Category:Geometry - Factbites

Category:Geometry - Factbites
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Topic: Category:Geometry

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In the News (Tue 29 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	The Math Forum - Math Library - Topology
		An list of suggestions for projects in geometry, topology, symmetry, making geometric solids, calendars, spherical and hyperbolic trigonometry, puzzles, models, etc. Projects were to be exhibited at the Geometry Fair at the end of the course.
		Thus it is a kind of generalized geometry (we are still interested in spheres and cubes, for example, but we might consider them to be "the same", yet distinct from a bicycle tire, which has a "hole") or a kind of generalized analysis...
		Clocks and Curvature (Geometry and the Imagination) - Conway, Doyle, Gilman, Thurston; The Geometry Center
	www.mathforum.org /library/topics/topology

	Descent (category theory)
		From the point of view of category theory the work of comonad s of Beck was a summation of those ideas.
		The reason for abstraction here is, at a fundamental level, that passage to a quotient space is not very well-behaved in topology: more accurately, it is a tribute to the efforts to use category theory to get round the alleged 'brutality' of imposing equivalence relations within geometric categories.
		The difficulties of algebraic geometry with passage to the quotient are acute: it is like doing the non-commutative geometry of Connes, to mention the currently-fashionable theory in the area of 'bad quotients', but with polynomials to separate points, rather than general continuous functions.
	www.worldhistory.com /wiki/D/Descent-(category-theory).htm

	sci_eng_edu_resource.list
		Contact: The contact name is Julian Smith or Gordon Baker jsmith@epas.utoronto.ca gbaker@epas.utoronto.ca (416) 978-5047 Electronic Journal on Virtual Culture Listserver UserLevel/Category: Teachers and Students Brief: Periodical which has articles concerned with the new culture created by electronic media.
		Contact: Owner: Deborah Economidis: DMartin@wcu.bitnet or DelValnet Intern: DMartin@mainvm.wcupa.edu Educational Center for Earth Observational Systems West Chester University SCHOOLNET Listserver User Level/Category: Teachers Brief: The SCHOOLNET Listserver is a forum for discussion regarding educational networking in Canada, the Canadian Educational Networking Coalition, and thoughts and ideas on the SchoolNet initiative.
		Contact: If you have any comments, please direct them to the author Katalin Harkanyi harkanyi@ucssun1.sdsu.edu CHEMED-L: Chemistry Education Listserver UserLevel/Category: Teacher Brief: Chemistry Education Discussion Mail List Map: Send a message to LISTSERV@UWF.BITNET, no subject.
	www.eff.org /Net_culture/Net_info/Resources/sci_eng_edu_resource.list

	Scheme (mathematics)
		Around 1942 Oscar Zariski had defined an abstract Zariski space from the function field of an algebraic variety, for the needs of birational geometry : this is like a direct limit of ordinary varieties (under 'blowing up'), and the construction, reminiscent of locale theory, used valuation rings as points.
		Recent ideas about higher algebraic stacks and derived algebraic geometry promise to further expand the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to algebraic topology and homotopy theory.
		Grothendieck and Dieudonné studied the category of all schemes, and Grothendieck's student Pierre Deligne later wrote that admitting bizarre schemes made the whole category of schemes much nicer.
	www.sciencedaily.com /encyclopedia/scheme__mathematics_

	Stefan Muller-Stach
		algebra over the category of structured spectra), we are developing a kind of algebraic geometry in homotopical contexts (like for example in the category C(k) of complexes of k-modules).
		This could also be seen as a first step in a more general theory of algebraic geometry over monoidal infinity categories; as an instance of this we compare and unify the previously described approaches using Simpson's Segal categories.
		When the model category has a compatible monoidal structure we also define, following Simpson, a notion of geometric or algebraic stack on it.
	www.informatik.uni-mainz.de /~stefan/homotopy.html (616 words)

