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。 该项目是一项联合培训工作，涉及三所大学（哥伦比亚-纽约市立大学-纽约大学），有八名主要研究人员和二十五名其他教师，他们都在数论和相关领域工作。哥伦比亚-纽约市立大学-纽约大学团队将开发九门新的研究生课程，用于设计全市范围的研究生数论课程，并采用更积极的培训方法，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（尖端研究所需），如：算术几何，自守表示，谱理论和组合数论。 此外，将开发六门新的或重组的本科课程，并将创建年度本科夏季研究计划和研究生暑期学校（每3年举行一次会议）。最近创建的哥伦比亚-纽约市立大学-纽约大学数论研讨会将重新设计，使其更容易为学生和更有活力的开创性研究。一个中心目标是提供一个环境，博士后，研究生，本科生，和教师从整个纽约大都市地区的研究和工作在一个合作的气氛，促进研究和发展。数论是数学的最古老和最基本的分支之一。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。最近在这个问题上的惊人突破，如泰勒-怀尔斯对费马大定理的证明，需要对大量数学的理解，其中大部分是数论之外的。对于初学者来说，仅仅学习一小部分数论是不够的。八PI的这一建议（戈德菲尔德，Kolyvagin，克雷默，Szpiro，Tschinkel，城市，Venkatesh，张）有广泛的重叠利益，但他们的综合专业知识涵盖了所有的现代数论。该团队的科学研究兴趣包括代数/算术几何，双曲几何， 自守形式和表示，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。该项目将在提高参与校园的数论水平方面发挥决定性作用：交叉传播已经存在的成功想法，并通过集体互动创造方法。我们希望，这一RTG将成为其他此类方案的国家模式。

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.