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

RTG提案的目标是使纽约大都市区成为数论研究的首要世界中心和模型。该项目是由三所大学（哥伦比亚大学-纽约市立大学-纽约大学）联合培训的成果，共有8名主要研究人员和25名其他教员从事数论和相关领域的工作。哥伦比亚大学-纽约市立大学-纽约大学的团队将开发9门新的研究生课程，以设计全市范围的研究生数论课程，其中包括更积极的培训方法，如研讨会，学生演讲，以及发展技术讲座，计算机编程和写作技能的机会。新的研究生课程将超越现有的第一年课程（交换代数/代数几何，代数数论），并将涵盖广泛的核心主题（前沿研究所需），如：算术几何，自同构表示，谱理论和组合数论。此外，将开发6门新的或重组的本科课程，并创建每年一次的本科生暑期研究计划和研究生暑期学校（每3年召开一次会议）。最近成立的哥伦比亚大学-纽约市立大学-纽约大学数论研讨会将被重新设计，使其更容易为学生和更具活力的开创性研究。中心目标是提供一个环境，让来自整个纽约大都会区的博士后、研究生、本科生和教师在促进研究和发展的协作氛围中共同学习和工作。数论是数学中最古老、最基本的分支之一。数论的基础研究在计算机科学和密码学中有着重要的应用。最近在数学领域取得的重大突破，如泰勒-怀尔斯对费马大定理的证明，需要对大量数学知识的理解，其中大部分是数论之外的知识。对于初学的学生来说，仅仅学习数论的一小部分已经不够了。该提案的8位PI （Goldfeld, Kolyvagin, Kramer, Szpiro, Tschinkel, Urban, Venkatesh, Zhang）有着广泛的重叠兴趣，但他们的综合专业知识涵盖了所有现代数论。该团队的科学研究兴趣包括代数/算术几何、双曲几何、自同构形式和表示、朗兰兹程序、解析数论、谱理论、遍历理论、李代数、代数数论、椭圆曲线、动力系统和密码学。这个项目将对提高参与校园的数论水平起决定性作用：交叉传粉已经存在的成功想法，并通过集体互动创造方法。希望这个RTG能成为全国其他类似项目的样板。

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.