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New Frontiers of Algebraic Geometry

代数几何新领域

基本信息

批准号：
2101726
负责人：
Evgueni Tevelev
金额：
$ 27.2万
依托单位：
University of Massachusetts Amherst
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101726&HistoricalAwards=false
关键词：
New Frontiers Algebraic Geometry

项目摘要

Algebraic geometry is one of the most advanced and multifaceted areas of modern mathematics. The objects of study are algebraic varieties: shapes defined by systems of polynomial equations. Polynomial constraints on variables are commonplace both in mathematics and in applications ranging from physics, where varieties are used to describe configuration spaces (and even the shape of the universe!), to computer science and statistics. The Principal Investigator's research focuses on describing algebraic varieties in a concise coordinate-free way that reveals hidden essential features of their geometry. The projects connect many different areas: birational geometry, derived categories, arithmetic geometry, mathematical physics, toric geometry, deformation theory, and transformation groups. The research program contributes to the development of a diverse, globally competitive STEM workforce through recruiting, training, and supervising of graduate and undergraduate students, including those from underrepresented groups, and providing resources for their research, teaching, and professional development. In particular, the project includes two projects designed for summer REUs in Algebraic Geometry. The Principal Investigator will organize and lecture at conferences, professional development events, and summer schools for graduate students and junior researchers including AGNES (Algebraic Geometry Northeastern Series), and schools in Latin America attended by U.S. graduate students. The research will advance inter-university and international cooperation. This research program includes collaborations with high energy physicists as well as software development of interest to mathematical cryptography (arithmetic of elliptic curves). The Principal Investigator will continue to develop graduate and undergraduate courses, including courses designed to introduce the broader public to mathematics and science.This project builds on the earlier work of the Principal Investigator (PI) on a broad range of foundational topics in algebraic geometry, including tropical compactifications, birational geometry of moduli spaces of curves and surfaces, compactifications of moduli spaces and derived categories of moduli spaces. The PI will advance modern algebraic geometry on several important frontiers: (1) Applications of derived categories to geometry, cohomology and K-theory of algebraic varieties, including moduli spaces of algebraic curves and vector bundles, toric varieties and Mori Dream Spaces. (2) A new interface between birational and arithmetic geometry exploring effective cones of algebraic surfaces via arithmetic geometry of curves. (3) A novel geometric interpretation and generalization of leading singularities of scattering amplitudes of elementary particles. (4) Development of kinematics, scattering equations and propagation of their solutions. (5) A new class of polyhedral approximations of geometric varieties via spherical tropicalization with the potential to develop new compactification techniques. (6) A new approach to parametrizing moduli spaces of surfaces near the stable limit using deformations of exceptional collections of vector bundles and semi-orthogonal decompositions for Q-Gorenstein degenerations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数几何是现代数学中最先进和最多方面的领域之一。研究的对象是代数变量：由多项式方程组定义的形状。变量的多项式约束在数学和从物理学到计算机科学和统计学的应用中都很常见，在物理学中，变量被用来描述配置空间（甚至宇宙的形状！）。首席研究员的研究重点是用一种简洁的无坐标方式描述代数变量，从而揭示其几何结构中隐藏的基本特征。这些项目连接了许多不同的领域：二分几何、派生类别、算术几何、数学物理、环形几何、变形理论和变换群。该研究项目通过招聘、培训和监督研究生和本科生（包括来自代表性不足群体的学生），并为他们的研究、教学和专业发展提供资源，为培养多元化、具有全球竞争力的STEM劳动力做出贡献。特别地，该项目包括为代数几何夏季reu设计的两个项目。首席研究员将为研究生和初级研究人员（包括AGNES（代数几何东北系列））以及美国研究生参加的拉丁美洲学校组织会议、专业发展活动和暑期学校并发表演讲。这项研究将促进大学间和国际合作。该研究项目包括与高能物理学家的合作，以及对数学密码学（椭圆曲线算法）感兴趣的软件开发。首席研究员将继续开发研究生和本科生课程，包括旨在向更广泛的公众介绍数学和科学的课程。该项目建立在首席研究员（PI）早期在代数几何中广泛的基础课题上的工作基础上，包括热带紧化、曲线和曲面的模空间的双空间几何、模空间的紧化和模空间的派生范畴。PI将在几个重要的前沿领域推进现代代数几何：(1)衍生范畴在几何、代数变体的上同调和k理论中的应用，包括代数曲线和向量束的模空间、环型变体和森梦空间。(2)通过曲线的算术几何探索代数曲面的有效锥，建立了算术几何与算术几何之间的新界面。(3)对基本粒子散射振幅先导奇点的一种新的几何解释和推广。(4)运动学、散射方程的发展及其解的传播。(5)通过球面热带化得到一类新的几何变量多面体逼近，具有发展新的紧化技术的潜力。(6)利用向量束异常集合的变形和Q-Gorenstein退化的半正交分解，给出了稳定极限附近曲面模空间参数化的新方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。