	Stefan Muller-Stach
		The idea is to mimic the usual construction of Grothendieck topologies, sheaves, schemes and stacks in algebraic geometry but taking properly into account the additional homotopical informations (like the notion of quasi-isomorphisms in the category C(k) above).
		This could also be seen as a first step in a more general theory of algebraic geometry over monoidal infinity categories; as an instance of this we compare and unify the previously described approaches using Simpson's Segal categories.
		Using the language and techniques of abstract homotopical algebra due to Quillen, Bousfield, Kan and others, we define pre-stacks over a model category, topologies on model categories and the associated categories of stacks.
	www.mathematik.uni-mainz.de /~stefan/homotopy.html (616 words)

	abstralggeo
		The opposite of a Zariski category is a strict spatial analytic geometry, whose analytic topology coincides with the Zariski topology defined by Diers.
		(h) The opposite of the category of commutative rings is a Zariski geometry; its analytic topology is the Zariski topology.
		(g) The category of Hausdorff spaces is a strict reduced disjunctable spatial analytic geometry; its analytic topology is the Hausdorff topology.
	www.mta.ca /~cat-dist/catlist/1999/abstralggeo (616 words)

	The Math Forum - Math Library - Topology
		A multi-purpose center for electronic distribution of information related to topology, the mathematical study of surfaces, sometimes called "rubber sheet geometry" because topologists consider geometric figures as though they were drawn on infinitely stretchable rubber sheets.
		An list of suggestions for projects in geometry, topology, symmetry, making geometric solids, calendars, spherical and hyperbolic trigonometry, puzzles, models, etc. Projects were to be exhibited at the Geometry Fair at the end of the course.
		A short article designed to provide an introduction to general topology, the study of sets on which one has a notion of "closeness" - enough to decide which functions defined on it are continuous.
	mathforum.org /library/topics/topology (616 words)

	University of Toronto -- Events@UofT -- Event Listing
		For example, interpreting correlators of QFT as Massey products in appropriate triangulated category of geometric nature, we can rigorously compute them using finite DG-resolutions presented by algebraic geometry.
		Stability data in a triangulated category (t-stability for shortness) generalize the concept of stability coming from geometric invariant theory and provide the category with a functorial filtration.
		Speaker: Alexei Gorodentsev, ITEP, Moscow - Abstract: Triangulated categories provide us with common framework for linking algebraic geometry, differential geometry and field theories via homological algebra.
	www.events.utoronto.ca /event.asp?ID=10632 (616 words)

	Descent (category theory) - Wikipedia, the free encyclopedia
		From the point of view of category theory the work of comonads of Beck was a summation of those ideas.
		The reason for abstraction here is, at a fundamental level, that passage to a quotient space is not very well-behaved in topology: more accurately, it is a tribute to the efforts to use category theory to get round the alleged 'brutality' of imposing equivalence relations within geometric categories.
		The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.
	en.wikipedia.org /wiki/Descent_(category_theory) (616 words)

	Real Algebraic and Analytic Geometry
		The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension).
		It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets.
		We then prove a result on uniform embeddings of analytic subsets of S-manifolds into a projective space, which extends theorems of Campana ([1]) and Fujiki ([6]) on compact complex manifolds.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG/preprints/0139.html (616 words)

	agmod-web.txt
		Furtherm* ore, as relative algebraic geometry* has found interesting applications in the T* annakian formalism (see [De1 ]), it should not be surprising that algebraic geometr y over the 1-category* of complexes is relevant to higher Tannakian theory.
		This has led to the theory of relative* * algebraic ge- ometry, which allows one to do algebraic geometry over well behaved symmetric m* onoidal base categories (see [De1, De2, Ha ]); usual algebraic geometry* corresponds th* en to the ä bsolute" case where the base category* is the category of Z-modules.
		Therefore, algebraic geometry is a theory which is based on the two funda* *mental notions of affine scheme and Grothendieck topology.
	hopf.math.purdue.edu /Toen-Vezzosi/agmod-web.txt (616 words)

	The Magma Philosophy
		Magma is a Computer Algebra system designed to solve problems in algebra, number theory, geometry and combinatorics that may involve sophisticated mathematics and which are computationally hard.
		The kernel of Magma contains implementations of many of the important concrete classes of structure in five fundamental branches of algebra, namely group theory, ring theory, field theory, module theory and the theory of algebras.
		Most of the major algorithms currently installed in the Magma kernel are state-of-the-art and give performance similar to, or better than, specialized programs.
	magma.maths.usyd.edu.au /magma/Features/node2.html (616 words)

