Homological Properties of Commutative Rings and Applications

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In this project, several questions in commutative algebra will be studied. They are concrete statements about homological properties of rings and modules, but have unexpected and intriguing connections to other areas of mathematics. One part of the project proposes to investigate the vanishing behavior of Ext and Tor functors over Noetherian rings. Better understanding of such behavior can be used to attack a diverse set of problems such as Gabber 's conjecture about Picard groups of punctured spectra of complete intersections, or the existence of non-commutative resolution of singularities. Other topics include classification of subcategories of maximal Cohen-Macaulay modules, solving equations in the semi-ring of vector bundles on algebraic varieties and regularity of Stanley-Reisner ideals associated to simplicial complexes. Commutative algebra deals with some of the most ubiquitous objects in mathematics, such as sets of equations in several variables. It has both natural and deep connections to fields such as algebraic and arithmetic geometry, number theory, representation theory and combinatorics. Such characteristics allow one to phrase many interesting questions at the front of modern research in the form accessible to even early graduate students. Many topics proposed in this project are about such connections, and their investigation are especially suitable for training and collaboration between young researchers.