41. Re: Category Theory the characteristic p case and other such theories For example, instead of justhaving a set theoretic topology, one uses a grothendieck topology defined in http://www.dcs.gla.ac.uk/mail-www/haskell/msg00975.html

[Prev] [Next] [Index] [Thread] Re: category theory From haberg@matematik.su.se Subject : Re: category theory Date : Thu, 15 Oct 1998 16:42:08 +0200 To haskell@dcs.gla.ac.uk mailto:haberg@member.ams.org http://www.matematik.su.se/~haberg/ http://www.ams.org/cml/ References Re: category theory From: Prev: Re: category theory Next: Re: category theory Index(es): Main Thread Post to "haskell": haskell@dcs.gla.ac.uk

42. Rigid Analytic Geometry And Its Applications 15, (7). 2.3 The residue theorem. 22, (3). 2.4 The grothendieck topologyon P. 25, (5). 2.5 Some sheaves on P. 30, (3). 2.6 Analytic subspaces ofP. 33, (4). http://www.booksmatter.com/b0817642064.htm

Home Search Browse Shopping Cart ... Help QuickSearch (Words, Author, Subject, ISBN) Publishers Email a friend about this book Rigid Analytic Geometry and Its Applications Format Hardcover Subject Mathematics / Geometry / Algebraic ISBN/SKU Author Jean Fresnel Publisher Springer Verlag Publish Date December 2003 Price Qualified Frequent Buyer Price Add to cart Ships from our store in 7 - 15 business days More delivery info here Table of Contents Preface ix 1 Valued Fields and Normed Spaces 1.1 Valued fields 1.2 Banach spaces and Banach algebras 2 The Projective Line 2.1 Some definitions 2.2 Holomorphic functions on an affinoid subset 2.3 The residue theorem 2.4 The Grothendieck topology on P 2.5 Some sheaves on P 2.6 Analytic subspaces of P 2.7 Cohomology on an analytic subspace of P 3 Affinoid Algebras 3.1 Definition of an affinoid algebra 3.2 Consequences of the Weierstrass theorem 3.3 Affinoid spaces, Examples 3.4 Properties of the spectral (semi-)norm 3.5 Integral extensions of affinoid algebras 3.7 Products of affinoid spaces, Picard groups 4 Rigid Spaces 4.1 Rational subsets

43. From E-prints@e-math.ams.org Fri Jun 5 124855 1998 Date Fri, 5 can be applied to define a canonical functor from an analytic category to the categoryof locales, which is a special type of grothendieck topology (ie framed http://felix.unife.it/Root/d-Mathematics/d-Algebraic-geometry/t-Categories-in-al

From e-prints@e-math.ams.org Fri Jun 5 12:48:55 1998 Date: Fri, 5 Jun 1998 01:00:33 -0400 From: AMSPPS

44. Re: Replacing Principal And Fiber Bundles With Sheaves? OK, so maybe this is just for the expository introduction, but the next section,I m hit with schemes, etale topology, grothendieck topology, Gtorsor, etc. http://www.lns.cornell.edu/spr/2003-10/msg0055006.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Replacing principal and fiber bundles with sheaves? Subject : Re: Replacing principal and fiber bundles with sheaves? From Date : 12 Oct 2003 23:44:07 -0400 Approved : baez@math.removethis.ucr.andthis.edu (s.p.research moderator) In-Reply-To abergman-3FD650.18011210102003@localhost Message-ID bmarbk$4ic$1@news.udel.edu Newsgroups : sci.physics.research Organization References bl9cbu$4of$1@news.udel.edu abergman-E4A385.02011804102003@localhost bm4c8n$q2b$1@news.udel.edu abergman-3FD650.18011210102003@localhost User-Agent : Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.4) Gecko/20030624 Follow-Ups Re: Replacing principal and fiber bundles with sheaves? From: kevin@inky.its.caltech.edu (Kevin A. Scaldeferri) Re: Replacing principal and fiber bundles with sheaves? From: References Re: Replacing principal and fiber bundles with sheaves? From: Re: Replacing principal and fiber bundles with sheaves? From: Re: Replacing principal and fiber bundles with sheaves? From: Re: Replacing principal and fiber bundles with sheaves? From: Prev by Date: Re: The string theory crackup, continued

