David Berman, 1978, U. of Texas. Currently David is Associate Professor in the Department of Mathematical Sciences at University of North Carolina at Wilmington.

Ph.D. related Articles:

David shows first the easier part, that given a Hilbert function $T$, of length N, then the monomial ideal having the first $T_i$ monomials of degree i as cobasis has the most number of generators. Then he shows that among these ideals, the one closest to a power of the maximal ideal (generators in two adjacent degrees) has the most number of generators. The first part has been significantly generalized to higher Betti numbers by A. Bigatti and H.A. Hulett, and in further directions by A. Aramova and J. Herzog, and others. .

David here developed an invariant of a vector space V of degree-j forms in a polynomial ring R, that includes all the information about dimensions of sequences U\in W(V) of vector spaces obtained from V by operations of R_i\cdot U (the ideal generated by U), and U:R_i (a partial saturation); shows that it is a finite invariant. This article promulgated the "persistence conjecture" that was seen and solved by G. Gotzmann.

David Berman's homepage

Abderrahim Miri, 1986, NU "Compressed Modules". Currently Abderrahim is Associate Professor of Mathematics at University of Rabat, Morocco. He has had several Ph.D. students, including A. Cherrabi.

Ph.D. related Article:

Abderrahim generalized the constructions of compressed algebras to modules, showing that there are some type one modules that cannot be "smoothed", and finding the dimensions of families of compressed modules of given socle type, rank, embedding dimension.

Susan J. Diesel: 1992 NU "Determinantal minors of catalecticant matrices". Currently Susan is working as a manager in mathematics related software development.

Ph.D. related Article:

Susan showed that the family GOR(T) parametrizing graded Gorenstein quotients A of R=k[x,y,z] having hilbert function H(A)=T, is irreducible. Uses the Buchsbaum-Eisenbud structure theorem for height three Gorensteins. The proof works by deforming from algebras having a higher number of generators to lower. The specialization diagram for Gor(T,D) (subfamily where the generator degrees D are fixed) is opposite to the inclusion diagram of the generator degrees. Also clarified the possible generator sets D given T.

More recently, J.O. Kleppe - after partial results from a number of groups - showed that in this height three case GOR(T) is smooth, which also implies the irreducibility result.

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