CAREER: Applications of Koszul duality to local algebra

This is a project in commutative algebra, with connections to algebraic geometry, algebraic topology, and representation theory. Classically, commutative algebra is concerned with the solutions to systems of polynomial equations like those defining lines, circles, parabolas, planes, spheres, and other familiar geometric objects. As polynomial equations are ubiquitous in mathematics, commutative algebra is fundamental to many fields in pure math like algebraic geometry, number theory, and topology, and it also has applications in diverse areas such as cryptography, statistics, and physics. Depending on the number of equations, the number of variables, and the complexity of the equations defining a solution set, the corresponding geometric object can be difficult or often impossible to understand in its entirety. Instead, it is often effective to study the local behavior of these objects and “glue” this local information together to gain global insights. The research carried out will investigate the local algebraic behavior of solution sets arising in very general, abstract settings using a variety of homological and homotopical techniques. The project also includes the organization of masterclasses and immersive graduate summer schools aimed at developing in-depth, communal learning activities centered on cutting-edge mathematics that connects commutative algebra to other areas. The research program has two central long-term goals with the common thread of Koszul duality, a pervasive phenomenon in algebra, geometry, and topology. Techniques toward both goals include the application of differential graded and simplicial methods, and of A-infinity structures. The first research direction investigates several related constructions in local algebra that measure singularities. A focus here is on a problem of Avramov from the late 1980s that predicts a relative Koszul duality between deformations and the homotopy Lie algebra; the former connects to classical deformation theory, while the latter allows one to draw on a wealth of ideas from rational homotopy theory. Infinitesimal deformations, Koszul homology, and cohomological support varieties will also be investigated. The second overarching goal is to better understand the asymptotic nature of free resolutions over singular rings, exploiting a crucial feature of Koszul duality: it supplies explicit, computable constructions for understanding the relevant homological algebra. The PI aims to develop machinery—and in turn, an effective algorithm—for constructing minimal free resolutions over a large class of rings. This framework has the potential to recover known universal constructions, such as those over complete intersection and Golod rings, while also applying to many new examples, including generic Gorenstein rings and monomially-defined examples. The main strategy, building on previous work of the PI, applies homotopical deformations of Koszul algebras and Koszul modules. 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. NSF Award ID: 2543090 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT,01003031DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Joshua Pollitz | Institution: Syracuse University, SYRACUSE, NY | Award Amount: $235,363 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2543090 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2543090.html