Non-Commutative Desingularizations and Representation Theory

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There is a classical dictionary between geometry and algebra, dating back at least to Descartes and Fermat in the seventeenth century, which has been developed and sharpened to allow the detailed study of geometric spaces in terms of algebraic objects known as commutative rings. (Commutativity means that order of multiplication doesn't matter: x times y is equal to y times x.) As our understanding of the physical world has grown, however, in such areas as quantum mechanics, string theory, and the study of fundamental particles, we have come to understand that the fine structure of the universe is essentially non-commutative. New dictionaries are being built to understand the connections between geometry and the non-commutative world. This research project concerns non-commutative analogues of a geometric operation called resolution of singularities, which "unfolds" a pinched or creased geometry to replace it with a smooth one.The investigator will work on problems at the intersection of commutative algebra, algebraic geometry, and non-commutative algebraic geometry. The research plan involves applying tools and techniques from the representation theory of local rings, Artin algebras, and algebraic groups, to the problems of constructing non-commutative analogues of resolutions of singularities, studying their structure, and applying their existence to study further problems in representation theory. The foundations of the theory of non-commutative resolutions (NCRs) are still in active development, and this proposal would contribute to the maturation of the theory. Furthermore the investigator proposes constructions of new examples of NCRs, which would expand our understanding of the limits of the theory. Some of these new examples would rely on new constructions in geometric tilting theory over homogeneous varieties.