	week83
		The idea of algebraic geometry is that we can study a space by studying the functions on that space --- which typically form some kind of commutative algebra.
		Hopefully lots of you know that Connes is the wizard of operator theory who turned to inventing a new branch of geometry, "noncommutative geometry".
		If you now steal a peek at "week79", you'll see that these two equations are just the same equations used to define adjoint functors in category theory!
	math.ucr.edu /home/baez/week83.html (1752 words)

	abstralggeo
		(h) The opposite of the category of commutative rings is a Zariski geometry; its analytic topology is the Zariski topology.
		The opposite of a Zariski category is a strict spatial analytic geometry, whose analytic topology coincides with the Zariski topology defined by Diers.
		(g) The category of Hausdorff spaces is a strict reduced disjunctable spatial analytic geometry; its analytic topology is the Hausdorff topology.
	www.mta.ca /~cat-dist/catlist/1999/abstralggeo (1752 words)

	The MathsLinker : préprints servers
		Algebraic Geometry ; Algebraic Topology ; Analysis of PDEs ; Category Theory ; Classical Analysis ;
		Quantum Algebra ; Representation Theory ; Rings and Algebras ; Scientific Computation ; Spectral Theory ; Symplectic Geometry
		Metric Geometry ; Number Theory ; Numerical Analysis ; Operator Algebras ; Optimization and Control ; Probability Theory ;
	mathslinker.chez.tiscali.fr /R_pre.htm (1752 words)

	MathGuide - Simple Search
		Field theory and polynomials; Commutative rings and algebras; Algebraic geometry; Linear and multilinear algebra, matrix theory; Associative rings and algebras; Nonassociative rings and algebras; Category theory, homological algebra
		Commutative rings and algebras; Nonassociative rings and algebras; Algebraic geometry; Combinatorics; Algebraic topology
		Commutative rings and algebras; Differential geometry; Functional analysis; Mathematical logic and foundations
	www.mathguide.de /cgi-bin/ssgfi/suche.pl?db=math&tag=SUC&words=13-XX&sort=&dsp=minitemp&COL=SUB (1752 words)

	topos
		In mathematics, a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it.
		The historical origin of topos theory is algebraic geometry.
		There is one major class of examples of topoi that wasn't listed in the introduction: if C is a small category, then the functor category Set
	www.fact-library.com /topos.html (624 words)

	Ring theory - The Encyclopedia
		Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.
		Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings.
		Following the model of algebraic geometry, attempts have been made recently at defining non-commutative geometry based on non-commutative rings.
	www.the-encyclopedia.com /description/Ring_theory (624 words)

	Open Questions: Geometry and Topology
		This was a "top down" or "wholistic" view of geometry, in that it did not seek to analyze geometric objects in terms of their constituent parts (such as points or lines).
		The maps establishing equivalence between differentiable manifolds are called diffeomorphisms, and the category is known as the category of differentiable manifolds, or alternatively, smooth manifolds.
		A vast generalization of traditional geometry which is concerned with studying various types of algebras that may be noncommutative.
	www.openquestions.com /oq-ma003.htm (624 words)

	LICS2001 Full Abstraction/Completeness Workshop
		An extension of these categories, introduced by the author, called unique decomposition categories (UDC) have been very successful in modelling and axiomatising the "particle-style" Geometry of Interaction semantics.
		These models live in the co-Kleisli categories of the linear categories of PERs over suitable Linear (Affine) Combinatory Algebras, which arise in the context of Geometry of Interaction.
		The theorem says that every dinatural transformation of definable functors over the double-gluing category GHCoh arises from the denotation of a cut-free MALL proof.
	aix1.uottawa.ca /~scpsg/Logic/LICS01 (1124 words)

	Xiuping Su's home page
		the representation theory of quivers, Hall algebras and quantum groups, geometry of representations, geometry of derived categories, preprojective algebras and cluster algebras.
		Singularities in derived categories( with Bernt Tore Jensen), preprint 12.
		Degeneration for derived categories( with Bernt Tore Jensen and Alexander Zimmermann), Journal of Pure and Applied Algebra 198(2005), 281-295.
	www.lamfa.u-picardie.fr /xiupingsu (1124 words)

