The Mathematics of F. J. Almgren Jr

Frederick Justin Almgren Jr., one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, began his graduate work at Brown in 1958. It was a very exciting place and time for geometric measure theory. Wendell Fleming had just arrived and begun his collaboration with Herbert Federer, leading to their seminal paper “Normal and Integral Currents” in 1960 [19]. Among the major results in “Normal and Integral Currents” was a compactness theorem which implied existence of k-dimensional rectifiable area minimizing varieties with prescribed boundaries in Rn. (Similar existence theorems were proved independently by Reifenberg and, in case n = k + 1, by De Giorgi.) Shortly afterward, Fleming (using earlier work of Reifenberg) proved that if k = 2 and n = 3, then the varieties are in fact smooth surfaces. Meanwhile De Giorgi (and subsequently, by a different argument, Reifenberg) did work implying that the varieties were smooth almost everywhere when n = k + 1, the case of hypersurfaces. Fred came to Brown with an unusual background. He had just spent three years as a Navy pilot, and before that his undergraduate degree at Princeton had been not in mathematics but in engineering. In fact, as an undergraduate he took only three math courses: two semesters of honors calculus and one semester of differential equations and infinite series. Later he would jokingly accuse various mathematicians at Brown of calling him “the most ignorant person” that they had ever met. “It was clear that he had great raw talent and good intuition,” says Federer, “but indeed he knew very little mathematics then. There were even basic things in group theory, for example, that he had never heard of. When he asked me to be his advisor, I suggested the problem I did because he didn’t know enough analysis for most problems in geometric measure theory.” Federer suggested a problem that was as much topological as measure theoretic. Four years earlier Dold and Thom had shown that there was a natural isomorphism between the homology groups of a compact manifold M and the homotopy groups of their associated symmetric product spaces. Federer realized that this could be interpreted as a statement about the homotopy groups of the space of 0-dimensional integral cycles of M. He conjectured a generalization to kdimensional cycles, namely, that the mth homotopy group of the space of integral k-cycles in M is naturally isomorphic to the (m + k)th homology group of M. Almgren’s thesis, published in the first volume of the journal Topology [2], proved that this was indeed the case. Oddly enough, its publication caused trouble. Almgren was supposed to sign the copyright of his thesis over to Brown University, but could not since he had already signed it over to Topology. To the dean that meant Almgren could not graduate. Brian White, the eighth of Fred Almgren’s doctoral students, received his Ph.D. in 1982. After a two-year postdoctoral fellowship at the Courant Institute in New York he went to Stanford University, where he is a professor of mathematics. His e-mail address is white@math. stanford.edu.