Local Algebra and Local Representation Theory

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One of the myriad functions of Mathematics is that it provides language to formulate, and tools to solve, equations that describe the physical world. Often the equations that one encounters are algebraic in nature, like those describing lines, circles, parabolas and the like, in contrast with, say, equations involving the trigonometric functions, or logarithms, or derivatives. Typically, the equations have infinitely many solutions---think about the equation defining a circle---and it is usually not possible to write down a complete list of solutions. Rather, the objective is to find ways to study the structure of the collection of the solution set, which is called a variety. A fruitful approach has been to consider the (algebraic) functions on the variety. These functions form a mathematical structure called a commutative ring, and my research has been dedicated to understanding these structures; not in the abstract, but in their various manifestations, which are galore, they arise in quite diverse contexts across mathematics and physics. Mathematics has also been remarkably successful in describing and studying phenomenon related to symmetry. This leads to another mathematical structure called a group. Intriguingly, in certain contexts, there is a way to attach a commutative ring---and a variety---to a group and in the past few years various researchers, including the PI, have been able to solve problems related to groups using tools that had been developed to study varieties. A part of the current project deals with these aspects.Two major themes weave through this project. One is the study of invariants of finite free complexes over commutative noetherian local rings; the second is the modular representation theory of finite groups and group schemes. The former too is connected to groups via certain conjectures in the theory of transformation groups. These conjectures---due to Adem, Avramov, Browder, Buchweitz, Carlsson, Swan, Halperin and others---postulated lower bounds on the length of the homology modules, and on the total rank, of such complexes, when the homology has nonzero finite length. Recently the PI and Mark Walker found counterexamples for many of these conjectures. One set of problems outlined in this project are aimed at discovering and establishing the ``correct" bounds for these invariants. Another set of projects seek to probe the structure of the stable module category of representations of a finite group, or finite group scheme, over a field of positive characteristic. The focus is on ``local strata" and various finiteness conditions for modules in this strata; in particular, on dualizability and cohomological finiteness. The multiplicative structure of Hochschild cohomology of commutative algebras is a third main topic of this proposal. The goal here is to characterize locally complete intersection algebras in terms of their Hochschild cohomology. This award will support the training of students in a very relevant area of mathematics that has applications to several fields.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.