COLLABORATIVE RESEARCH: EMSW21-RTG: JOINT COLUMBIA-CUNY-NYU RESEARCH TRAINING GROUP IN NUMBER THEORY

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The goal of the RTG proposal is to make the New York metropolitan area a premier world center and model for the study of number theory. The project is a joint training effort involving three universities (Columbia-CUNY-NYU) with eight principal investigators and twenty-five other faculty all working in number theory and related areas. The Columbia-CUNY-NYU team will develop nine new graduate courses for the design of a city-wide graduate number theory curriculum with more active training methods such as workshops, presentations by students, and opportunities to develop technical lecturing, computer programming, and writing skills. The new graduate courses will go beyond the existing first year courses (commutative algebra/algebraic geometry, algebraic number theory) and will cover a broad range of core topics (required for cutting edge research) such as: arithmetic geometry, automorphic representations, spectral theory, and combinatorial number theory. In addition, six new or restructured undergraduate courses will be developed and an annual undergraduate summer research program and graduate summer school (meeting once every 3 years) will be created. The recently created Columbia-CUNY-NYU Number Theory Seminar will be redesigned, making it more accessible to students and more vibrant for seminal research. A central goal is to provide an environment where postdocs, graduate students, undergraduates, and faculty from the entire New York metropolitan area study and work together in a collaborative atmosphere that fosters research and development.Number theory is one of the oldest and most fundamental branches of mathematics. Basic research in number theory has led to important applications in computer science and cryptography. Recent spectacular breakthroughs in the subject such as the proof of Fermat's Last Theorem by Taylor-Wiles require an understanding of an enormous amount of mathematics, much of it outside number theory. It is not sufficient any more for beginning students to study a small segment of number theory. The eight PI's of this proposal (Goldfeld, Kolyvagin, Kramer, Szpiro, Tschinkel, Urban, Venkatesh, Zhang) have extensive overlapping interests but their combined expertise covers all of modern number theory. The scientific research interests of the team include algebraic/arithmetic geometry, hyperbolic geometry, automorphic forms and representations, Langlands program, analytic number theory, spectral theory, ergodic theory, Lie algebras, algebraic number theory, elliptic curves, dynamical systems, and cryptography. This project will be decisive in raising the level of number theory across participating campuses: cross-pollinating successful ideas already in existence, and creating approaches through collective interaction. It is hoped that this RTG will become a national model for other programs of this kind.