Herb Clemens

Aaron Bertram, James A. Carlson, Holger Kley

Herb Clemens received his A.B. in mathematics from Holy Cross College in

Worcester, Massachusetts in 1961 and his Ph.D. from the University of California

at Berkeley in 1966 under the direction of Phillip Griffiths. In his thesis, Herb

gave a proof of the monodromy theorem using direct topological methods. Shortly

thereafter, he left to work for the Peace Corps in Chile, where he spent the years

1966-68 helping

to establish a master's program in mathematics at the Universidad

Tecnica del Estado in Santiago. During the next few years Herb held appointments

at the Institute for Advanced Study in Princeton, in Chile as a Senior Fulbright

Lecturer, and at Columbia University, where he remained until he came to the

University of Utah in 1975. He received the University of Utah's Distinguished

Research Award in 1983, and was named Distinguished Professor at the University

of Utah in March of 2000.

It was shortly after his return from Chile that Clemens and Griffiths began

working on the classical question of whether the cubic threefold is rational. In land-

mark work published in 1972 they answered the question in the negative, thereby

disproving Liiroth's conjecture: that a subfield of a rational function field is ratio-

nal. While other disproofs of Liiroth's conjecture were obtained about the same

time (Iskovskih-Manin 1971, Artin-Mumford 1972), the Clemens-Griffiths work was

especially influential because of the deep insights it provided into the geometry and

Hodge theory of cubic threefolds. Chief among these is a gemstone in the classi-

cal theory of three-dimensional algebraic varieties: the fact that a smooth cubic

threefold is determined by its intermediate Jacobian.

In 1983 Herb published, after years of persistent work, the surprising result that

the Griffiths group, which measures algebraic modulo homological equivalence, can

be non finitely generated. Especially beautiful was the specificity of the theorem:

non finite generation holds for the generic quintic threefold. Both this paper and

the work on cubic threefolds were the subject of invited addresses at the Interna-

tional Congress of Mathematicians- in Vancouver (1974) and in Berkeley (1986).

Another milestone was his 1986 paper which showed that a generic quintic sur-

face carries no rational curves. In recent years the focus of Clemens' work has

been to fashion higher-order methods with which to attack deformation problems

in algebraic geometry.

Herb's influence on mathematics extends far beyond his published research

papers. First, there are his twelve formal Ph.D. students to date, listed below.

Second, there were Herb's informal students, such as Enrico Arbarello and Arnaud

Beauville, for whom he was an important mentor and life-long friend. And finally

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