Bott as a teacher

Bott's lectures are legendary for their seeming ease of comprehension. His style is typically the antithesis of the Definition-Theorem-Proof approach so favored among mathematical speakers. Usually he likes to discuss a simple key example that encapsulates the essence of the problem. Often, as if by magic, a concrete formula with transparent significance appears.

At a reception for new graduate students at Harvard, he once shared his view of the process of writing a Ph.D. thesis. He said it is like doing a homework problem; it's just a harder problem. You try to understand the problem thoroughly, from every conceivable angle. Much of the thesis work is perseverance, as opposed to inspiration. Above all, ``make the problem your own.''

Many of his students testify to his warmth and humanity, but he also expects the students to meet an exacting standard. He once banned the word ``basically'' from an advisee's vocabulary, because that word to Bott signifies that some details are about to be swept under a rug.

This insistence on thoroughness and clarity applies to his own work as well. I. M. Singer remarked that in their younger days, whenever they had a mathematical discussion, the most common phrase Bott uttered was ``I don't understand,'' and that a few months later Bott would emerge with a beautiful paper on precisely the subject he had repeatedly not understood.

Seminar speakers at Harvard tend to address themselves to the experts in the audience. But like Steenrod, Bott often interrupts the speakers with the most basic questions, with the salutary effect of slowing down the speakers and making them more intelligible to lesser mortals.

At Michigan and Harvard, Bott directed over 36 Ph.D. theses. Some of his students have become luminaries in their own right: Stephen Smale and Daniel Quillen received the Fields Medal in 1966 and 1978 respectively, and Robert MacPherson the National Academy of Science Award in Mathematics in 1992. The following is, I hope, the complete list of his Ph.D. students:

Year         Ph.D. Student        Dissertation Title 




1957 		 Smale, Stephen 		 $\parbox{2.5in}{\raggedright Regular Curves on Riemannian Manifolds}$ 


1961 		 Edwards, Harold 		 $\parbox{2.5in}{\raggedright Application of Intersection Theory to Boundary Value Problems}$ 


Curtis, Edward 		 $\parbox{2.5in}{\raggedright The Lower Central Series for Free Group Complexes}$


1963 		 Conlon, Lawrence 		 $\parbox{2.5in}{\raggedright Spaces of Paths on a Symmetric Space}$ 


Zilber, Joseph Abraham 		 $\parbox{2.5in}{\raggedright Categories in Homotopy Theory}$


1964 		 Holzsager, Richard Allan 		  $\parbox{2.5in}{\raggedrightClassification of Certain Types of Spaces}$


Quillen, Daniel 		 $\parbox{2.5in}{\raggedright Formal Properties of Over-Determined Systems Of Linear Partial Differential Equations}$


1965 		 Landweber, Peter S. 		  $\parbox{2.5in}{\raggedrightK\uml unneth Formulas for Bordism Theories}$


Lazarov, Connor 		 $\parbox{2.5in}{\raggedright Secondary Characteristic Classes in $K$-theory}$ 


1969 		 Brooks, Morris William 		 $\parbox{2.5in}{\raggedright The Cohomology of the Complement of a Submanifold}$ 


Brown, Richard Lawrence 		 $\parbox{2.5in}{\raggedright Cobordism Embeddings and Fibrations of Manifolds}$


1970 		 Blass, Andreas R. 		 Orderings of Ultrafilters


MacPherson, Robert D. 		 $\parbox{2.5in}{\raggedright Singularities of Maps and Characteristic Classes}$


1973 		 Miller, Edward Y. 		 $\parbox{2.5in}{\raggedright Local Isomorphisms of RiemannianHermitian and Combinatorial Manifolds}$


1974 		 Garberson, John Dayton 		 $\parbox{2.5in}{\raggedright The Cohomology of Certain Algebraic Varieties}$


1975 		 Mostow, Mark 		 $\parbox{2.5in}{\raggedright Continuous Cohomology of Spaces with Two Topologies}$


Perchik, James 		  $\parbox{2.5in}{\raggedrightCohomology of Hamiltonian and Related Formal Vector Field Lie Algebras}$


1976 		 Weiss, Richard Simon 		  $\parbox{2.5in}{\raggedrightRefined Chern Classes for Foliations}$


1977 		 Brooks, Robert 		 $\parbox{2.5in}{\raggedright On the Smooth Cohomology of Groups ofDiffeomorphisms}$ 


1981 		 Hingston, Nancy 		  $\parbox{2.5in}{\raggedrightEquivariant Morse Theory and Closed Geodesics}$ 


1982 		 Gunther, Nicholas 		 $\parbox{2.5in}{\raggedright Hamiltonian Mechanics and Optimal Control}$ 


Laquer, Turner Henry 		 $\parbox{2.5in}{\raggedright Homogeneous Connections and Yang-MillsTheory on Homogeneous Spaces}$ 


1984 		 Ticciati, Robin 		 $\parbox{2.5in}{\raggedright Singular Points in Moduli Spaces of Yang-Mills Fields}$ 


1985 		 Forman, Robin 		 $\parbox{2.5in}{\raggedright Functional Determinants and Applications to Geometry}$


1986 		 Corlette, Kevin 		 $\parbox{2.5in}{\raggedright Stability and Canonical Metrics in Infinite Dimensions}$


1987 		 Block, Jonathan 		 $\parbox{2.5in}{\raggedright Excision in Cyclic Homology of Topological Algebras}$ 


1989 		 Kocherlakota, Rama 		 $\parbox{2.5in}{\raggedright Integral Homology of Real Flag Manifolds and Loop Spaces of Symmetric Spaces}$ 


Morelli, Robert 		 $\parbox{2.5in}{\raggedright Hilbert's Third Problem and the $K$-Theory of Toric Varieties}$


Bressler, Paul 		  $\parbox{2.5in}{\raggedrightSchubert Calculus in Generalized Cohomology}$ 


1991 		 Grossberg, Michael 		 $\parbox{2.5in}{\raggedright Complete Integrability and Geometrically Induced Representations}$ 


1992 		 Weinstein, Eric 		 $\parbox{2.5in}{\raggedright Extension of Self-Dual Yang-Mills Equations across the Eighth Dimension}$ 


Szenes, Andras 		 $\parbox{2.5in}{\raggedright The Verlinde Formulas and Moduli Spaces of Vector Bundles}$ 


1993 		 Tolman, Susan 		  $\parbox{2.5in}{\raggedrightGroup Actions and Cohomology}$ 


1994 		 Teleman, Constantine 		 $\parbox{2.5in}{\raggedright Lie Algebra Cohomology and the Fusion Rules}$


Costes, Constantine 		 $\parbox{2.5in}{\raggedright Some Explicit Cocycles for Cohomology Classes Of Group Diffeomorphisms Preserving a $G$-Structure}$


2000 		 Bernhard, James 		 $\parbox{2.5in}{\raggedright Equivariant de Rham Theory and Stationary Phase Expansions}$