Preface
The study of Hilbert schemes and vector bundles is a fundamental problem
in algebraic geometry. Their connections with physics and representation theory
were pioneered in the work of Penrose and Atiyah during the 1970s. They played
central roles in the Donaldson theory and the Seiberg-Witten theory from 1983
to 1995. From 1995, physicists working in string theory have speculated many
surprising but deep results concerning Hilbert schemes, vector bundles and their
interplay with representation theory. For instance, a mathematical version of the
S-duality conjecture formulated by C. Vafa and E. Witten revealed a beautiful
connection between stable vector bundles on algebraic surfaces and representations
of certain infinite-dimensional Lie algebras. In the case of rank one, this leads to an
elegant relation between the Hilbert schemes of points on algebraic surfaces and the
representations of the infinite-dimensional Heisenberg algebras. Another example
of recent interplay between vector bundles and representation theory motivated by
physics is the relation among principal bundles over elliptic Calabi-Yau manifolds,
representation of compact Lie groups and the physics F-theory. These recent physics
discoveries have been leading to intensive studies of and rapid advances in the theory
of Hilbert schemes, vector bundles and representation theory.
It
is under such a background that in April 2002, the Department of Math-
ematics at the University of Missouri hosted a conference in Columbia, Missouri
on Hilbert Schemes, Vector Bundles and Their Interplay with Representation The-
ory. The meeting brought together both senior and young researchers, including a
number of graduate students, in algebraic geometry and representation theory.
Main speakers at the conference were W.-P. Li (HKUST), E. Izadi (University
of Georgia), D. Morrison (Duke), K. Oguiso (University of Tokyo), J. Li (Stan-
ford), W. Wang (MSRI & University of Virginia), R. Friedman (Columbia Uni-
versity), and H. Nakajima (Kyoto University). There were nineteen short com-
munications presented by E. Gasparim (New Mexico State University), C.-H. Liu
(Harvard), T. Nevins (MSRI), A. Iarrobino (Northeastern), G. J. Pearlstein (UC-
Irvine), B. Kotzev (University of Missouri), Y. Kimiko (Kyoto University), D. Ar-
cara (University of Georgia), A. Vitter (Tulane), W. Li (Oklahoma State Uni-
versity), C. Madonna (Univ. Roma 2), A. Caldararu (U. Mass), A. Mavlyutov
(Indiana University), T. Luo (UT-Arlington), Y. Kachi (University of Tennessee),
E. Markman (U. Mass), B. Purnaprajna (University of Kansas), P. Rao (University
of Missouri-St. Louis), and X. Wu (University of South Carolina).
The meeting provided a forum for mathematicians in algebraic geometry and
representation theory to meet with colleagues and learn of recent research devel-
opments in the focused areas. It also gave graduate students and recent Ph.D's an
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