45. Grothendieck's Reconstruction Principle And 2-dimensional Topology grothendieck's Reconstruction Principle and 2dimensional topology and Geometry This paper attempts to relate some ideas of grothendieck in his Esquisse d'un programme and some of the recent http://rdre1.inktomi.com/click?u=http://citebase.eprints.org/cgi-bin/citations?i

46. Grothendieck's Séminaire De Géométrie Algébrique - Encyclopedia Article Abou After completing a doctorate, he worked with Alexander grothendieck at the Institut thefundamental group is one of the basic concepts of algebraic topology. http://encyclopedia.thefreedictionary.com/Grothendieck's Séminaire de géométr

Dictionaries: General Computing Medical Legal Encyclopedia Grothendieck's Séminaire de géométrie algébrique Word: Word Starts with Ends with Definition In mathematics Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. Click the link for more information. Alexander Grothendieck Alexander Grothendieck (born March 28 1928 in Berlin), is one of the leading mathematicians of the twentieth century, with major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966 and coawarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds. Because of his mastery of abstract approaches to mathematics, but also because of the many stories told about his retirement and his alleged mental disorders, he is one of the most intriguing scientific personalities of the 20th century.

47. Wikipedia Grothendieck's Galois Theory Wikipedia Free Encyclopedia's article on 'grothendieck's Galois theory' In mathematics, grothendieck's Galois theory is a highly abstract approach to the Galois theory of the fundamental group of algebraic topology in the setting of algebraic geometry http://rdre1.inktomi.com/click?u=http://en.wikipedia.org/wiki/Grothendieck's_Gal

48. Grothendieck During this period grothendieck s work provided unifying themes in geometry,number theory, topology and complex analysis. He introduced http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html

Alexander Grothendieck Born: 28 March 1928 in Berlin, Germany Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Alexander Grothendieck In 1949 Grothendieck moved to the University of Nancy where he worked on functional analysis with . He became one of the Bourbaki group of mathematicians which included Weil Henri Cartan and . He presented his doctoral thesis Grothendieck spent the years 1953-55 at the University of Sao Paulo and then he spent the following year at the University of Kansas. However it was during this period that his research interests changed and they moved towards topology and geometry. In fact during this period Grothendieck had been supported by the Centre National de la Recherche Scientifique, the support beginning in 1950. After leaving Kansas in 1956 he therefore returned to the Centre National de la Recherche Scientifique. However in 1959 he was offered a chair in the newly formed Institut des Hautes Etudes Scientifiques which he accepted. In [2] the next period in Grothendieck's career is described as follows:- It is no exaggeration to speak of Grothendieck's years algebraic geometry , and him as its driving force. He received the

49. Grothendieck Topology Alexander grothendieck InformationBlast Motives and the motivic Galois group (and grothendieck categories); Crystals andcrystalline cohomology, yoga of De Rham and Hodge coefficients. Tame topology. http://www.worldhistory.com/wiki/G/Grothendieck-topology.htm

World History (home) Encyclopedia Index Localities Companies Surnames ... This Week in History Grothendieck topology Grothendieck topology in the news In mathematics , a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C , and with that the definition of general cohomology theories. A category together with a Grothendieck topology on it is called a site . This tool is used in algebraic number theory and algebraic geometry scheme s, but also for flat cohomology and crystalline cohomology. Note that a Grothendieck topology is not a topology in the classical sense. History and idea At a time when cohomology for sheaves on topological spaces was well established, Alexander Grothendieck wanted to define cohomology theories for other structures, his scheme s. He thought of a sheaf on a topological space as a "measuring rod" for that space, and the cohomology of such a measuring rod as a rough measure for the underlying space. His goal was thus to produce a structure which would allow the definition of more general sheaves or "measuring rods"; once that was done, the model of topological cohomology theories could be followed almost verbatim. Motivating example Start with a topological space X and consider the sheaf of all continuous real-valued functions defined on X . This associates to every open set U in X the set F U ) of real-valued continuous functions defined on U . Whenver U is a subset of V , we have a "restriction map" from F V ) to F U ). If we interpret the topological space

