A. Iarrobino Ph.D. students:
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:
The number of generators of a colength $N$ ideal in a power series ring. J. Algebra
73 (1981), no. 1, 156--166.(reviewed in MR 83d:13002).
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. .
Simplicity of a vector space of forms: finiteness of the number of complete Hilbert
functions. J. Algebra 45 (1977), no. 1, 88--93.(reviewed in MR56 #8572).
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:
Compressed Gorenstein modules: Artin modules of type one having extremal
Hilbert functions, Comm. Algebra 21 (1993), no. 8, 2837--2857.(reviewed in MR94j:13019).
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:
Some irreducibility and dimension theorems for
families of height 3 Gorenstein algebras", Pacific J. Math. 172 (1996),
365--397.(reviewed in MR99f:13016).
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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