New Structures in Homological Commutative Algebra

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Solving polynomial equations is one of the oldest and most fundamental topics in mathematics. One might expect that equations would grow ever more complex as the number of variables increases; in fact, this sort of phenomenon is known as the "curse of dimensionality" and it is very common in mathematics. However, recent work of Ananyan and Hochster has shown that this does not happen: under a certain regime, the complexity of solving equations does not increase as the number of variables increases. In fact in some ways, the problem even becomes simpler. In this project, the PI will try to carry the core insights of Ananyan and Hochster to new types of equations. This would sharpen our understanding of the structure of systems of equations, with the potential for both theoretical and computational applications.The recent progress on Stillman's Conjecture has led to a plethora of new bounded results in algebra, many of which are modern twists on classical results of Hilbert. In previous work, the PI had constructed new limit rings, involving inverse limits and ultraproducts, and applied these limit rings to homological questions in commutative algebra. This led to two new proofs of Stillman's Conjecture. The PI proposes developing similar new frameworks for regular local rings and for coherent sheaves on projective space. The intellectual merit of this project would primarily come through the broad array of boundedness results this would yield for regular local rings and for cohomology of coherent sheaves on projective space This project will also have impacts on K-12 education through the PI's leadership of the Madison Math Circle, an outreach program that provides a taste of exciting ideas in math and science to high school and advanced middle school students.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.