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Topic: Geometric invariant theory

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	php-deluxe.net - description Geometric invariant theory
		In mathematics, geometric invariant theory in algebraic geometry is a (technically complex) development building on nineteenth century invariant theory.
		Mumford s motivation was to develop a concrete, geometrical theory of moduli spaces of algebraic varieties.
		The theory has been very influential, and the technical concept of stability used has been basic in much later research, for example on moduli spaces of vector bundles.
	www.php-deluxe.net /encyclopedia,index.page,Geometric-invariant-theory.htm (434 words)

	Alibris: Invariants
		"Geometric Invariant Theory" by Mumford/Fogarty (the first edition waspublished in 1965, a second, enlarged editon appeared in 1982) is the standard reference on applications of invariant theory to the construction of moduli spaces.
		Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrvdinger Equations
		This is an introduction to the theory of affine Lie Algebras, to the theory of quantum groups, and to the interrelationships between these two fields that are encountered in conformal field theory.
	www.alibris.com /search/books/subject/Invariants (962 words)

	Invariant theory
		In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations.
		Current theories relating to the symmetric group and symmetric functions, commutative algebra, moduli spaces and the representations of Lie groups are rooted in this area.
		The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring.
	www.fact-index.com /i/in/invariant_theory.html (378 words)

	MAT 615 - Topics in Algebraic Geometry -- Spring 2004 (Site not responding. Last check: 2007-11-05)
		Geometric invariant theory, D. Mumford, J. Fogarty, and F. Kirwan, 3rd ed.Springer
		Lectures on invariant theory, I. Dolgachev, Cambridge 2003.
		A conjectural description of the tautological ring of the moduli space of curves, C. Faber, math.AG/9711218
	www.math.sunysb.edu /~sorin/615 (124 words)

	David C. Murphy: Research
		Invariant theory - stability of principal bundles on Riemann surfaces, alternatives to GIT quotients using Hilbert schemes
		In this seminar, I presented work on the applications of invariant theory to computer vision and assisted Robert Fossum's undergraduate student as she studied computer vision under his supervision as part of the McNair Mentoring Program.
		One-parameter subgroups play a critical role in the affine case, which is similar to their application to instability problems in geometric invariant theory.
	max.cs.kzoo.edu /~dmurphy/researchindex.html (1030 words)

	3.2.4 Number Theory
		The number theory group at Oklahoma State University has established a thriving program of research, including a regular seminar series featuring lectures of both a research and expository nature by the resident number theorists, as well as frequent lectures by distinguished young and senior number theorists from around the country.
		Number theory is famed not just for the beauty of its theorems, but for the enormous wealth and variety of techniques involved in discovering and proving these theorems.
		Our faculty is prepared to offer courses in algebraic number theory, class field theory, analytic number theory, the arithmetic of elliptic curves as well as other arithmetic algebraic varieties, p-adic analysis, automorphic and modular forms, discrete subgroups of algebraic groups, computational number theory, as well as many other subfields of number theory.
	www.math.okstate.edu /grad/brief-hbk/3_2_4Number_Theory.html (395 words)

	Matches for:
		Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space.
		One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold.
		The extra geometric richness of the $SU(2)$-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date.
	www.mathaware.org /bookstore?fn=20&arg1=pcmsseries&item=PCMS-4 (279 words)

	Algebraic Geometry
		Many questions in number theory concern the solutions of polynomials with integer coefficients over the integers or rational numbers, or modulo n for all natural numbers n, or over finite fields (which may be viewed as some kind of approximate solutions).
		Some of the spectacular recent developments in number theory, such as the solution of the Mordell conjecture (which is a statement about rational points on algebraic curves) or the role of elliptic and modular curves in the proof of Fermat's last theorem, indicate the degree to which number theory and algebraic geometry are linked.
		Among the many areas of interest are the study of curves, surfaces, threefolds and vector bundles; geometric invariant theory; toric geometry; singularities; algebraic geometry in characteristic p and arithmetic algebraic geometry; connections between algebraic geometry and topology, mathematical physics, integrable systems, and differential geometry.
	www.math.columbia.edu /research/main/alggeom/index.html (949 words)

	Calf Abstracts
		Geometric invariant theory is a powerful tool for creating quotients in algebraic geometry.
		In the study of 3-dimensional manifolds, two of the most useful structures to have are hyperbolic geometric structures and essential surfaces lying in the manifold.
		Actually the theory of schemes is far more general than Algebraic Geometry, and many concepts arising in the geometric context make sense only for a particular kind of schemes usually called "algebraic schemes".
	www.maths.bath.ac.uk /~ntb20/Calf/CalfAbs.html (4262 words)

