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Commutative Algebra and Algebraic Geometry

交换代数和代数几何

基本信息

批准号：
2001649
负责人：
David Eisenbud
金额：
$ 18万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001649&HistoricalAwards=false
关键词：
Commutative Algebra Algebraic Geometry

项目摘要

Algebraic Geometry is the study of systems of polynomial equations. As such, it has broad applications not only within mathematics but also in many fields of science from medicine to physics. The subject has developed for over 200 years, but there are many fresh problems and new directions. Commutative algebra is a subject that lies at the intersection of algebraic geometry and number theory. Since the advent of powerful and easily available computing resources, the possibilities for experimentation within commutative algebra, algebraic geometry, and their applications has multiplied. A byproduct of this project will be the further development of these computational tools. Cohen-Macaulay modules and sheaves play a role in commutative algebra and algebraic geometry that is a natural analogue of the role of finite dimensional representations in the case of finite dimensional algebras. The Principal Investigator will work on projects in commutative algebra, algebraic geometry, and computational methods that center around the theory of Cohen-Macaulay modules over particularly interesting classes of varieties: Toric varieties, complete intersections, and residual intersections. The PI will also continue to train graduate students in related research fields and actively be involved in several highly recognized outreach activities.The Principal Investigator will work on Ulrich modules and Clifford Algebras. Cohen-Macaulay and Ulrich modules over quadratic hypersurfaces are well-understood from work of Knoerrer (over algebraically closed fields) and Buchweitz-Eisenbud-Schreyer over arbitrary fields. The PI will investigate deeper questions about Ulrich modules and other maximal Cohen-Macaulay modules on complete intersections of two quadrics using Clifford algebra techniques, extending Miles Reid's thesis, and making explicit work of Bondal-Orlov and Kapranov, as well as the theory of maximal Cohen-Macaulay modules over complete intersections developed by the proposer with Irena Peeva. Finally the PI will work on the cohomology of sheaves on toric varieties, extending techniques from exterior algebra algebras introduced in joint work with Daniel Erman and Frank-Olaf Schreyer for cohomology of sheaves on projective space and successfully extended to products of projective space to all toric varieties.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.

代数几何是研究多项式方程组的学科。因此，它不仅在数学中而且在从医学到物理学的许多科学领域中具有广泛的应用。这门学科已经发展了200多年，但也有许多新的问题和新的方向。交换代数是代数几何和数论的交叉学科。自从强大且容易获得的计算资源出现以来，交换代数、代数几何及其应用中的实验的可能性成倍增加。这个项目的副产品将是这些计算工具的进一步发展。Cohen-Macaulay模和层在交换代数和代数几何中扮演的角色，是有限维代数中有限维表示的自然模拟。主要研究者将致力于交换代数，代数几何和计算方法的项目，这些项目围绕着科恩-麦考利模块的理论，特别是有趣的品种：复曲面品种，完整的交叉点和剩余交叉点。PI还将继续培养相关研究领域的研究生，并积极参与几项高度认可的外展活动。首席研究员将从事Ulrich模块和Clifford代数的工作。二次超曲面上的Cohen-Macaulay模和Ulrich模可以从Knoerrer（代数闭域上）和Buchweitz-Eisenbud-Schreyer（任意域上）的工作中得到很好的理解。 PI将使用Clifford代数技术研究关于两个二次曲面的完全相交上的Ulrich模和其他极大Cohen-Macaulay模的更深层次的问题，扩展Miles Reid的论文，并明确Bondal-Orlov和Kapranov的工作，以及由提议者与Irena Peeva开发的完全相交上的极大Cohen-Macaulay模的理论。最后，PI将研究复曲面簇上层的上同调，扩展技术从外部代数代数介绍了在联合工作与丹尼尔埃尔曼和弗兰克-Olaf Schreyer因射影空间上的层的上同调而获奖，并成功地将其推广到所有环面类的射影空间的乘积。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识产权进行评估来支持。优点和更广泛的影响审查标准。