ABELIAN VARIETIES
in Fourier-Mukai Transforms in Algebraic Geometry
April 2006; published online September 2007.
Chapter. Subjects: geometry. 18902 words.
Historically, Mukai's equivalence with the Poincare bundle on the product of an abelian variety and its dual as kernel was the fist Fourier-Mukai transform. The first section in this chapter...
Algebraic and Geometric Surgery
September 2002; published online September 2007.
Book. Subjects: geometry. 386 pages.
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic...
Almost Contact and Contact Geometry
in Sasakian Geometry
October 2007; published online January 2008.
Chapter. Subjects: geometry. 17163 words.
This chapter presents the necessary foundational material on almost contact, contact, and metric contact structures. The key result is an orbifold version of the famous Boothby-Wang fibration. An...
Applications of Stochastic Geometry in Image Analysis
in New Perspectives in Stochastic Geometry
November 2009; published online February 2010.
Chapter. Subjects: geometry. 11154 words.
A discussion is given of various stochastic geometry models (random fields, sequential object processes, polygonal field models) which can be used in intermediate‐ and high‐level image analysis. Two...
Branes on Poisson Varieties
in The Many Facets of Geometry
July 2010; published online September 2010.
Chapter. Subjects: geometry. 13285 words.
This chapter takes a second look at a classical structure in differential and algebraic geometry — that of a holomorphic Poisson structure — which is a complex manifold with a holomorphic Poisson...
BUNDLES
in Algebraic and Geometric Surgery
September 2002; published online September 2007.
Chapter. Subjects: geometry. 17268 words.
This chapter brings together the basic properties of fibre bundles, fibrations, and vector bundles required in surgery theory. The tangent and normal bundles of a manifold are introduced. The...
Classical Stochastic Geometry
in New Perspectives in Stochastic Geometry
November 2009; published online February 2010.
Chapter. Subjects: geometry. 21902 words.
The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect...
COBORDISM THEORY
in Algebraic and Geometric Surgery
September 2002; published online September 2007.
Chapter. Subjects: geometry. 7558 words.
Two manifolds are cobordant if, and only if, one can be transformed to the other by a sequence of surgeries. This chapter discusses the Pontrjagin-Thom transversality isomorphism between the...
Confined divisors
in Flips for 3-folds and 4-folds
June 2007; published online September 2007.
Chapter. Subjects: geometry. 6002 words.
Shokurov proposed the CCS conjecture as an intermediate step leading to his finite generation conjecture implying the existence of pl flips in all dimensions. After the work of Hacon and McKernan,...
Consistent Orientation of Moduli Spaces
in The Many Facets of Geometry
July 2010; published online September 2010.
Chapter. Subjects: geometry. 11915 words.
In a series of papers by Freed, Hopkins, and Teleman (2003, 2005, 2007a) the relationship between positive energy representations of the loop group of a compact Lie group G and the twisted...