50. Alexander Grothendieck :: Online Encyclopedia :: Information Genius Motivess and the motivic Galois group (and grothendieck categories); Crystalss andcrystalline cohomology, yoga of De Rham and Hodge coefficients Tame topology. http://www.informationgenius.com/encyclopedia/a/al/alexander_grothendieck.html

Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Alexander Grothendieck Online Encyclopedia Alexander Grothendieck (born March 28 in Berlin ), is one of the leading mathematicians of the twentieth century, with major contributions to algebraic geometry homological algebra , and functional analysis . He was awarded the Fields Medal in and Crafoord Prize in Table of contents 1 Mathematical achievements 1.1 Major mathematical topics (from Récoltes et Semailles) 2 Life 2.2 Childhood and studies ... 3 External links Mathematical achievements Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre , after sheaves had be invented by Jean Leray . Grothendieck took them to a higher level, changing the tools and the level of abstraction. Amongst his insights, he shifted attention from the study of individual varieties to the relative point of view (pairs of varieties related by a morphism ), allowing a broad generalization of many classical theorems. This he applied first to the Riemann-Roch theorem , around 1954, which had already recently been generalized to any dimension by Hirzebruch).

51. 2 The Grothendieck-Riemann-Roch Theorem If K 0 (X) is the grothendieck ring of vector bundles on X, the Chern character(defined using Chern classes by the same formula as in topology) gives a ring http://www.imsc.ernet.in/~kapil/papers/harishconf/node3.html

Next: 3 Divisors on varieties Up: Algebraic Cycles Previous: 1 Model case of 2 The Grothendieck-Riemann-Roch theorem Let X be a non-singular variety over k . An algebraic cycle of codimension p is an element of the free Abelian group on irreducible subvarieties of X of codimension p ; the group of these cycles is denoted Z p X ). As in the case of curves one can introduce the effective cycles Z p X which is the sub-semigroup of Z p X ) consisting of non-negative linear combinations. There is a subgroup R p X Z p X ), defined to be the subgroup generated by all the cycles div f W where W ranges over irreducible subvarieties of codimension p - 1 in X , and f k W . The quotient X Z p X R p X ) is called the Chow group of codimension p cycles on X modulo rational equivalence; if n = dim X then we use the notation X X ). For p = 1 and X a smooth projective curve the Chow group X ) is precisely the class group X ) introduced above. The generalisation of Schubert calculus on the Grassmannians is the intersection product X X X making X X ) into an associative, commutative, graded ring, where X Z , and X ) = for p X . The Chow ring is thus an algebraic analogue for the even cohomology ring X Z ) in topology. A refined version of this analogy is examined in Section 6. In any case we note the following `cohomology-like' properties.

52. Grothendieck land. During this period grothendieck s work provided unifying themesin geometry, number theory, topology and complex analysis. He http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Grthndck.htm

Alexander Grothendieck Born: 28 March 1928 in Berlin, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Alexander Grothendieck In 1949 Grothendieck moved to the University of Nancy where he worked on functional analysis with . He became one of the Bourbaki group of mathematicians which included Weil Henri Cartan and . He presented his doctoral thesis Grothendieck spent the years 1953-55 at the University of San Paulo and then he spent the following year at the University of Kansas. However it was during this period that his research interests changed and they moved towards topology and geometry. In fact during this period Grothendieck had been supported by the Centre National de la Recherche Scientifique, the support beginning in 1950. After leaving Kansas in 1956 he therefore returned to the Centre National de la Recherche Scientifique. However in 1959 he was offered a chair in the newly formed Institut des Hautes Etudes Scientifique which he accepted. In [1] the next period in Grothendieck's career is described as follows:- It is no exaggeration to speak of Grothendieck's years Fields Medal in . In looking back at this period, one marvels at the generosity with which Grothendieck shared his ideas with colleagues and students, the energy he and his collaborators devoted to meticulous redaction, the excitement with which they set out to explore a new land.

