CAREER: Combinatorial Categories and Commutative

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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.