CAREER: Combinatorial Categories and Commutative

这个项目的主要目的是研究两个不同的数学领域之间的联系：组合范畴的表示论和交换代数。这是一个有价值的目标，因为它允许一个人将工具和观点从一个领域应用到另一个领域，这可能导致新的见解。这一策略已经成功地应用了很多次，导致在不同的领域，包括化学，统计学和几何学的新成果。该项目也非常注重教育。PI将与本科生合作，他将组织一个关于这个主题的会议，向研究生介绍这个相对较新的数学领域。PI还将在互联网上免费提供各种数学材料（如视频讲座）。该项目中有两类主要问题。首先是确定在何种程度上的想法，从交换代数可以适用于表示的组合范畴一般。例如，PI和一个合作者将Grobner基理论带到了范畴方面。这导致了许多有趣的结果（例如解决了Lannes-Schwartz Artinian猜想），但也提出了大量的问题。第二类问题是更深入地发展感兴趣的特定组合范畴的表示论。一些主要的例子是扭曲交换代数，其中与交换代数的联系特别强。

The main purpose of this project is to investigate the connection between two different areas of mathematics: representation theory of combinatorial categories and commutative algebra. This is a worthwhile goal, since it allows one to apply tools and perspectives from one area to the other, which can lead to new insights. This strategy has been applied successfully a number of times, leading to new results in a diverse range of areas, including chemistry, statistics, and geometry. The project also has a strong focus on education. The PI will work with undergraduate students, and he will organize a conference on this topic that will introduce graduate students to this relatively new field of mathematics. The PI will also make various mathematical materials (such as filmed lectures) available freely on the internet.There are two main classes of problems within the project. The first is to determine the extent to which ideas from commutative algebra can be applied to representations of combinatorial categories in general. For instance, the PI and a collaborator carried over the theory of Grobner bases to the categorical side. This led to many interesting results (such as the resolution of the Lannes--Schwartz Artinian conjecture), but raised a large number of questions as well. The second class of problems is to develop more deeply the representation theory of particular combinatorial categories of interest. Some of the main examples are twisted commutative algebras, where the connection to commutative algebra is particularly strong.