Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups

Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups
Author :
Publisher : American Mathematical Soc.
Total Pages : 154
Release :
ISBN-10 : 9780821840764
ISBN-13 : 0821840762
Rating : 4/5 (64 Downloads)

Synopsis Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups by : John Rognes

The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Galois Extensions of Structured Ring Spectra

Galois Extensions of Structured Ring Spectra
Author :
Publisher : American Mathematical Society(RI)
Total Pages : 137
Release :
ISBN-10 : 1470405040
ISBN-13 : 9781470405045
Rating : 4/5 (40 Downloads)

Synopsis Galois Extensions of Structured Ring Spectra by : John Rognes

Galois Extensions of Structured Ring Spectra: Abstract Introduction Galois extensions in algebra Closed categories of structured module spectra Galois extensions in topology Examples of Galois extensions Dualizability and alternate characterizations Galois theory I Pro-Galois extensions and the Amitsur complex Separable and etale extensions Mapping spaces of commutative $S$-algebras Galois theory II Hopf-Galois extensions in topology References Stably Dualizable Groups: Abstract Introduction The dualizing spectrum Duality theory Computations Norm and transfer maps References Index.

Geometric and Topological Aspects of the Representation Theory of Finite Groups

Geometric and Topological Aspects of the Representation Theory of Finite Groups
Author :
Publisher : Springer
Total Pages : 493
Release :
ISBN-10 : 9783319940335
ISBN-13 : 3319940333
Rating : 4/5 (35 Downloads)

Synopsis Geometric and Topological Aspects of the Representation Theory of Finite Groups by : Jon F. Carlson

These proceedings comprise two workshops celebrating the accomplishments of David J. Benson on the occasion of his sixtieth birthday. The papers presented at the meetings were representative of the many mathematical subjects he has worked on, with an emphasis on group prepresentations and cohomology. The first workshop was titled "Groups, Representations, and Cohomology" and held from June 22 to June 27, 2015 at Sabhal Mòr Ostaig on the Isle of Skye, Scotland. The second was a combination of a summer school and workshop on the subject of "Geometric Methods in the Representation Theory of Finite Groups" and took place at the Pacific Institute for the Mathematical Sciences at the University of British Columbia in Vancouver from July 27 to August 5, 2016. The contents of the volume include a composite of both summer school material and workshop-derived survey articles on geometric and topological aspects of the representation theory of finite groups. The mission of the annually sponsored Summer Schools is to train and draw new students, and help Ph.D students transition to independent research.

Handbook of Homotopy Theory

Handbook of Homotopy Theory
Author :
Publisher : CRC Press
Total Pages : 1142
Release :
ISBN-10 : 9781351251600
ISBN-13 : 1351251600
Rating : 4/5 (00 Downloads)

Synopsis Handbook of Homotopy Theory by : Haynes Miller

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings
Author :
Publisher : American Mathematical Soc.
Total Pages : 218
Release :
ISBN-10 : 9780821840917
ISBN-13 : 0821840916
Rating : 4/5 (17 Downloads)

Synopsis Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings by : Wolfgang Bertram

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations

The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations
Author :
Publisher : American Mathematical Soc.
Total Pages : 120
Release :
ISBN-10 : 9780821842508
ISBN-13 : 0821842501
Rating : 4/5 (08 Downloads)

Synopsis The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations by : Salah-Eldin Mohammed

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations and stochastic partial differential equations near stationary solutions.

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces
Author :
Publisher : American Mathematical Soc.
Total Pages : 86
Release :
ISBN-10 : 9780821841365
ISBN-13 : 082184136X
Rating : 4/5 (65 Downloads)

Synopsis Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces by : William Mark Goldman

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$.

Abstract" Homomorphisms of Split Kac-Moody Groups"

Abstract
Author :
Publisher : American Mathematical Soc.
Total Pages : 108
Release :
ISBN-10 : 9780821842584
ISBN-13 : 0821842587
Rating : 4/5 (84 Downloads)

Synopsis Abstract" Homomorphisms of Split Kac-Moody Groups" by : Pierre-Emmanuel Caprace

This work is devoted to the isomorphism problem for split Kac-Moody groups over arbitrary fields. This problem turns out to be a special case of a more general problem, which consists in determining homomorphisms of isotropic semisimple algebraic groups to Kac-Moody groups, whose image is bounded. Since Kac-Moody groups possess natural actions on twin buildings, and since their bounded subgroups can be characterized by fixed point properties for these actions, the latter is actually a rigidity problem for algebraic group actions on twin buildings. The author establishes some partial rigidity results, which we use to prove an isomorphism theorem for Kac-Moody groups over arbitrary fields of cardinality at least $4$. In particular, he obtains a detailed description of automorphisms of Kac-Moody groups. This provides a complete understanding of the structure of the automorphism group of Kac-Moody groups over ground fields of characteristic $0$. The same arguments allow to treat unitary forms of complex Kac-Moody groups. In particular, the author shows that the Hausdorff topology that these groups carry is an invariant of the abstract group structure. Finally, the author proves the non-existence of cocentral homomorphisms of Kac-Moody groups of indefinite type over infinite fields with finite-dimensional target. This provides a partial solution to the linearity problem for Kac-Moody groups.

Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds

Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds
Author :
Publisher : American Mathematical Soc.
Total Pages : 150
Release :
ISBN-10 : 9780821841488
ISBN-13 : 0821841483
Rating : 4/5 (88 Downloads)

Synopsis Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds by : Raphael Ponge

This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, including CR and contact manifolds. In this context the main differential operators at stake include the Hormander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian, the CR conformal operators of Gover-Graham and the contact Laplacian. These operators cannot be elliptic and the relevant pseudodifferential calculus to study them is provided by the Heisenberg calculus of Beals-Greiner andTaylor.