53. Alexander Grothendieck / © Gruppe EM Translate this page topology editor, topology navigator, topological visualisation, euch3, eM, gruppeeM, Leibniz, Leibnizianer, AG, Alexander grothendieck, NWDKS, NDKS, NWEKS http://www.eucheucheuch.de/em/ag.htm

g ruppe eM A lexander G rothendieck A G . tells you how to solve every problem ... last teaching at Montpellier again leaving ..., it's said towards the Pyrenees, ... His wish: "please let me alone". "Toleranz" Quote of one of the last correspondences with Alexander (translated by rdb): A.G. "La Clef des Songes" ("The Key to Dreams"): A.G. abstract / Zusammenfassung / Kurztext cat top elegant math More info on Grothendieck www.math.univ-montp2.fr/agata/malgoire.html Jean Malgoire (la longue marche) www.math.jussieu.fr/~leila/index.php Grothendieck Circle wwwmath.uni-muenster.de/math/u/scharlau/scharlau/ Winfried Scharlau, biographical texts http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html www.ihes.fr/IHES/Presentation/Historique/ancien.html brandt.kurowski.net/projects/lsa/wiki/view.cgi?doc=429 in english www.lacitoyennete.com/magazine/retro/grothendiecka.php in french www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html in english www.britannica.com/eb/article?eu=99003 in english homepages.feis.herts.ac.uk/~nehaniv/grothendieck.html in english modular.fas.harvard.edu/sga/ SGA web pages

54. Zeittafel/Übersicht advises grothendieck to do his doctorate in Nancy with Laurent Schwartz. He goesthere and finishes his doctorate within two years, finding a new topology on http://www.math.jussieu.fr/~leila/biog.html

A brief timeline for the life of Alexander Grothendieck (which has the advantage of accuracy) Sascha Schapiro and Hanka Grothendieck, together with Hanka's daughter Frode (called Maidi, born 1924 from an earlier marriage) and Alexander (called Schurik), live together in Berlin. Schapiro works as a street photographer, one of the only jobs his anarchist beliefs allow him, Hanka works irregularly as a journalist. In May 1933, Sascha leaves Berlin for Paris, where he waits for Hanka to join him. In December, Hanka Grothendieck leaves Schurik with a foster family, the Heydorns, in Hamburg, and joins Sascha in France. Schurik's foster mother, Dagmar Heydorn, decides that given the political situation and her own family's dangerous involvement with the Resistance, it would be safer for Schurik to be sent to join his parents in France. She locates the parents and sends him away by train. He first joins his father, then spends the summer with his mother in N^icirc;mes. Hanka Grothendieck is interned, as an ``undesirable'' (German), in the Rieucros Camp near Mende, together with Alexander. From there, he goes to school in the village four or five kilometers distant. During this time Sascha is interned in the camp of Le Vernet, from which he is deported to Auschwitz in 1942. Grothendieck and his mother move to a small village near Montpellier; he works irregularly on a farm while studying mathematics at the university of Montpellier.

