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Topic: Classification theorems of surfaces

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	Cubic surfaces
		Clebsch described the plane representations of various rational surfaces, he was especially interested in that of the general cubic surface.
		Other results on cubic surfaces were proved by Clebsch which included: there exists a covariant of order nine which intersects the cubic surface in exactly 27 lines; and every smooth cubic surface can be represented in the plane using four plane cubic surfaces through six points and vice-versa.
		Starting from the construction of a cubic surface given by a straight line, three groups of three points on a line, and six other points, Le Paige was led to the construction of a cubic surface given by a line, three points on a line and twelve other points.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Cubic_surfaces.html (1002 words)

	Mathematics
		This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis).
		Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.
		However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers.
	www.brainyencyclopedia.com /encyclopedia/m/ma/mathematics.html (3081 words)

	PlanetMath: Pappus's centroid theorem
		Theorem 1 The surface of revolution generated by a smooth curve
		In English-speaking countries, these two theorems are known as Pappus's theorems, after the ancient Greek geometer Pappus of Alexandria.
		This is version 4 of Pappus's centroid theorem, born on 2005-08-15, modified 2005-08-27.
	planetmath.org /encyclopedia/PappussTheoremForSurfacesOfRevolution.html (419 words)

	UC Davis Math: Glossary (Site not responding. Last check: 2007-10-18)
		A theorem of Koebe, revived by Thurston, states that given any planar graph, there is a circle packing with a circle for each vertex of the graph and kissing circles for each edge.
		In the proof of the Geometrization Theorem, when the skinning map is applied to a conformal structure A to produce a new one B, B is made up of many (unrolled) copies of A that resemble leopard spots.
		A surface (or manifold) which locally minimizes surface area (or surface volume), which means that one cannot replace small patches of the surface and decrease the area.
	math.ucdavis.edu /profiles/glossary.html (9932 words)

	classification theorems of surfaces
		In topology, the classification theorem for surfaces gives us a complete list of compact spaces that are locally the same as the Euclidean plane.
		Among other things, the theorem implies that compact 2-manifolds are characterized by the Euler characteristic together with the genus.
		A closely related example to the classification of compact 2-manifolds is the classification of compact Riemann surfaces, i.e., compact complex 2-manifolds.
	www.mcfly.org /classification_theorems_of_surfaces (298 words)

	MAS231, Geometry II: Knots and Surfaces
		Surfaces, doughnuts and pretzels (classification by number of holes).
		Gauss-Bonnet theorem for integral of geodesic curvature in terms of integral of Gauss curvature in the interior, for simple closed curves and for curvilinear n-gons.
		The student should be able to state some of the main theorems and be able to reproduce some of the shorter proofs or parts of proofs.
	www.maths.qmw.ac.uk /undergraduate/modules/MAS231.html (429 words)

	Giuseppe Tinaglia
		Surfaces which are critical points for the area functional under a volume constraint are instead called constant mean curvature (CMC) surfaces and in fact the average of the two principal curvatures is constant.
		We prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical.
		We show that, for a minimal surface of any genus, whose boundary is "almost graphical" in some sense, that the surface must be graphical once we move sufficiently far from the boundary.
	www.nd.edu /~gtinagli/research.htm (996 words)

	Mathematics
		Outline of the topological classification of compact surfaces, vector fields, geodesics, and Jacobi fields.
		The global differential geometry of surfaces and the elementary theory of Riemann surfaces.
		Theorems of Church, Kleene, Godel, Mostowski and Turing.
	www.njit.edu /v2/archivecatalog/graduate/96/68.html (2081 words)

	"blog" for math380
		Classification of critical points in terms of Hessians (the test involving principal minors of the corresponding matrix).
		We then proved it for a rectangle (this is a simpler computation than the one in the book), then for regions that are unions of rectangles and finally for general regions by a limiting argument.
		Dependencs of surface and flux integrals on parameterizations.
	www.math.uiuc.edu /~lerman/380/blog.html (1384 words)

	Mathematics
		Solovay’s theorem on the consistency of “all sets of reals are Lebesgue measurable.” Additional topics in set theory depending on the audience.
		The classification of simple Lie algebras proceeds in terms of the associated reflection groups and a classification of them in terms of their Dynkin diagrams.
		Classification of irreducible characters for the infinite symmetric group and the infinite-dimensional unitary group.
	pr.caltech.edu /catalog/courses/listing/ma.html (2506 words)

	UCSC General Catalog 2006-08 - Programs and Courses
		Starting with the fundamental theorem of calculus and related techniques, the integral of functions of a single variable is developed and applied to problems in geometry, probability, physics, and differential equations.
		Surfaces of constant curvature; the theorems of Bonnet and Hadamard.
		The fundamental theorems relating to critical points to the topology of a manifold are treated in detail.
	reg.ucsc.edu /catalog/html/programs_courses/mathCourses.htm (3953 words)

