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.

代数几何形状是现代数学的最先进和多方面的领域之一。研究对象是代数品种：由多项式方程系统定义的形状。变量的多项式约束在数学和应用中都很普遍，从物理学（用于描述配置空间（甚至是宇宙的形状！）到计算机科学和统计）的物理学范围。首席研究者的研究重点是以简洁的无坐标方式描述代数品种，该品种揭示了其几何形状的隐藏基本特征。这些项目连接了许多不同的领域：生育几何形状，衍生类别，算术几何，数学物理学，曲折的几何形状，变形理论和转化组。该研究计划通过招募，培训和监督研究生和本科生，包括来自代表性不足的群体，并为其研究，教学和专业发展提供资源，从而有助于发展多样化的全球竞争性STEM劳动力。特别是，该项目包括两个为代数几何形状设计的夏季REUS的项目。首席调查员将在会议，专业发展活动和暑期学校组织和演讲，为研究生和初级研究人员（包括Agnes（代数几何）东北系列赛），以及在拉丁美洲的学校由美国研究生参加。这项研究将提高大学间和国际合作。该研究计划包括与高能量物理学家的合作以及数学加密兴趣的软件开发（椭圆曲线的算术）。 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模量空间的派生类别。 PI将在几个重要边界上推进现代代数几何形状：（1）派生类别在代数品种的几何形状，共同体学和K理论中的应用，包括代数曲线和矢量包裹的模量空间，曲折的品种，曲折的品种和Mori Dream Dreams。（2）通过曲线的算术几何形状探索代数表面的有效锥之间的偶像几何形状和算术几何形状之间的新接口。（3）基本颗粒散射幅度的主要奇异性的新型几何解释和概括。（4）运动学，散射方程式和溶液的传播的发展。（5）通过球形热带化的几何品种的一类新的多面体近似，具有开发新的紧凑技术的潜力。（6）一种新的方法，通过使用Q-Gorenstein脱生的矢量捆绑包和半正交分解的特殊集合的形态来参数稳定限制的模量空间。这奖反映了NSF的立法任务，并被认为是通过基金会的智力评论来评估的，并且值得通过评估的支持。