Syzygies

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This project is in the field of commutative algebra. The main research goal is to understand the structure of minimal free resolutions over complete intersection ring. The idea is to describe the asymptotic structure of the generic Eisenbud operators, and then use it in order to obtain the differentials. The proposed research also includes a topic involving interdisciplinary approaches connecting commutative algebra with the fields of combinatorics and topology. The PI will co-organize two conferences in commutative algebra, which will provide a forum for discussion of recent developments and foster research in new directions.Research on free resolutions is a core and beautiful area in commutative algebra. It contains a number of challenging and important conjectures and open problems. The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in two famous papers in 1890 and 1893. He proved that over a polynomial ring (over a field) every finitely generated module has a finite resolution. In the local and the graded cases there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution of the resolved module. Its properties are closely related to the invariants of the resolved module. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra; they also have applications in other mathematical fields.