55. Steve Vickers's WWW Home Page pour les vraiment nuls is a further introduction to grothendieck s idea of toposesas generalized topological spaces, using geometric logic to hide topology. http://mcs.open.ac.uk/sjv22/

Steven Vickers Department of Pure Maths The Open University , Walton Hall, Milton Keynes, MK7 6AA, England. s.j.vickers@open.ac.uk Tel: 01908 653144 I work in the area of Theoretical Computer Science, with particular interests in geometric logic, topology and topos theory. I have recently moved from the Department of Computing at Imperial College , where I worked in the Theory and Formal Methods group. You can download many of my papers , and also theses of some of my PhD students You can view the talk " Schemas as Toposes " that I gave at the OU on 2nd July 2002. Geometric Logic Most of my work is in investigating "geometric logic" (so-called from its origins in algebraic geometry). This has some unusual properties. First, it makes a hard distinction between formulae and axioms: a logical formula is restricted in the connectives it can use, to conjunction, disjunction, equality and existential quantification, and an axiom (for a geometric theory) expresses relationships between formulae in the form "for all x, y, z, ... (formula1 - formula2)". Hence the missing connectives can be introduced in just one layer - no nesting - in axioms. Second, it allows infinite disjunctions in formulae. Though these features may look weird, they have some conceptual justification in the idea of observation: formulae correspond to observable facts (and infinite disjunction is not a problem observationally) while the axioms correspond to background assumptions or scientific hypotheses.

56. Topology - Technology Services You learn methods of constructing algebraic objects on your basic topological space(X,T I always liked the sound of AtiyahSinger and grothendieck (a great name http://www.physicsforums.com/archive/t-3458

Physics Help and Math Help - Physics Forums Mathematics General Math Archives View Thread : Topology Topology Lonewolf What are the main differences in approach between standard? topology and algebraic topology? Register Now! Free! Talk Science! marcus [i]Originally posted by Lonewolf [/i] [B]What are the main differences in approach between standard? topology and algebraic topology? [/B] Lonewolf here is something amazing: http://www.math.niu.edu/~rusin/known-math/index/mathmap.html It is a map of maths. It is multilevel clickable. You click on topology and it gives you a list of different branches of topology, including algebraic, then you click on algebraic and it tells you what it is it also tells some history in some cases, or what it might be good for (always a problem) Register Now! Free! Talk Science! Lonewolf Great link, Marcus. Thanks. It's much better than mathworld's superficial explanations. Register Now! Free! Talk Science! marcus That site is called "The Mathematical Atlasa clickable index map of mathematics" the actual answer to your question might be less fun than using this map to find it I will try to answer for my own moral and spiritual improvement (since you can get the answer by yourself without my help) what you call "standard? topology" is ordinarily called

57. ANU - Mathematical Sciences Institute (MSI) - Research Groups - Algebra And Topo Algebra and topology Seminar. 1pm Tuesday 14 January 2002 Theatre 4 Manning ClarkCentre Mark Kisin University of Muenster grothendieck s pcurvature Conjecture http://wwwmaths.anu.edu.au/research.programs/aat/seminar.abs/02.01.15.html

Skip Navigation ANU Home Search ANU Mathematical Sciences Institute (MSI) Research Groups - Algebra and Topology MSI Home People Research Study ... Jobs Algebra and Topology Home People Research Seminars ... Software MSI Intranet Internal pages Quick Links Search MSI Contact us Algebra and Topology Seminar 1pm Tuesday 14 January 2002 Theatre 4 Manning Clark Centre Mark Kisin University of Muenster Grothendieck's p -curvature Conjecture and Generic curves Let X be a complex curve, and let r be a finite dimensional complex representation of its fundamental group. Grothendieck's conjecture gives an arithmetic criterion which predicts when the image of r is finite. I will explain this conjecture, give some simple examples, and present some joint work with Matthew Emerton which gives an approach to the conjecture when X is sufficiently generic. Privacy Contact ANU Page last updated: 2 February, 2004 Please direct all enquiries to: MSI webmaster Page authorised by: Dean, MSI The Australian National University - CRICOS Provider Number 00120C

58. Topology Of Singular Spaces And Constructible Sheaves Email a friend about this book, topology of Singular Spaces and ConstructibleSheaves. 3.3 Localization results for grothendieck groups and trace formulae. http://www.booksmatter.com/b376432189X.htm

