Equivariant Orthogonal Spectra and $S$-Modules

Equivariant Orthogonal Spectra and $S$-Modules PDF Author: M. A. Mandell
Publisher: American Mathematical Soc.
ISBN: 082182936X
Category : Mathematics
Languages : en
Pages : 125

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Book Description
The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory.For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.

From Representation Theory to Homotopy Groups

From Representation Theory to Homotopy Groups PDF Author: Donald M. Davis
Publisher: American Mathematical Soc.
ISBN: 0821829874
Category : Mathematics
Languages : en
Pages : 65

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Book Description
A formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations has been obtained by Bousfield. This work applys this theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989.

Equivariant Analytic Localization of Group Representations

Equivariant Analytic Localization of Group Representations PDF Author: Laura Ann Smithies
Publisher: American Mathematical Soc.
ISBN: 0821827251
Category : Mathematics
Languages : en
Pages : 106

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Book Description
This book is intended for graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

The Submanifold Geometries Associated to Grassmannian Systems

The Submanifold Geometries Associated to Grassmannian Systems PDF Author: Martina Brück
Publisher: American Mathematical Soc.
ISBN: 0821827537
Category : Mathematics
Languages : en
Pages : 111

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Book Description
This work is intended for graduate students and research mathematicians interested in differential geometry and partial differential equations.

The $AB$ Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems

The $AB$ Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems PDF Author: Olivier Druet
Publisher: American Mathematical Soc.
ISBN: 0821829890
Category : Mathematics
Languages : en
Pages : 113

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Book Description
Function theory and Sobolev inequalities have been the target of investigation for many years. Sharp constants in these inequalities constitute a critical tool in geometric analysis. The $AB$ programme is concerned with sharp Sobolev inequalities on compact Riemannian manifolds. This text summarizes the results of contemporary research and gives an up-to-date report on the field.

Lie Algebras Graded by the Root Systems BC$_r$, $r\geq 2$

Lie Algebras Graded by the Root Systems BC$_r$, $r\geq 2$ PDF Author: Bruce Normansell Allison
Publisher: American Mathematical Soc.
ISBN: 0821828118
Category : Mathematics
Languages : en
Pages : 175

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Book Description
Introduction The $\mathfrak{g}$-module decomposition of a $\mathrm{BC}_r$-graded Lie algebra, $r\ge 3$ (excluding type $\mathrm{D}_3)$ Models for $\mathrm{BC}_r$-graded Lie algebras, $r\ge 3$ (excluding type $\mathrm{D}_3)$ The $\mathfrak{g}$-module decomposition of a $\mathrm{BC}_r$-graded Lie algebra with grading subalgebra of type $\mathrm{B}_2$, $\mathrm{C}_2$, $\mathrm{D}_2$, or $\mathrm{D}_3$ Central extensions, derivations and invariant forms Models of $\mathrm{BC}_r$-graded Lie algebras with grading subalgebra of type $\mathrm{B}_2$, $\mathrm{C}_2$, $\mathrm{D}_2$, or $\mathrm{D}_3$ Appendix: Peirce decompositions in structurable algebras References.

Homotopy Theory of Diagrams

Homotopy Theory of Diagrams PDF Author: Wojciech Chachólski
Publisher: American Mathematical Soc.
ISBN: 0821827596
Category : Mathematics
Languages : en
Pages : 106

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Book Description
In this paper the authors develop homotopy theoretical methods for studying diagrams. In particular they explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept introduced is that of a model approximation. A model approximation of a category $\mathcal{C}$ with a given class of weak equivalences is a model category $\mathcal{M}$ together with a pair of adjoint functors $\mathcal{M} \rightleftarrows \mathcal{C}$ which satisfy certain properties. The key result says that if $\mathcal{C}$ admits a model approximation then so does the functor category $Fun(I, \mathcal{C})$.

On the Foundations of Nonlinear Generalized Functions I and II

On the Foundations of Nonlinear Generalized Functions I and II PDF Author: Michael Grosser
Publisher: American Mathematical Soc.
ISBN: 0821827294
Category : Mathematics
Languages : en
Pages : 113

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Book Description
In part 1 of this title the authors construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given.

Almost Commuting Elements in Compact Lie Groups

Almost Commuting Elements in Compact Lie Groups PDF Author: Armand Borel
Publisher: American Mathematical Soc.
ISBN: 0821827928
Category : Mathematics
Languages : en
Pages : 153

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Book Description
This text describes the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in the extended Dynkin diagram of the simply connected cover, together with the co-root integers and the action of the fundamental group. In the case of three commuting elements, we compute Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons invariants and the dimensions of the components of the moduli space.

Representation Theory and Complex Analysis

Representation Theory and Complex Analysis PDF Author: Michael Cowling
Publisher: Springer Science & Business Media
ISBN: 3540768912
Category : Mathematics
Languages : en
Pages : 400

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Book Description
Six leading experts lecture on a wide spectrum of recent results on the subject of the title. They present a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces, and recall the concept of amenability. They further illustrate how representation theory is related to quantum computing; and much more. Taken together, this volume provides both a solid reference and deep insights on current research activity.