Description of research

Note: numbers in braces refer to publication list.

Dr. Hoffman's graduate training was concentrated in algebraic topology and related fields. He became interested in certain questions about fixed-point-free self-maps of manifolds, and this led to a study of cohomology endomorphisms of complex flag manifolds. His thesis on this topic (written under the supervision of F. Peterson) was the basis of two papers [2,3] that appeared in 1984. After one year at Ohio State University, he took a year's leave of absence to accept a visiting research position at the Memorial University of Newfoundland. There he collaborated with W. Homer and S. A. Broughton in research on complex flag manifolds [1,4] while continuing his research on fixed-point-free self-maps [5]. He essentially resolved all his original questions about the latter topic a few years after coming to the Naval Academy, and his results appear in a 1989 paper [10] in the Pacific Journal of Mathematics. He also established several results on homological restrictions on free group actions [7,9]. He has continued to study group-theoretic implications of some of his work on cohomology endomorphisms.

Dr. Hoffman is recognized as an expert on flag manifolds, and his work in this area has been cited in a number of papers. These include that of H. Glover and G. Mislin on Grassmann manifolds, that of H. Shiga and M. Tezuka on cohomology automorphisms of homogeneous spaces, those of S. Papadima on rigidity properties of homogeneous spaces, and that of Duan Haibo and E. Rees on non-focal embeddings. His work on group actions has been cited by L. W. Cusick and J. F. Davis.

In 1986-88 Dr. Hoffman collaborated with his colleague W. D. Withers, a dynamicist, in the investigation of certain polynomials that arise from affine Weyl groups. This resulted in two papers published in 1988: an expository article Withers wrote for the American Mathematical Monthly and a joint paper [8] in the Transactions of the AMS.

In 1988, Dr. Hoffman's colleague C. Moen brought to his attention a conjecture (the "sum conjecture") involving certain multiple series. In studying these series, Dr. Hoffman soon formulated a second conjecture (the "duality conjecture") and obtained partial proofs of both conjectures, which he published in the Pacific Journal in 1992 [11]. Moen had proved the first previously unknown case of the sum conjecture by a lengthy argument; in 1995 Dr. Hoffman found a short proof of this case by combining a result of his own with one from a 1994 paper of C. Markett. A joint note on this result [13] appeared in the Journal of Number Theory; subsequently the sum conjecture was proved in full generality by A. Granville and D. Zagier (independently).

In fact, there is currently considerable interest in these series, which have been called "multiple zeta values" by D. Zagier. They appear in several areas of active research, including the work of M. Kontsevich on knot invariants, that of V. G. Drinfeld and others on quantum groups, and that of D. J. Broadhurst on perturbative quantum field theory. Kontsevich's work provided a representation of multiple zeta values as iterated integrals, giving an easy proof of the duality conjecture. In the summer of 1996, Dr. Hoffman initiated an algebraic approach to the study of multiple zeta values. His main result provides an explicit set of values irreducible with respect to algebraic operations (though not with respect to other relations involving limiting processes). His approach also clarifies the relation between the algebra of multiple zeta values and the (well-known) algebra of symmetric functions. His paper detailing this work [14] appeared in the Journal of Algebra. Some aspects of the problem suggested a general Hopf algebra construction, which was the subject of a later paper [17]. In 1999, Dr. Hoffman conjectured a new identity for multiple zeta values. This "cyclic sum conjecture" was proved the following year by Y. Ohno. The identity and various related algebraic questions are explored in their joint paper [20]. Dr. Hoffman also discovered an occurrence of multiple zeta values in mirror symmetry [19].

In [18], Dr. Hoffman described a theory of "covering spaces" for partially ordered sets (posets) in which each covering relation is assigned a weight. One example of such a poset is the set of rooted trees, which serve as generators for the Hopf algebra defined by D. Kreimer in connection with renormalization in quantum field theory. Dr. Hoffman developed these ideas further in [21], where he gave an isomorphism between the dual of Kreimer's Hopf algebra and the Hopf algebra of rooted trees defined earlier by R. Grossman and R. G. Larson.

In a 1995 article in the American Mathematical Monthly, [12], Dr. Hoffman studied "derivative polynomials" (polynomials arising from repeated differentiation of the tangent and secant functions) and their relation to certain improper integrals and infinite series. He developed these results further and related derivative polynomials to Springer numbers and other combinatorial sequences, as well as to Euler polynomials, in a later article for the Electronic Journal of Combinatorics [16].

Two brief expository writings of Dr. Hoffman which appeared in the American Mathematical Monthly have been cited in the literature. His solution to an advanced problem there was cited in a 1990 paper by the distinguished algebraist I. Kaplansky, and his note on finite abelian groups [6] was used by J. M. Burns and B. Goldsmith to classify maximal abelian subgroups of the symmetric group and to compute the digraph number of a finite abelian group. A note on curvature appeared in the January 1998 Monthly [15].

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