Geometry.Net - Pure_And_Applied_Math: Homological Algebra

Homological Algebra And Category Theory
70), 1999 2000 UNDERGRADUATE CALENDAR. Carleton University. Mathematics70.418*. homological algebra and Category Theory. Axioms of
http://www.carleton.ca/cu9900uc/course/70/418.html

Extractions: 1999-2000 Undergraduate Calendar Course Offerings Mathematics Honours(70) 1999 - 2000 UNDERGRADUATE CALENDAR Carleton University Axioms of set theory; categories, functors, natural transformations; free, projective, injective and flat modules; tensor products and homology functors, derived functors; dimension theory. Also offered at the graduate level, with additional or different requirements, as Mathematics 70.518, for which additional credit is precluded.
The Theory Of Everything
Category Theory and homological algebra. Clarendon Press,Oxford 1992;Gelfand, SI and Manin, YIhomological algebra Springer, 2000;
http://www.mmsysgrp.com/QIS/category.htm

Extractions: 'Categorification' is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called `coherence laws'. Iterating this process requires a theory of `n-categories', algebraic structures having objects, morphisms between objects, 2-morphisms between morphisms and so on up to n-morphisms. After a brief introduction to n-categories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. John C. Baez (University of California, Riverside): From Finite Sets to Feynman Diagrams This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' and `state', and similarly those of `spacetime' and `process', which we argue is to be expected in any background-free quantum theory. We sketch how higher-dimensional algebra provides the mathematical tools to make this link precise. Finally, we comment on attempts to formulate a theory of quantum gravity in 4-dimensional spacetime using `spin networks' and `spin foams'.
A Digression - Homological Algebra
Construction and Properties Previous Construction and Properties ContentsA Digression homological algebra. Rob Thompson 2000-02-08.
http://math.hunter.cuny.edu/~rthompso/homotopy/node22.html
Alphamusic - Basic Homological
Translate this page Freitag, den 07. Februar 2003. Osborne, M. Scott Basic homological algebraBuch Springer-Verlag Telos Bestell-Nr. 0-387-98934-X 54.65 EUR. 384.
http://www.alphamusic-shop.de/340/038798934x.html

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18Gxx
18Gxx homological algebra. 18G20 Homological dimension, See also {13D05,16E10}; 18G25 Relative homological algebra, projective classes;
http://www.ma.hw.ac.uk/~chris/MR/18Gxx.html
Tutorial:Homological Algebra 2
Tutorialhomological algebra 2. section{Problems in computational homologicalalgebra}. \section{Basic concepts}. \prob{1}{Modules and homomorphisms}.
http://www.theory.cs.ritsumei.ac.jp/~takayama/M2/815.html

Extractions: To determine a method of implementing modules and homomorphisms: how we should represent these objects. To determine how to compute the kernel of a homomorphism or $R$-modules, given that we know how to compute the kernel of a matrix of free modules. To construct the mapping cone of a map of chain complexes. To construct a free resolution (not usually minimal) of an ideal generated by monomials in a polynomial ring. We use successive mapping cones to construct the resolution. To construct a minimal free resolution of a so-called ``stable'' monomial ideal. The method is almost identical to the construction of the Taylor resolution. The horseshoe resolution is used to construct the connecting homomorphism for any left-exact, or contravariant right exact functor, e.g. Tor and Ext. We apply the technique to a specific example i1 : R = ZZ/101[quote a..quote f]
Some Selected Publications
Some Selected Publications. lambe@math.su.se. Perturbation Theory in Differentialhomological algebra I, with VKAM Gugenheim, IL. J. Math., 33 (1989), 566582.
http://www.matematik.su.se/~lambe/public/pubs.html

Extractions: (Note that all files are "gzipped". Need gzip? Click on gzip.org and scroll down to "Executables") Perturbation Theory in Differential Homological Algebra I, with V.K.A.M. Gugenheim, IL. J. Math., 33 (1989), 566-582. ps or pdf Keywords: distinguished resolution; twisting cochains; free resolutions over the group ring of two-stage nilpotent groups; homological perturbation; differential tor Perturbation Theory in Differential Homological Algebra II, with V.K.A.M. Gugenheim and Jim Stasheff, IL. J. Math., 35 (1991), 359-373. ps or pdf Keywords: homological perturbation theory; twisting cochain; differential Tor; differential complexes representing a given chain homotopy type; coalgebra; SDR-data; coalgebra homotopy; coderivation; tensor coalgebra; bar construction differential; perturbation lemma
Encyclopædia Britannica
transformations; Universal constructions; The duality principle;Conclusions. homological algebra Homology groups of groups; Derived
http://www.britannica.com/eb/article?eu=120643
Algebra And Algebraic Topology Home Page
School of Informatics, Algebra and Algebraic Topology.Category Science Math Topology Research Groups At Bangor our interest in homological algebra continues in this sametradition. New ALGEBRAIC TOPOLOGY AND homological algebra. A
http://www.informatics.bangor.ac.uk/public/mathematics/research/algtop/algtop2.h

Extractions: Introduction. The research in algebra at Bangor has to a large extent been motivated by problems in algebraic topology and homological algebra. The recent spate of new and exciting concepts (crossed modules, crossed n-cubes, nonabelian tensor products, etc.) originating in those two areas has opened out many algebraic aspects of the theory and applications which are waiting to be investigated. Many final year pure mathematics courses have some Algebraic Topology in them. Typically the fundamental group and/or the homology groups are defined, studied and applied to various problems such as the existence of certain types of continuous maps, the classification of knots, and other problems of classification usually in low dimensions. These problems usually involve two basic questions: In both cases the method of algebraic topology is to model certain important aspects of each space by some algebraic gadget, perhaps a group or a set of interrelated groups, use these to translate the problem to an algebraic context; try to solve that algebraic problem and finally to reinterpret the results back in terms of spaces.
Mathlinks.info - Abstract Algebra - 2
homological algebra. Articles / Courses / Lectures / Texts / Tutorials;A Course in homological algebra Notes (L Lady - University of Hawaii);
http://www.mathlinks.info/em002_abstract_algebra2.htm

Extractions: Abstract Algebra: The Basic Graduate Year Downloadable Textbook (R Ash - University of Illinois at Urbana-Champaign) Abstract Algebra Course Notes (D Wilkins -Trinity College) Abstract Algebra On-Line Course Materials (J Beachey - Northern Illinois University) Algebra Topics Articles (MathPages) Elements of Abstract and Linear Algebra Downloadable Textbook (E Connell - University of Miami) Graduate Course in Algebra Course Materials (L Lady - University of Hawaii) Introduction to Abstract Algebra Course Notes (P Garrett - University of Minnesota) Algebraic Areas of Mathematics Pages (D Rusin - The Mathematical Atlas) Algebraic Systems Pages (C Blomqvist) Catalogue of Algebraic Systems Pages (E Clark - University of South Florida) Group Theory and Algebra in Physics Resources (The Net Advance of Physics) Home Page of J. S. Milne. Pages Linear Algebraic Groups and Related Structures Preprint Server (University of Bielefeld) Permutation Loops Article (MathPages: Algebra) Quasi-Groups Article (MathPages: Algebra) 3-Element Groupoids Page (J Berman and S Burris - University of Waterloo) Classical Groups Lecture Notes (P Cameron - University of London) The Development of Group Theory Article (MacTutor History of Mathematics Archives - University of St. Andrews)

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