POWRE: Differentials, Singularities and Applications

University Of North Texas, Denton TX

Investigators

Abstract

This POWRE award supports a Visiting Professorship at the Department of Mathematics at Northeastern University to study differentials of isolated hypersurface singularities. This award will allow the PI to interact with several leading researchers in her field. As part of the human resource development activities she will also sponsor the local graduate student seminar at Northeastern. This research project combines theoretical expertise from cyclic homology and K-theory and computer algebra techniques to gather information about invariants of isolated hypersurface singularities. The most salient invariant of isolated hypersurface singularities is the so-called Tjurina number, or the dimension of the torsion module of differentials. She has identified the torsion module of differentials both as a Hodge-component of cyclic homology and as an ideal quotient. The latter identification has led to an efficient algorithm for computation of the number of generators exploiting Matlis duality in Gorenstein Artin Algebras. As part of her research activities the PI will undertake the following three-part project: 1. The structure of isolated singularities: The anticipated outcome of this research will be an upper bound for the number of generators and the length of the module of differentials for 3 dimensional hypersurface singularities. 2. Residues and Duality for isolated hypersurface singularities: The goal will be an explicit description of residues and an investigation of the connection with inverse systems. 3. Description of the module of logarithmic differentials and connection with hyperplane/surface arrangements: This part of the project will be computational in nature and will focus on the use and development of algorithms for Groebner basis computations in exterior algebras. This POWRE project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).