Home Search Browse Shopping Cart ... Help QuickSearch (Words, Author, Subject, ISBN) Publishers Email a friend about this book Topology of Singular Spaces and Constructible Sheaves Format Hardcover Subject Mathematics / Reference ISBN/SKU Author Jorg Schurmann Publisher Springer Verlag Publish Date February 2004 Price Qualified Frequent Buyer Price Add to cart Ships from our store in 7 - 15 business days More delivery info here Table of Contents Introduction 1 Thom-Sebastian Theorem for constructible sheaves Introduction 1.1 Milnor fibration 1.1.1 Cohomological version of a Milnor fibration 1.1.2 Examples 1.2 Thom-Sebastian Theorem 1.2.1 Preliminaries and Thom-Sebastiani for additive functions 1.2.2 Thom-Sebastiani Theorem for sheaves 1.3 The Thom-Sebastiani Isomorphism in the derived category 1.4 Appendix: Kunneth formula 2 Constructible sheaves in geometric categories Introduction 2.0.1 The basic results 2.0.2 Definable spaces 2.1 Geometric categories 2.2 Constructible sheaves 2.3 Constructible functions 3 Localization results for equivariant constructible sheaves Introduction 3.1 Equivariant sheaves

59. Partners: Functional Analysis And Topology By Lawrence Narici 6 N. Bourbaki, General topology, Part 2, Hermann, Paris and AddisonWesley, Reading,Mass., 1966. 10 A. grothendieck, Produits tensoriels topologiques et http://at.yorku.ca/t/a/i/c/42.htm

Topology Atlas Document # taic-42 Topology Atlas Invited Contributions vol. 6 issue 1 (2001) p. 4-6 Partners: Functional Analysis and Topology Lawrence Narici Department of Mathematics and Computer Science, St. John's University, Jamaica, NY 11439, USA NARICIL@STJOHNS.EDU Amazon.com page for Functional Analysis by G. Bachman and L. Narici, Dover, Mineola, New York, 2000, a reprint of the the 1966 Academic Press book of the same title. See also the invited contribution What is functional analysis? by the same author. Introduction Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space-the space of continuous linear functionals-of a normed space played an especially important role in the formative years of functional analysis. To further this endeavor, many new kinds (weak, strong, etc.) of convergence and compactness were introduced . Metric and general topological spaces evolved in order to provide a framework in which to treat these types of convergence. As general topology gestated, many concepts were greatly clarified and simplified. (For example, "continuous" meant transforming convergent sequences into convergent sequences until about 1935.) These clarifications led to the development of general topological vector spaces in the 1930's. Beginnings As set theory developed at the end of the nineteenth century, its paradoxes revealed that mathematics had a disturbingly shaky foundation. With the aim of placing set theory in particular and mathematics generally on a firmer logical pedestal, Hilbert and others looked to Euclidean geometry for a model [

60. Historia Matematica Mailing List Archive: [HM] Weil Conjectures Well, maybe more famous are the Weil conjectures, which Weil and Serre and grothendieckunderstood as algebraic topology (perhaps grothendieck had the http://sunsite.utk.edu/math_archives/.http/hypermail/historia/mar00/0182.html

[HM] Weil conjectures and algebraic topology Subject: [HM] Weil conjectures and algebraic topology From: Volker Eisermann ( eiserman@math.uni-bonn.de Date: Fri Mar 31 2000 - 05:01:13 EST Next message: Volker Eisermann: "Re: [HM] Bourbaki, theory, and problems" Previous message: Emili Bifet: "[HM] Leibniz, Lagrange, Hudde and Moebius" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] I would like to make a remark on Colin McLartys challanging answer to my posting: "20th century mathematics and knot theory": That's very interesting and I would like to hear more about this. I have never dreamed of considering the Weil conjectures as algebraic topology. It would be great, if you can give me a reference for Weil/Serre/Grothendieck having this point of view. Let me just explain, how I wanted to use the term algebraic topology: I did not want to make any claims about something like "the true nature of algebraic topology", or about what algebraic topology should be, but about algebraic topology as a part of reality: There are textbooks on algebraic topology (like Dold, Spanier, Bredon)

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