	Dual (category theory) -- Facts, Info, and Encyclopedia article
		A ((geometry) the interchangeability of the roles of points and planes in the theorems of projective geometry) duality between categories C and D is defined as an (Essential equality and interchangeability) equivalence between C and the opposite of D.
		Hence, the dual of a dual of a category is itself.
		The category of (Click link for more info and facts about Stone space) Stone spaces and (Click link for more info and facts about continuous function) continuous functions is equivalent to the opposite of the category of (A system of symbolic logic devised by George Boole; used in computers) Boolean algebras and homomorphisms.
	www.absoluteastronomy.com /encyclopedia/d/du/dual_(category_theory).htm (452 words)

	Xiuping Su's home page
		the representation theory of quivers, Hall algebras and quantum groups, geometry of representations, geometry of derived categories, preprojective algebras and cluster algebras.
		Singularities in derived categories( with Bernt Tore Jensen), preprint 12.
		Degeneration for derived categories( with Bernt Tore Jensen and Alexander Zimmermann), Journal of Pure and Applied Algebra 198(2005), 281-295.
	www.lamfa.u-picardie.fr /xiupingsu (452 words)

	Logic Research, Department of Mathematics, Univ. of Manchester, UK
		Attached to the Ziegler spectrum is a localisation theory and corresponding sheaf theory, which may be regarded as a kind of non-commutative geometry attached to any category of modules (and to more general categories, arising from algebra, geometry and topology, which share some properties with categories of modules).
		A key structure which was discovered during the model-theoretic investigation of modules is a topological space, the Ziegler spectrum, which is associated with a category of modules.
		A particular attraction of this area is the interaction of ideas from various parts of mathematics, namely algebra, logic (in the form of model theory) and (abelian) category theory.
	www.ma.man.ac.uk /DeptWeb/Groups/Logic/ModelTheory.html (620 words)

	Geometry.Net - Pure_And_Applied_Math: Grothendieck Topology
		See list of category theory topics for a breakdown of the relevant Encyclopedia pages.
		This tool is used in algebraic number theory and algebraic geometry, mainly to define étale cohomology of schemes, but also for flat cohomology and crystalline cohomology.
		This tool is used in algebraic number theory and algebraic geometry schemess, but also for flat cohomology and crystalline cohomology.
	www4.geometry.net /pure_and_applied_math/grothendieck_topology.html (620 words)

	Open Questions: Geometry and Topology
		This was a "top down" or "wholistic" view of geometry, in that it did not seek to analyze geometric objects in terms of their constituent parts (such as points or lines).
		The maps establishing equivalence between differentiable manifolds are called diffeomorphisms, and the category is known as the category of differentiable manifolds, or alternatively, smooth manifolds.
		A vast generalization of traditional geometry which is concerned with studying various types of algebras that may be noncommutative.
	www.openquestions.com /oq-ma003.htm (14549 words)

	Untitled Document
		Mark Levine explained the significance of Grothendieck’s notion of a motive in algebraic geometry, of the search for Deligne’s category of mixed motives and of the significance of the recent construction of the derived category of mixed motives by Levine, Hanamura and others, explaining how it fits in with the motivic cohomology of Voevodsky-Suslin-Friedlander.
		The accompanying Spitalfields Day consisted of three lectures designed to show how algebraic K-theory impinges on group cohomology, number theory and algebraic geometry.
		The case of the general linear group is particularly closely related to algebraic K-theory where such a comparison isomorphism, due originally to Suslin and Jardine independently, states that the mod p Quillen algebraic K-theory of the complex numbers and the topological K-theory coincide.
	www.lms.ac.uk /newsletter/329/329_02.html (14549 words)

	Mathematics Archives - Topics in Mathematics - Abstract Algebra
		Course Notes, Group Theory, Fields and Galois Theory, Algebraic Geometry, Algebraic Number Theory, Modular Functions and Modular Forms, Elliptic Curves, Abelian Varieties, Lectures on Etale Cohomology, Class Field Theory, Preprints
		Graduate courses on Group Theory, Fields and Galois Theory, Algebraic Number Theory, Class Field Theory, Modular Functions and Modular Forms, Elliptic Curves, Algebraic Geometry, Lectures on Etale Cohomology, Abelian Varieties
		Elementary Number Theory, Lucas' Theorem, Pascal's triangle via cellular automata, Bernoulli numbers and polynomials, Theorems of Morley and Emma Lehmer and their generalizations, Some useful p-adic numbers
	archives.math.utk.edu /topics/abstractAlgebra.html (1342 words)

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Topic: Category:Geometry

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In the News (Tue 29 Dec 09)