Applications of Homological Algebra in Algebra, Geometry, and Physics

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I propose to work on the following topics.***(1) Construction and Classification of (graded) Matrix Factorizations and Maximal Cohen-Macaulay Modules over Gorenstein Rings. These objects from commutative algebra are now exploited in singularity theory and string theory with tools from (noncommutative) algebraic geometry, and, more generally, homological algebra.***(a) With Iyama (Nagoya) and Yamaura (Yamanuishi) we will describe how to construct all graded maximal Cohen-Macaulay modules over reduced commutative graded Gorenstein curve singularities.***(b) With my former student Alexander Pavlov (Madison) we will write down all matrix factorizations of smooth plane cubics that have linear entries, corresponding to so-called Ulrich bundles. Based on the determination in Pavlov's thesis of all possible graded Betti numbers of indecomposable graded maximal Cohen-Macaulay modules on homogeneous coordinate rings of elliptic curves, we will then determine all graded matrix factorizations in this case.***(2) Properties and Role of Hochschild-Tate Cohomology in Algebra and Geometry.***This is the appropriate version of Hochschild cohomology for the stable category of maximal Cohen-Macaulay Modules. I collaborate with some of my current students to study the Hochschild(-Tate) cohomology of cohomology rings of flag varieties, essentially equal to the classical homology of the free loop space over these manifolds.***(3) Representation Theory of Algebras and Non-commutative Desingularizations.***With Hille (Muenster) we study higher representation-infinite algebras that arise from (weakly) Fano varieties and with him and Iyama we investigate tilting and cluster tilting for projective varieties. I will also continue to study tilting theory on determinantal and other varieties; Castelnuovo-Mumford regularity for varying t-structures; Maximal Cohen-Macaulay endomorphism rings; Rigidity of maximal Cohen-Macaulay modules.***(4) McKay Correspondence for Reflection Groups.***In collaboration with Eleonore Faber (Ann Arbor/Leeds) and Ingalls (UNB) we will extend the classical McKay correspondence for finite subgroups of SL(2,C) to finite reflection subgroups of GL(n,K), thereby obtaining noncommutative desingularizations of the highly singular discriminants of these group actions. This relates intimately to the next point:***(5) Continued Study of Free Divisors and Discriminants.***These hypersurfaces are singular in codimension one, but with highly structured singular locus. I am particularly interested in rank one maximal Cohen-Macaulay modules on such hypersurfaces as those provide compact determinantal expressions of their equations.******This research requires expertise in (Homological) Algebra, Algebraic and Complex Geometry, and in***Representation Theory, primarily applying and investigating homological methods.