	Search SPIE Papers - Publications - SPIE Web
		We propose triplet-quaternion-valued invariants, which are related to the descriptions of objects as the zero sets of implicit polynomials.
		Besides a systematic mathematical foundation for a remarkably general framework, the advantages of the Gestalt theory of natural surfaces include a concrete computational approach to simulate or recreate images whose geometric invariants and quantities might be perceived and estimated by an observer.
		Eghbalnia, Hamid; Assadi, Amir H. The aims of this series of papers are: (a) to formulate a geometric framework for non-linear analysis of global features of massive data sets; and (b) to quantify non-linear dependencies among (possibly) uncorrelated parameters that describe the data.
	www.spie.org /app/publications/index.cfm?fuseaction=toc&volume=4476&view=1 (3794 words)

	Abstracts of Technical Talks (Day one) (Site not responding. Last check: 2007-11-05)
		Geometric invariant theory is a well-established field in its own right with many applications in diverse areas.
		The theory underlying the different mechanisms of washing and providing deeper insights into the complexities of washing of different inks will also be outlined.
		The agreement between the theory and experiments is unsatisfactory, especially in the adiabatic region, for the one-dimensional model, as axial conduction is neglected.
	chemeng.iisc.ernet.in /tja/html/brochure/node15.html (3436 words)

	Arithmetic and Geometry Seminar: Abstract (Site not responding. Last check: 2007-11-05)
		A basis for moduli problems is the theory of quotient varieties whose constructions are oftentimes very delicate and their geometry and topology, often unifying several subjects all together, are of great interest and use.
		A first systematic account of a quotient theory is Mumford's geometric invariant theory (GIT) which was based upon some of Hilbert's earlier work.
		GIT gives a method for constructing projective quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.).
	www.math.ucsb.edu /~mckernan/98-99/yi.html (305 words)

	Gatorsports.com :: 100 years of Gator Football (Site not responding. Last check: 2007-11-05)
		That is far from being true: the classical epoch in the subject may have continued to the final publications of Alfred Young, more than 50 years later.
		In a separate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, has been rehabilitated.
		An undergraduate level introduction to the classical theory of invariants of binary forms (but not the Omega process!).
	www.gatorsports.com /apps/pbcs.dll/section?template=wiki&text=invariant_theory (785 words)

	An Introduction to Geometric Control Theory (Site not responding. Last check: 2007-11-05)
		In this series of lectures an elementary introduction to the mathematical theory of input/output systems is given.
		These concepts are thoroughly discussed for the class of linear finite dimensional time invariant control systems in state space form [(A,B,C,D)-systems].
		A geometric theory for this class of systems is developed together with some applications to control and observation problems.
	www.nd.edu /~cam/seminars/trumpf.htm (97 words)

	Ian Morrison: Selected Publications
		My specialty is algebraic geometry, especially moduli theory but I have also done a lot of work with a computational flavor and have taken a few excursions into number theory and topology.
		Relates the initial forms of the ideal of a projective variety to the geometric invariant theory of its Hilbert point(s).
		Conjectures a classification of such singularities, proves the terminality of the candidate singularities, and outlines various geometric consequences of the conjectures with computer based evidence.
	www.fordham.edu /morrison/publications.html (833 words)

	A geometric invariant-based framework for the analysis of protein conformational space -- Tendulkar et al. 21 (18): ...
		A geometric invariant-based framework for the analysis of protein conformational space -- Tendulkar et al.
		The geometric invariants on the extreme left and extreme right carry a greater absolute weight toward a respective principal component.
		For each peak, the consensus secondary structure is assigned based on the octapeptide members of the corresponding cluster (Berman et al., 2000), and is represented as H: helix, B: beta-strand and L: loop with the subscripts indicating the number of amino acids of the octapeptide with the given secondary structure.
	bioinformatics.oxfordjournals.org /cgi/content/full/21/18/3622 (3622 words)

	Geometric Invariant Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge)
		'Geometric Invariant Theory' by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged editon appeared in 1982) is the standard reference on applications of invariant theory to the construction of moduli spaces.
		The book deals firstly with actions of algebraic groups on algebraic varieties, separating orbits by invariants and constructionquotient spaces; and secondly with applications of this theory to the construction of moduli spaces.
		It is a systematic exposition of the geometric aspects of the classical theory of polynomial invariants.
	www.literacyconnections.com /cgi-bin/apf4/amazon_products_feed.cgi?Operation=ItemLookup&ItemId=3540569634 (253 words)

	Amazon.com: invariant (Site not responding. Last check: 2007-11-05)
		Polynomials and Quadratic Forms Classical invariant theory is the study of...
		The algebra of invariants, by J. Grace and A. Young.
		I The elements of invariant theory Lecture I (April 27,...
	www.amazon.com /s?ie=UTF8&keywords=invariant&tag=lexico&index=blended&link_code=qs&page=1 (790 words)

