This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

Elements of Homotopy Theory

We define for a topological group G and a family of subgroups \( \mathcal{F} \) two versions for the classifying space for the family \( \mathcal{F} \) , the G-CW-version \( E_\mathcal{F} \) (G) and the numerable G-space version \( J_\mathcal{F} \) (G). They agree if G is discrete, or if G is a Lie group and each element in \( \mathcal{F} \) compact, or if \( \mathcal{F} \) is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C*-algebra, for the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.

Survey on Classifying Spaces for Families of Subgroups

We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G.

The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite-dimensional CAT(0)-space.

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The Borel Conjecture for hyperbolic and CAT(0)-groups

For a normal covering over a closed oriented topological manifold we give a proof of the L 2 -signature theorem with twisted coecients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C max-version of the Baum-Connes conjecture imply the L 2 -signature theorem for a normal covering over a Poincar e space, provided that the group of deck transformations is torsion-free. We discuss the various possible denitions of L 2 -signatures (using the signature operator, using the cap product of dierential forms, using a cap product in cellular L 2 -cohomology, . . . ) in this situation, and prove that they all coincide.

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Various L 2 -signatures and a topological L 2 -signature theorem

We give a survey on L 2 -invariants such as L 2 -Betti numbers and L 2 torsion taking an algebraic point of view. We discuss their basic denitions, properties and applications to problems arising in topology, geometry, group theory and K-theory.

L 2 -Invariants from the Algebraic Point of View

After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-G-set-version, the inverselimit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the zero-th equivariant stable cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant stable cohomotopy groups of finite proper equivariant CW-complexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological K-theory of the classifying space for proper actions.

https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2005/0001/0003/PAMQ-2005-0001-0003-a004.pdf

The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups

Let be a group together with a sequence of normal subgroups 1 2 ::: of nite index [ : k] such that T k k =f1g. Let (X;Y )

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Fachbereich Mathematik

Papers

The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory

Various L 2 -signatures and a topological L 2 -signature theorem

L 2 -Invariants from the Algebraic Point of View

The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups

Approximating L 2 -signatures by their compact analogues