# Floer Homology, Gauge Theory, and Low-Dimensional Topology

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*David A. Ellwood; Peter S. Ozsváth; András I. Stipsicz; Zoltán Szabó*

A co-publication of the AMS and Clay Mathematics Institute

Mathematical gauge theory studies connections
on principal bundles, or, more precisely, the solution spaces of
certain partial differential equations for such
connections. Historically, these equations have come from mathematical
physics, and play an important role in the description of the
electro-weak and strong nuclear forces. The use of gauge theory as a
tool for studying topological properties of four-manifolds was
pioneered by the fundamental work of Simon Donaldson in the early
1980s, and was revolutionized by the introduction of the
Seiberg–Witten equations in the mid-1990s. Since the birth of
the subject, it has retained its close connection with symplectic
topology. The analogy between these two fields of study was further
underscored by Andreas Floer's construction of an infinite-dimensional
variant of Morse theory that applies in two a priori different
contexts: either to define symplectic invariants for pairs of
Lagrangian submanifolds of a symplectic manifold, or to define
topological invariants for three-manifolds, which fit into a framework
for calculating invariants for smooth four-manifolds. “Heegaard
Floer homology”, the recently-discovered invariant for three-
and four-manifolds, comes from an application of Lagrangian Floer
homology to spaces associated to Heegaard diagrams. Although this
theory is conjecturally isomorphic to Seiberg–Witten theory, it
is more topological and combinatorial in flavor and thus easier to
work with in certain contexts. The interaction between gauge theory,
low-dimensional topology, and symplectic geometry has led to a number
of striking new developments in these fields. The aim of this volume
is to introduce graduate students and researchers in other fields to
some of these exciting developments, with a special emphasis on the
very fruitful interplay between disciplines.

This volume is based on lecture courses and advanced seminars given at the
2004 Clay Mathematics Institute Summer School at the Alfréd
Rényi Institute of Mathematics in Budapest, Hungary. Several of the authors have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art
introduction to current research, covering material from Heegaard Floer
homology, contact geometry, smooth four-manifold topology, and symplectic
four-manifolds.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in low dimensional, contact and symplectic topology, and gauge theory.