Mathematical Sciences: Multiplicative Structures on Free Resolutions & Local Algebra

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The principal investigator will study three areas of commutative algebra. The first area is concerned with multiplicative structures on free resolutions. She will find a criterion for the existence of an associative commutative differential graded algebra structure on the minimal free resolution on cyclic modules. She will also determine if such a structure exists for the minimal resolutions of the cyclic modules defined by the ideals in the linkage class of a complete intersection. The second area is concerned with fundamental groups and equivalences of singularities. In particular, she will determine a number k such that for a given determinantal ideal I in a power series ring R and for any other ideal J of the same height, J equivalent to I modulo the kth power of the Jacobean ideal of I guarantees the existence of an isomorphism of R which maps I onto J so that J is also a determinantal ideal. The third area is to determine, in positive characteristics, if the symbolic algebra of a monomial prime is finitely generated. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.