The $K$-book

The $K$-book
Author: Charles A. Weibel
Publisher: American Mathematical Soc.
Total Pages: 634
Release: 2013-06-13
Genre: Mathematics
ISBN: 0821891324

Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr

My 'k' Sound Box

My 'k' Sound Box
Author: Jane Belk Moncure
Publisher:
Total Pages: 0
Release: 2018-08
Genre: Alphabet
ISBN: 9781503831360

Little k has an adventure with items beginning with his letter's sound, such as kittens, kingbirds, a koala, and kids in a kindergarten. Additional features to aid comprehension include rebus icons for word recognition, a word list for review, activities for further learning, a note to parents and educators, and an introduction to the author and illustrator.

The Brothers K

The Brothers K
Author: David James Duncan
Publisher: Dial Press
Total Pages: 654
Release: 2010-07-28
Genre: Fiction
ISBN: 030775524X

A NEW YORK TIMES NOTABLE BOOK Once in a great while a writer comes along who can truly capture the drama and passion of the life of a family. David James Duncan, author of the novel The River Why and the collection River Teeth, is just such a writer. And in The Brothers K he tells a story both striking and in its originality and poignant in its universality. This touching, uplifting novel spans decades of loyalty, anger, regret, and love in the lives of the Chance family. A father whose dreams of glory on a baseball field are shattered by a mill accident. A mother who clings obsessively to religion as a ward against the darkest hour of her past. Four brothers who come of age during the seismic upheavals of the sixties and who each choose their own way to deal with what the world has become. By turns uproariously funny and deeply moving, and beautifully written throughout, The Brothers K is one of the finest chronicles of our lives in many years. Praise for The Brothers K “The pages of The Brothers K sparkle.”—The New York Times Book Review “Duncan is a wonderfully engaging writer.”—Los Angeles Times “This ambitious book succeeds on almost every level and every page.”—USA Today “Duncan’s prose is a blend of lyrical rhapsody, sassy hyperbole and all-American vernacular.”—San Francisco Chronicle “The Brothers K affords the . . . deep pleasures of novels that exhaustively create, and alter, complex worlds. . . . One always senses an enthusiastic and abundantly talented and versatile writer at work.”—The Washington Post Book World “Duncan . . . tells the larger story of an entire popular culture struggling to redefine itself—something he does with the comic excitement and depth of feeling one expects from Tom Robbins.”—Chicago Tribune

K-theory

K-theory
Author: Michael Atiyah
Publisher: CRC Press
Total Pages: 181
Release: 2018-03-05
Genre: Mathematics
ISBN: 0429973179

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.

The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions

The Poset of $k$-Shapes and Branching Rules for $k$-Schur Functions
Author: Thomas Lam
Publisher: American Mathematical Soc.
Total Pages: 113
Release: 2013-04-22
Genre: Mathematics
ISBN: 082187294X

The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.

Certain Notions of Single-Valued Neutrosophic K-Algebras

Certain Notions of Single-Valued Neutrosophic K-Algebras
Author: Muhammad Akram
Publisher: Infinite Study
Total Pages: 15
Release:
Genre: Mathematics
ISBN:

We apply the notion of single-valued neutrosophic sets to K-algebras. We develop the concept of single-valued neutrosophic K-subalgebras, and present some of their properties. Moreover, we study the behavior of single-valued neutrosophic K-subalgebras under homomorphism.

Algebraic K-theory: The Homotopy Approach Of Quillen And An Approach From Commutative Algebra

Algebraic K-theory: The Homotopy Approach Of Quillen And An Approach From Commutative Algebra
Author: Satya Mandal
Publisher: World Scientific
Total Pages: 680
Release: 2023-06-22
Genre: Mathematics
ISBN: 9811269408

In this book the author takes a pedagogic approach to Algebraic K-theory. He tried to find the shortest route possible, with complete details, to arrive at the homotopy approach of Quillen [Q] to Algebraic K-theory, with a simple goal to produce a self-contained and comprehensive pedagogic document in Algebraic K-theory, that is accessible to upper level graduate students. That is precisely what this book faithfully executes and achieves.The contents of this book can be divided into three parts — (1) The main body (Chapters 2-8), (2) Epilogue Chapters (Chapters 9, 10, 11) and (3) the Background and preliminaries (Chapters A, B, C, 1). The main body deals with Quillen's definition of K-theory and the K-theory of schemes. Chapters 2, 3, 5, 6, and 7 provide expositions of the paper of Quillen [Q], and chapter 4 is on agreement of Classical K-theory and Quillen K-theory. Chapter 8 is an exposition of the work of Swan [Sw1] on K-theory of quadrics.The Epilogue chapters can be viewed as a natural progression of Quillen's work and methods. These represent significant benchmarks and include Waldhausen K-theory, Negative K-theory, Hermitian K-theory, 𝕂-theory spectra, Grothendieck-Witt theory spectra, Triangulated categories, Nori-Homotopy and its relationships with Chow-Witt obstructions for projective modules. In most cases, the proofs are improvisation of methods of Quillen [Q].The background, preliminaries and tools needed in chapters 2-11, are developed in chapters A on Category Theory and Exact Categories, B on Homotopy, C on CW Complexes, and 1 on Simplicial Sets.

Algebraic K-Groups as Galois Modules

Algebraic K-Groups as Galois Modules
Author: Victor P. Snaith
Publisher: Springer Science & Business Media
Total Pages: 332
Release: 2002-03-01
Genre: Mathematics
ISBN: 9783764367176

This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993. The course was one of four associated with the 1993-94 Fields Institute programme, which I helped to organise, entitled "Artin L-functions". Published as [132]' the final chapter of the course introduced a manner in which to construct class-group valued invariants from Galois actions on the algebraic K-groups, in dimensions two and three, of number rings. These invariants were inspired by the analogous Chin burg invariants of [34], which correspond to dimensions zero and one. The classical Chinburg invariants measure the Galois structure of classical objects such as units in rings of algebraic integers. However, at the "Galois Module Structure" workshop in February 1994, discussions about my invariant (0,1 (L/ K, 3) in the notation of Chapter 5) after my lecture revealed that a number of other higher-dimensional co homological and motivic invariants of a similar nature were beginning to surface in the work of several authors. Encouraged by this trend and convinced that K-theory is the archetypical motivic cohomology theory, I gratefully took the opportunity of collaboration on computing and generalizing these K-theoretic invariants. These generalizations took several forms - local and global, for example - as I followed part of number theory and the prevalent trends in the "Galois Module Structure" arithmetic geometry.

Topological Triviality and Versality for Subgroups $A$ and $K$

Topological Triviality and Versality for Subgroups $A$ and $K$
Author: James Damon
Publisher: American Mathematical Soc.
Total Pages: 121
Release: 1988
Genre: Mathematics
ISBN: 082182452X

In this paper we shall prove two theorems which together allow the infinitesimal methods of Thom and Mather in singularity theory to be applied to problems of topological equivalence of mappings.