Cycles, Transfers, and Motivic Homology Theories

by Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander, Annals of Mathematics Studies, No. 143,

Princeton University Press, NJ, 254 pages, ISBN 0-691-04815-0

When thinking about this book, three questions come to mind: What are mo-

tives? What is motivic cohomology? How does this book fit into these frameworks?

Let me begin with a very brief answer to these questions. The theory of motives

is a branch of algebraic geometry dealing with algebraic varieties over a fixed field

k. The basic idea is simple: enlarge the category of varieties into

one which is abelian, meaning that it resembles the category of abelian groups: we

should be able to add morphisms, take kernels and cokernels of maps, etc. The

objects in this enlarged category are to be called motives, whence the notion of the

motive associated to an algebraic variety.

Any reasonable cohomology theory on varieties should factor through this cate-

gory of motives. Even better, the abelian nature of motives allows us to do homo-

logical algebra. In particular, we can use Ext groups to form a universal cohomology

theory for varieties. Not only does this motivate our interest in motives, but the

universal property also gives rise to the play on words "motivic cohomology".

Different ways of thinking about varieties leads to different classes of motives

and different aspects of the theory. And each aspect of the theory quickly leads us

into conjectural territory.

The best understood part of the theory is the abelian category of pure motives.

This is what we get by restricting our attention to smooth projective varieties, iden-

tifying numerically equivalent maps and taking coefficients in the rational numbers

Q. The pure motives of smooth projective varieties are the analogues of semisimple

modules over a finite-dimensional Q-algebra. The theory of pure motives is related

to many deep unsolved problems in algebraic geometry, via what are known as the

standard conjectures.

At the other extreme, we have the most general part of the theory of motives,

obtained by considering all varieties and taking coefficients in the integers Z...

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The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles.

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On the Scalar Curvature of Einstein Manifolds

Some applications of differential topology in general relativity

Gravity coupled with matter and foundation of non-commutative geometry

Aspherical gravitational monopoles

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Eigenvalue Estimates for the Dirac Operator on Quaternionic Kaehler Manifolds

Quaternionic Killing Spinors

Gravity from Dirac Eigenvalues

On index formulas for manifolds with metric horns.

Kodaira Dimension and the Yamabe Problem

An Introduction to Noncommutative Spaces and their Geometry

Duality Symmetries and Noncommutative Geometry of String Spacetime

A-hat Genus and Collapsing

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Moduli Spaces of Singular Yamabe Metrics

Constant scalar curvature metrics with isolated singularities

Gluing and moduli for noncompact geometric problems

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Inhomogeneous electronic structure probed by spin-echo experiments
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Imaging phase separation near the Mott boundary

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Dependent Magnetoresistance of the Layered Organic Superconductor

\kappa-(ET)2Cu(NCS)2: Simulation and Experiment

Constraints on Microscopic Theories of Superconductivity in Layered Organic Superconductors

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Physics' Greatest Puzzles 'Millennium Madness'

Dr. David Gross, a theoretical physicist at UCSB

#1. Are all the (measurable) dimensionless parameters that characterize the

physical Universe, calculable in principle, or are some merely determined by

historical or quantum mechanical accident and uncalculable?

#2. How can quantum gravity help explain the origin of the Universe?

#3. What is the lifetime of the proton and how do we understand it?

#4. Is Nature supersymmetric, and if so, how is supersymmetry broken?

#5. Why does the Universe appear to have one time and three space dimensions?

#6. Why does the cosmological constant have the value that it has?

Is it zero and is it really constant?

#7. What are the fundamental degrees of freedom of M Theory (the theory whose

low-energy limit is eleven-dimensional supergravity and that subsumes the

five consistent superstring theories) and does the theory describe nature?

#8. What is the resolution of the black hole information paradox?

#9. What physics explains the enormous disparity between the gravitational scale

and the typical mass scale of the elementary particles?

#10. Can we quantitatively understand quark and gluon confinement in quantum

chromodynamics and the existence of a mass gap?

Projects: View
ISR Space Elevator Animation
Ion Propulsion Plasma Propulsion

Russian and Asian Language Translators

Updated 10 December 2005 at Scientific Frontiers Extended







Physics of Superconductors December 2005