Bott later went to college at
McGill University in
Montreal, where he studied
electrical engineering. He then earned a
PhD in mathematics from
Carnegie Mellon University in
Pittsburgh in 1949. His thesis, titled
Electrical Network Theory, was written under the direction of
Richard Duffin. Afterward, he began teaching at the
University of Michigan in
Ann Arbor. Bott continued his study at the
Institute for Advanced Study in Princeton. He was a professor at
Harvard University from 1959 to 1999. In 2005 Bott died of
cancer in
San Diego. With
Richard Duffin at Carnegie Mellon, Bott studied existence of
electronic filters corresponding to given
positive-real functions. In 1949 they proved a fundamental theorem of
filter synthesis. Duffin and Bott extended earlier work by
Otto Brune that requisite functions of
complex frequency s could be realized by a
passive network of
inductors and
capacitors. The proof relied on
induction on the sum of the
degrees of the polynomials in the numerator and denominator of the rational function. In his 2000 interview with Allyn Jackson of the
American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with
network synthesis and
electronic filter topology introduced him to
algebraic topology. Bott met
Arnold S. Shapiro at the IAS and they worked together. He studied the
homotopy theory of
Lie groups, using methods from
Morse theory, leading to the
Bott periodicity theorem (1957). In the course of this work, he introduced
Morse–Bott functions, an important generalization of
Morse functions. This led to his role as collaborator over many years with
Michael Atiyah, initially via the part played by periodicity in
K-theory. Bott made important contributions towards the
index theorem, especially in formulating related
fixed-point theorems, in particular the so-called '
Woods Hole fixed-point theorem', a combination of the
Riemann–Roch theorem and
Lefschetz fixed-point theorem (it is named after
Woods Hole, Massachusetts, the site of a conference at which collective discussion formulated it). The major Atiyah–Bott papers on what is now the
Atiyah–Bott fixed-point theorem were written in the years up to 1968; they collaborated further in recovering in contemporary language
Ivan Petrovsky on
Petrovsky lacunas of
hyperbolic partial differential equations, prompted by
Lars Gårding. In the 1980s, Atiyah and Bott investigated
gauge theory, using the
Yang–Mills equations on a
Riemann surface to obtain topological information about the
moduli spaces of stable bundles on Riemann surfaces. In 1983 he spoke to the Canadian Mathematical Society in a talk he called "A topologist marvels at Physics". He is also well known in connection with the
Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic
sheaves and their cohomology groups; and for work on
foliations. With
Chern he worked on
Nevanlinna theory, studied
holomorphic vector bundles over
complex analytic manifolds and introduced the Bott-Chern classes, useful in the theory of
Arakelov geometry and also to
algebraic number theory. He introduced
Bott–Samelson varieties and the
Bott residue formula for complex manifolds and the
Bott cannibalistic class. ==Awards==