Preface
Quantization is an important topic in mathematics and physics. From the
physics point of view, methods of quantization are procedures for building models
for quantum mechanical systems from analogous and more intuitive classical me-
chanical systems, which provide strikingly precise experimental predictions. Much
of the development of theoretical physics in the 20th century may be regarded as
the process of refining quantization to give improved experimental predictions, and
the search for a unified field theory is an attempt to quantize general relativity
in a manner compatible with existing quantum theory. On the mathematics side,
problems related to quantization and quantum mechanics was a strong motivation
for the development of functional analysis, the representation theory of Lie groups,
and spectral geometry. More recent developments with much current activity in-
clude geometric quantization, deformation quantization, and quantum analogues of
various classical objects.
Geometric quantization seeks to give as natural as possible a procedure for as-
sociating a Hilbert space to each symplectic manifold satisfying an integrality con-
dition. Geometric quantization was developed by Kirillov, Kostant, and Souriau in
the 1960’s, and was energized by the attempt to prove the “quantization commutes
with reduction” conjecture of Guillemin and Sternberg in the 1980’s and 1990’s,
which was completed in part by work of Meinrenken, and is continued through the
study of its L2-analogues. Much work has been done on the semiclassical asymp-
totics of geometric quantization, such as the proof by Bordemann, Meinrenken, and
Schlichenmaier of a general asymptotic formula for Berezin–Toeplitz quantization
on compact ahler manifolds. Semiclassical analysis has resulted in applications in
number theory, as in the work of Borthwick and Uribe on relative Poincare series.
The theory of deformation quantization seeks to deform the commutative alge-
bra of functions on a Poisson manifold into a noncommutative algebra in which the
semi-classical limit is given by the Poisson bracket of functions. Kontsevich’s proof
of his formality conjecture showed that every Poisson manifold has a star product
on the formal level, and this work was one of the key results which earned him the
Fields medal. This work was later reinterpreted by Tamarkin and related to path
integrals by Cattaneo and Felder. Ideas from deformation quantization also play
a central role in recent work of Costello giving a rigorous geometric construction
of the Witten genus. Deformation quantization is used by Etingof and Ginzburg
to give a better geometric understanding of the rational Cherednik algebra and its
representations, as well as for other associative algebras.
The papers in this volume are based on talks given at the Center for Mathemat-
ics at Notre Dame program on quantization, which was held from May 31 to June
10 of 2011. The program consisted of a summer school on quantization, followed by
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