	Geometry of Quiver Varieties (Site not responding. Last check: 2007-11-05)
		The subject is chosen to be as narrow as possible to be attackable by a graduate student but broad enough to provide a wealth of worthy research problems.
		The problem involves understanding the metric structure of the quiver variety near infinity, and probably related to intersection cohomology of Goresky and MacPherson [90e:55013].
		These numbers seem to be fundamental invariants of quiver varieties and the methods should be - yet unexplored - hyperkähler analogues of the Duistermaat-Heckman-type localizations in symplectic geometry.
	www.ma.utexas.edu /~hausel/seminars/quiver (719 words)

	Department of Mathematics and Statisitcs - Faculty Members in Pure Math
		Infinite dimensional Lie algebras, representation theory, vertex operators, homology of algebras, Lie algebras associated with the other nonassociative structures, mathematical physics.
		Geometric modelling, solid modelling, spline surfaces, computer graphics, homotopy theory, history of mathematics.
		Category theory and its applications to algebra, topology and theoretical computer science.
	www.math.yorku.ca /new/people/math.htm (266 words)

	ICM 94: Abstract
		Many moduuli spaces in algebraic geometry can be expressed as quotients in the sense of Mumford's geoemtric invariant theory [9] of nonsingular complex projective varieties $X$ by actions of complex reductive groups $G$.
		Any such quotient can also be identified with a symplectic quotient or Marsden-Weinstein reduction of the variety $X$ by a maximal compact subgroup $K$ of the reductive group $G$ [8, 9, 10].
		M. Thaddeus, "Conformal field theory and the cohomology of the moduli space of stable bundles", J. Diff.
	e-math.ams.org /mathweb/icm94/04.kirwan.html (652 words)

	Personal Home Page (Site not responding. Last check: 2007-11-05)
		Regular functions on X which are invariant with respect to this action form a ring called the ring of invariants.
		When the ring of invariants is finitely generated, its spectrum defines an affine variety called the algebraic quotient and denoted X//Ga.
		Mumford, J. Fogarty, F.Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und Ihrer Grenzgebiete 2.
	www.math.unibas.ch /~bonnet/Research.html (626 words)

	Mat614 Birational geometry and geometric invariant theory (Site not responding. Last check: 2007-11-05)
		Geometric Invariant Theory (GIT) is one of the cornerstones of modern geometry, playing the key role in the solution to many moduli problems.
		Roughly put, if a reductive group acts on a projective variety then, after choosing a lift of the group action to an ample bundle on the variety, the GIT quotient exists and has many properties that one might expect in an orbit space.
		There will be some overlap with the latter parts of the versions of Mat615 - Topics in Algebraic Geometry given in Spring 2002 and Spring 2004 but I'll try to avoid repetition as much as possible.
	www.math.sunysb.edu /~craw/teaching/2004/mat614/index.html (400 words)

	[No title]
		Giancarlo Benettin, is KAM theory and its application to the problem of fast rotations of a rigid body, or, equivalently, to a small perturbation of the Euler system.
		I have worked in tight closure, a theory for equicharacteristic rings due to Melvin Hochster and Craig Huneke which has simplified and extended a lot of results in commutative algebra and algebraic geometry.
		My research is centred on the theory of nonlinear partial differential equations, with an emphasis on those arising in the Calculus of Variations.
	www.msri.org /people/members/pdinfo.html (2240 words)

	Bioinformatics research activities of Ashish Tendulkar at IIT Bombay India (Site not responding. Last check: 2007-11-05)
		In my Ph D thesis, I have analyzed protein structures using geometric and machine learning techniques with specific goals of protein structure prediction and functional classification of the protein structures.
		More specifically, we represented geometries of substructures in proteins with unilateral structure descriptors in form of Geometric Invariants and applied clustering to form groups of similar geometries.
		Geometric Invariant Theory applied to Protein Structure Classification,, Pramod Wangikar, Ashish V Tendulkar, Milind Sohoni.
	www.it.iitb.ac.in /~ashish/research (645 words)

	Graduate Programs in Mathematics
		An introduction to linear algebra and group theory, covering: vector spaces, linear maps, matrices and matrix algebra, row and column operations and their application to normal forms, determinants, characteristic subspaces, the characteristic and minimal polynomials, and symmetric groups.
		Measure theory is not a prerequisite for this course.
		Topics will be chosen from: computational group theory, computational number theory, algorithms for computing with finite fields, the discrete Fourier Transform and its applications, the Knuth-Bendix algorithm for finitely presented algebras, polynomial factorization and related topics in computer algebra.
	www.math.neu.edu /WWW_math/Grad/grad_program.html (6075 words)

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