	Grad course descriptions
		Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory.
		Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.
		Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.
	www.math.upenn.edu /grad/courses.html (2588 words)

	2006-2007 Course Register
		Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus.
		Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.
		Classification of vector bundles and fiber bundles. Characteristic classes and obstruction theory.
	www.upenn.edu /registrar/register/math.html (4476 words)

	Bäcklund and Darboux Transformations - Cambridge University Press
		It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned.
		Hasimoto Surfaces and the Nonlinear Schrödinger Equation: Geometry and associated soliton equations; 5.
		Isothermic surfaces: the Calapso and Zoomeron equations; 6.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=052181331X (347 words)

	Courses in the Department of Mathematics
		Hartogs’ Theorem), a deeper study of Riemann surfaces, the uniformization theorem, the Dirichlet problem in higher dimensions, differential equations in a complex domain and the Riemann-Hilbert problem, Hardy spaces.
		Inverse and implicit function theorems, transversality, Sard’s theorem and the Whitney embedding theorem.
		Geodesics and the associated variational formalism (formulas for the 1st and 2nd variation of length), the exponential map, completeness, and the influence of curvature on the structure of a manifold (positive versus negative curvature).
	catalogs.uchicago.edu /divisions/math-courses.html (2661 words)

	Referativni Zhurnal Classification
		Feedback to mathweb at MR This classification was prepared as a piece of the UDC (Universal Decimal Classification) which covers all knowledge in a fairly uniform way.
		Topology of manifolds 271.19.19.17 Topology of manifolds of lower dimensions 271.19.19.17.17 Topological surfaces 271.19.19.17.19 Three-dimensional topological manifolds 271.19.19.17.19.17 Classification of three-dimensional manifolds 271.19.19.17.19.17.19 Poincare conjecture and related problems 271.19.19.17.21 Four-dimensional topological manifolds 271.19.19.17.21.17 Classification of four-dimensional manifolds 271.19.19.17.21.17.19 Poincare conjecture for four-dimensional manifolds 271.19.19.17.27 Embeddings and immersions in lower dimensions 271.19.19.17.33 Knots.
		Existence theorems, uniqueness theorems and theorems on the differential properties of solutions 271.29.15.15.25 Differential equations with discontinuous and multivalued right-hand sides 271.29.15.15.31 Differential inequalities 271.29.15.17 Methods for solving various types of equations and systems of equations 271.29.15.19 First integrals 271.29.15.21 Equations that are not solved with respect to the highest derivative.
	www.ams.org /mathweb/Classif/RZhClassification.html (1545 words)

	[alg-geom/9602006] Chapters on algebraic surfaces
		It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion of some of the technical foundations, but with no pretence at an exhaustive treatment.
		Special features include the theory of minimal models of surfaces via Mori theory, a complete selfcontained proof of the theorems on classification of surfaces, and a clean treatment of the foundational results on rational and elliptic Gorenstein surface singularities.
		Surfaces and singularities p.80 Exercises to Chapter 4 p.106
	arxiv.org /abs/alg-geom/9602006 (258 words)

	Dept. of Mathematics: Academic Programs
		Sylow's theorems for finite groups, p-groups, abelian groups, group action on sets, domains, prime and maximal ideals, unique factorization domain.
		Limits of functions, continuity, uniform continuity, differentiation, the mean value theorem, Rolle's theorem, L'Hospital's rule, Taylor's theorem, Riemann Integral, properties of the Riemann Integral, the fundamental theorem of calculus, pointwise and uniform convergence, applications of uniform convergence.
		Banach spaces; the dual topology and weak topology; the Hahn-Banach, Krein- Milman and Alaoglu theorems; the Baire category theorem; the closed graph theorem; the open mapping theorem; the Banach-Steinhaus theorem; elementary spectral theory; and differential equations.
	www.coas.howard.edu /mathematics/programs_graduate_courses.html (1023 words)

	Math - Courses
		Theorems with historical significance will be studied as they relate to the development of modern mathematics.
		Study of geometry in Euclidean space by means of calculus, including theory of curves and surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.
		Definition and classification of differential equations; general, particular, and singular solutions; existence theorems; theory and technique of solving certain differential equations: phase plane analysis, elementary stability theory; applications.
	www.csufresno.edu /catoffice/current/mathcrs.html (1755 words)

	Lectures Abstracts (Site not responding. Last check: 2007-10-18)
		While many of the problems involving braid group factorizations remain open to this date (in particular the "Hurwitz problem", whose solution would in principle give a classification of symplectic manifolds), the braid monodromy data associated to a plane curve gives access to useful information, such as the fundamental group of the complement.
		Lefschetz pencils and differential and symplectic type of algebraic surfaces [Kas' amd Moisezon's approach to Lefschetz theorems, corollaries and applications].
		The first condition for using a group in cryptography is to be able to specify its elements, hence, for a presented group, to have an efficient solution to the word problem.
	www.cs.biu.ac.il /~eni/Eilat2005Abstracts.htm (424 words)

