The structure of free resolutions in commutative algebra and algebraic geometry

PI的研究旨在通过几个特定的项目提供对自由分辨率结构及其使用代数和几何的更深层次的理解。其中一个项目将继续系统地发展Boij-Soederberg理论。这个理论是最近由Eisenbud和Schreyer发展起来的，它为理解特定原子构建块的自由分辨率提供了结果。这是一个强有力的新理论，关于该理论的基础和范围仍有许多悬而未决的问题。另一个项目的目标是在光滑环变的背景下发展一个新的同调代数框架。David Cox和其他人的工作表明，可能会有一个影响深远的代数/几何字典，它在很大程度上与分级多项式环的代数和射影空间的几何之间的强大字典相似。然而，同调交换代数的许多结果在环变的情况下没有令人满意的类似物。一个更健壮的同调理论将依赖于多重交换代数、代数几何和组合学之间的联系。第三个项目涉及高维罗内塞子带自由分辨率的渐近结构。这是为了填补文献中的一个空白：尽管对曲线的高阶Veronese子带的自由分辨率的研究已经取得了丰硕的成果并得到了广泛的应用，但对高维变量在非常正嵌入下的自由分辨率知之甚少。自由分辨率是由多项式矩阵构建的。它们自然出现在许多代数上下文中，与交换代数、代数几何、计算代数等主题有关。多项式矩阵比线性代数中出现的标量矩阵具有更丰富的结构，因此关于自由分辨率的基本问题仍然没有解决。PI的研究项目旨在提供关于自由分辨率的总体结构结果，并将这些新的结构见解应用于交换代数和代数几何中的开放问题。这些研究项目都为显式计算提供了潜力，为本科生和研究生提供了研究机会，并为Macaulay2开发了新的软件包。PI还将继续他在K-12教育方面的推广项目，这些项目通常涉及研究数学家、教育工作者、公共或非营利组织之间的合作。

The PI's research aims to provide a deeper understanding of the structure of free resolutions and their use algebra and geometry via several specified projects. One such project will continue the systematic development of Boij-Soederberg theory. This theory was recently developed by Eisenbud and Schreyer, and it provides results for understanding free resolutions in terms of specified atomic building blocks. This is a powerful new theory, and there remain many open questions about both the foundations and the reach of the theory. Another project aims to develop a new framework for homological algebra in the context of smooth toric varieties. The work of David Cox and others suggests the potential for a far-reaching algebraic/geometric dictionary that largely parallels the powerful dictionary between the algebra of the graded polynomial ring and the geometry of projective space. Yet many of the results from homological commutative algebra have no satisfying analogue in the context of toric varieties. A more robust homological theory would rely on connections between multigraded commutative algebra, algebraic geometry, and combinatorics. A third project involves the asymptotic structure of the free resolutions of high degree Veronese subrings. This aims to fill a gap in the literature: whereas the study of the free resolutions of high degree Veronese subrings for curves has been fruitful and widely applied, little is known about the free resolutions of higher dimensional varieties under a very positive embedding. Free resolutions are built from matrices of polynomials. They arise naturally in many algebraic contexts, with connections to topics in commutative algebra, algebraic geometry, computational algebra, and more. Matrices of polynomials have a richer structure than the matrices of scalars that arise in linear algebra, and so basic questions about free resolutions remain unresolved. The PI's research projects aim to provide overarching structural results about free resolutions, and to apply these new structural insights to open problems in commutative algebra and algebraic geometry. These research projects all offer potential for explicit computations, providing research opportunities for undergraduate and graduate students, as well as leading to new software packages to be developed for Macaulay2. The PI will also continue his outreach projects on K-12 education, which often involve collaborative efforts between research mathematicians, educators, and public or nonprofit groups.