.
INTELLECTUAL MERITS The project outcomes of this award consist of two separate themes: I) Higher Rank Automorphic Representations and L-functions: The theory of automorphic forms, representations, and L-functions is a central theme in modern number theory, and has provided links between such diverse areas of mathematics as algebraic geometry, representation theory, probability, combinatorics, and mathematical physics. The Kuznetsov formula relates the spectrum of a certain differential operator to the geometry of the space on which the operator acts. In a collaborative effort with Dorian Goldfeld, the PI developed an explicit Kuznetsov formula for the higher rank group GL(3,Z), and used this formula to determine symmetry types of various families of automorphic L-functions. II) Primes and Local-Global Phenomena in Thin Orbits: The class of problems falling under the so-called Affine Sieve is a recent variant of more classical sieve problems (such as the Twin Prime and Goldbach Conjectures), in which polynomials are replaced by orbits under groups or semi-groups of affine-linear transformations. In many natural situations, the arising orbits are thin, meaning they are degenerate in their algebro-geometric closure, having many fewer points in an archimedean norm ball. Two quintessential examples are the Local-Global Conjecture for Integral Apollonian Gaskets and Zaremba's Conjecture on uniformly badly approximable rational numbers. The statements of both problems could be understood by the ancient Greeks, yet their solution evades us to this day. In joint work with Jean Bourgain, the PI proved density versions of both conjectures, that they hold for a density one set of numbers. A by-product of the developed technology is the ability to produce primes in these thin orbits. BROADER IMPACTS I) Training and development: The PI supervised the thesis of Ilya Vinogradov at Princeton (jointly with Yakov Sinai). The award supported Vinogradovâ€™s regular travel from Princeton to New Haven. Vinogradov defended his thesis in May 2012, and is now a postdoc at University of Bristol, UK. The PI is also supervising the Ph.D. theses of Shinnyih Huang and Liyang Zhang at Yale, and Xin Zhang at Stony Brook. The award supported Xinâ€™s travel between Stony Brook and New Haven, as well as a year at Yale as a Visiting Assistant in Research. The PI has also been overseeing the independent studies of other graduate and undergraduate students at Stony Brook and Yale, and is also mentoring a high school student for the Intel Science Talent Search. II) Dissemination Activities: The award has been used to partially support weekly seminars in number theory, dynamics, and related topics at Stony Brook and Yale. The award also subsidized the PIâ€™s travel: during the past five years, he has delivered 93 invited lectures at colloquia, seminars, and conferences in the US and abroad. The PI wrote survey paper, published by the Bulletin of the AMS, which collects under a common umbrella local-global phenomena in thin orbits. Since a picture is worth its proverbial 1000 words, the PI maintains a website (see http://users.math.yale.edu/~avk23) of pictures and animations related to his research, and makes freely available the numerous Mathematica programs he developed to make said graphics.