	VIGRE Components: Invitation to Research
		The idea of invariant measures will be stressed, various convergence theorems will be discussed, and connections with and applications to statistical mechanics, number theory, and combinatorics will be pointed out.
		The fourth lecture will be about Lefschetz fixed point theorem and about Hopf theorem on zeros of vector fields, as well as its improvement in the case of a Hamiltonian vector fields (Morse inequalities).
		In his address to the AMS in 1900, Hilbert proposed ten problems for the new century, one of which was to establish by ``finitistic'' means that basic mathematics was free of contradictions.
	www.math.ohio-state.edu /vigre/components-itr.php (4950 words)

	Guide to the Mathematics Subject Classification Scheme
		This classification scheme focuses on the former -- the study of the validity of the measurements one might make.
		The classification allows these topics to be included within each major heading at a secondary level, although there is always some material which cannot otherwise be classified.
		This classification also includes the study of polygons and polyhedra, and frequently overlaps discrete mathematics and group theory; through piece-wise linear manifolds, it intersects topology.
	www.math.niu.edu /~rusin/known-math/index/beginners.html (5525 words)

	University of Michigan Student Geometry/Topology Seminar
		Riemann surfaces arise naturally by analytically continuing functions on the complex plane and allow a precise formulation of the concept of a many-valued function.
		The classification problem for 4-manifolds is much harder than that of 3-manifolds mainly because not every topological 4-manifold admits a smooth structure and there may be infinitely many non-diffeomorphic smooth structures on a given 4-manifold whenever one exists.
		His resulting theory gave a complete classification of closed, simply connected 4-manifolds with respect to their "intersection forms", a unimodular integral bilinear form defined on the second homology of the manifold.
	www-personal.umich.edu /~magid/stugeomtop/previous.html (1372 words)

	Vol. 1 Ch. 4
		On the surface, this was merely a matter of semantics, a search for convenient cubbyholes in the case of a borderline subject.
		It meant, for example, that numerous innovations pertaining to perspective accrued to the larger categories to the extent that, when the artistic aspects of perspective came under fire, it could reasonably appear to some that perspective had died although the number of treatises on perspective in these other fields continued to grow.
		Perspective never became an independent concept and was therefore subject the the vagaries of classification of four branches of learning: optics, architecture, drawing and geometry.
	www.mmi.unimaas.nl /people/Veltman/books/vol1/ch4.htm (8936 words)

	MA475 Riemann Surfaces
		This leads to interesting geometric, analytic and topological theorems about Riemann surfaces, showing also their ubiquity in much of modern mathematics.
		Then I will define Riemann surfaces as abstract objects, and give examples from several sources: the Riemann sphere, complex tori, algebraic curves, solutions of differential equations and so on.
		Aims: to motivate the idea of a Riemann surface along the lines of Riemann's original reasoning; to introduce the abstract concepts supported by examples; to discuss analytic continuation as a bridge between intuitively obvious and mathematically precise notions; to discuss the basic theory of complex tori and other compact Riemann surfaces.
	www.maths.warwick.ac.uk /undergrad/pydc/mauve/mauve-MA475.html (300 words)

	Department of Mathematics
		Projective geometry: the real projective plane considered as the set of 1-dimensional subspaces of 3-dimensional euclidean space, collinearity and linear dependence, the collinearity lemma, the theorems of Desargues and Pappus, cross-ratios, harmonic conjugates, perspectivities and projectivities, the fundamental theorem of projective geometry, finite projective planes and combinatorial problems.
		Symmetry: isometries in 2- and 3-dimensional euclidean space, groups of symmetries and the classification of frieze and wallpaper patterns.
		Surfaces: an informal introduction to topological spaces and continuous functions and homeomorphisms, (compact) surfaces including possible boundaries considered as polygons with some or all edges identified in pairs, surgery, Euler characteristic, orientability, the classification theorem for surfaces, embedability of graphs in surfaces, maps on surfaces, chromatic number, Heawood's formula, the chromatic number of a graph.
	www.maths.mq.edu.au /undergraduate/math300s.html (201 words)

	1997-98 Undergraduate Bulletin
		The algebra of complex numbers, elementary functions and their mapping properties, complex limits, power series, analytic functions, contour integrals, Cauchy's theorem and formulae, Laurent series and residue calculus, elementary conformal mapping and boundary value problems, Poisson integral formula for the disk and the half plane.
		Introduction to topics in topology, particularly surface topology, including classification of compact surfaces, Euler characteristic, orientability, vector fields on surfaces, tesselations, and fundamental group.
		Classification of surfaces, fundamental group and covering spaces.
	www.unc.edu /ugradbulletin/1997-98/dept/math.htm (2856 words)

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