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首屈一指的世界数论研究中心和典范。 该项目是一项联合培训项目，涉及三所大学（哥伦比亚大学-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员，全部从事数论和相关领域的工作。哥伦比亚大学-纽约市立大学-纽约大学团队将开发九门新的研究生课程，用于设计全市研究生数论课程，并​​采用更积极的培训方法，例如研讨会、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），例如：算术几何、自守表示、谱论和组合数论。 此外，还将开发六门新的或重组的本科课程，并创建年度本科生暑期研究计划和研究生暑期学校（每三年一次）。最近创建的哥伦比亚大学-纽约市立大学-纽约大学数论研讨会将进行重新设计，使其更容易为学生所接受，并为开创性研究注入更多活力。中心目标是提供一个环境，让整个纽约都市区的博士后、研究生、本科生和教师在促进研究和开发的协作氛围中一起学习和工作。数论是数学最古老和最基本的分支之一。数论的基础研究导致了计算机科学和密码学的重要应用。最近该学科取得的重大突破，例如泰勒-怀尔斯对费马大定理的证明，需要理解大量的数学知识，其中大部分都在数论之外。对于初学者来说，学习一小部分数论已经不够了。该提案的八位 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。该团队的科研兴趣包括代数/算术几何、双曲几何、自守形式和表示、朗兰兹纲领、解析数论、谱论、遍历理论、李代数、代数数论、椭圆曲线、动力系统和密码学。该项目对于提高参与校园的数论水平具有决定性作用：交叉传播现有的成功想法，并通过集体互动创造方法。希望该RTG能够成为全国其他此类项目的典范。

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.