Commutative Algebra and Algebraic Geometry

代数几何是研究多项式方程组的学科。因此，它不仅在数学中有广泛的应用，而且在从医学到物理的许多科学领域都有广泛的应用。这门学科已经发展了200多年，但也出现了许多新的问题和新的方向。交换代数是一门介于代数几何和数论之间的学科。随着强大且容易获得的计算资源的出现，在交换代数、代数几何及其应用中进行实验的可能性成倍增加。该项目的一个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发挥作用，这是有限维表示在有限维代数中作用的自然类比。首席研究员将致力于交换代数、代数几何和计算方法方面的项目，这些项目以Cohen-Macaulay模的理论为中心，涉及特别有趣的种类：环状簇、完全交集和剩余交集。该中心还将继续培训相关研究领域的研究生，并积极参与几个广受认可的外展活动。首席调查员将致力于Ulrich模块和Clifford代数的工作。从Knoerrer(代数闭域上)和Buchweitz-Eisenbud-Schreyer在任意域上的工作可以很好地理解二次超曲面上的Cohen-Macaulay模和Ulrich模。PI将利用Clifford代数技巧研究两个二次曲面完全交上的Ulrich模和其他极大Cohen-Macaulay模的更深层次的问题，推广Miles Reid的命题，明确Bondal-Orlov和Kapranov的工作，以及作者与Irena Peeva发展的完全交上极大Cohen-Macaulay模的理论。最后，PI将致力于环变种上的层的上同调，将技术从外部代数扩展到与Daniel Erman和Frank-Olaf Schreyer共同研究射影空间上的层的上同调，并成功地扩展到所有环变体的射影空间的乘积。该奖项反映了NSF的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

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.