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Topic: Monodromy group

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Monodromy conjecture

Weight monodromy conjecture

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	Monodromy - Wikipedia, the free encyclopedia
		It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we 'run round' a path encircling a singularity.
		The failure of monodromy is best measured by defining a monodromy group: a group of transformations acting on the data that codes what does happen as we 'run round'.
		In this case the monodromy group is infinite cyclic.
	en.wikipedia.org /wiki/Monodromy (536 words)

	monodromy (Site not responding. Last check: 2007-10-07)
		It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functionss we may wish to define fail to be single-valued as we 'run round' a path encircling a singularity.
		The associated Galois group of the extension is called the monodromy group of the extension.
		In the case that the extension is already Galois, the associated monodromy group is sometimes called a group of deck transformations.
	www.yourencyclopedia.net /Monodromy (605 words)

	DIFFERENTIAL - Online Information article about DIFFERENTIAL (Site not responding. Last check: 2007-10-07)
		GRouPs, THEORY OF) when the introduction for the variables in the differen-Apptica tial equations of the new variables given by the t of theory of equations of the group leads, for all values of the cantina- parameters of the group, to the same differential equaousgrouPs tions in the new variables.
		This is the group called the rationality group, or the group of trans-formations of the original homogeneous linear differential equation.
		The group must not be confounded with a subgroup of itself, the monodromy group of the equation, often called simply the group of the equation, which is a set of transformations, not depending on arbitrary variable parameters, arising for one particular fundamental set of solutions of the linear equation see GROUrs, THEORY OF).
	encyclopedia.jrank.org /DEM_DIO/DIFFERENTIAL.html (6881 words)

	Monodromy -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		The failure of monodromy is best measured by defining a monodromy group: a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group of transformations acting on the data that codes what does happen as we 'run round'.
		Linear differential equations defined in an open, connected set S in the complex plane have a monodromy group, which (more precisely) is a (additional info and facts about linear representation) linear representation of the (additional info and facts about fundamental group) fundamental group of S, summarising all the analytic continuations round loops within S.
		In the case that the extension is already Galois, the associated monodromy group is sometimes called a (additional info and facts about group of deck transformations) group of deck transformations.
	www.absoluteastronomy.com /encyclopedia/m/mo/monodromy.htm (584 words)

	Monodromy - TheBestLinks.com - Complex analysis, Differential equation, Fundamental group, Field extension, ... (Site not responding. Last check: 2007-10-07)
		Monodromy, Complex analysis, Differential equation, Fundamental group, Field...
		The associated Galois group of the extension $L_f/\mathbb{F}(x) is called the$ monodromy group of the extension.
		In the case that the extension $\mathbb{C}(y) is already Galois, the associated$ monodromy group is sometimes called a group of deck transformations.
	www.thebestlinks.com /Monodromy.html (628 words)

	Monodromy (Site not responding. Last check: 2007-10-07)
		The failure of monodromy is best measured by defining a monodromy group : a group of transformations acting on the data that codes what doeshappen as we 'run round'.
		Linear differential equations defined in an open, connected set S inthe complex plane have a monodromy group, which (more precisely) is a linear representation of the fundamentalgroup of S, summarising all the analytic continuations round loops within S.
		The effect when applied to loops based at m is to define a holonomy group of translations of the fiber at m; if thestructure group of B is G, it is a subgroup of G that measures the deviation of B from theproduct bundle MxG.
	www.therfcc.org /monodromy-69645.html (441 words)

	Project-Team - Café (Site not responding. Last check: 2007-10-07)
		The Lie group, or symmetry group, of a differential system is the (biggest) group of point transformations leaving the solution set invariant.
		For a given group of transformations on a set of independent and dependent variables there exist invariant derivations and a finite set of differential invariants that generate all the differential invariants [66].
		We are engaged in investigating the algebraic and algorithmic aspect of the subject.
	www.inria.fr /rapportsactivite/RA2003/cafe/module4.html (993 words)

	Cornell Math - Rodrigo Perez (Site not responding. Last check: 2007-10-07)
		The concept of "fractal group" has recently opened a vast new area of research by bringing algebraic methods to bear on the geometry of fractal sets.
		The iterated monodromy group IMG(f) of a rational map f : \overline{C}\longrightarrow\overline{C} with postcritically finite orbits is a finitely generated, infinitely presented group that seems to encode a lot of information about the structure of the corresponding Julia set.
		Currently, we are extending this proof to iterated monodromy groups of more general rational maps.
	www.math.cornell.edu /People/Postdocs/perez.html (251 words)

	Orateurs
		It was extended to Burnside factors of word hyperbolic groups by Ivanov and Ol'shanskii and it has been recently shown to be useful in the study of Burnside factors of more complicated ``very'' non-hyperbolic groups, see a paper of Ol'shanskii and Sapir on the non-amenable finitely presented torsion-by-cyclic groups.
		Geometric group theory proposes to consider abstract groups as geometric objects, and to study algebraic properties from the point of view of large-scale geometry.
		In particular, we will show that the limit space of the iterated monodromy group of an expanding map is homeomorphic to the Julia set of the map, i.e., that the iterated monodromy group captures in this case all the essential dynamics of the map.
	mad.epfl.ch /villars04/projet/node1.html (1162 words)

	The Geometry Junkyard: Symmetry and Group Theory
		The MSRI Computing Group uses another horoball diagram as their logo.
		Jonathan A. Poritz and coworkers investigate the fundamental domains of cyclic group actions on hyperbolic 3-space, resulting in lots of pretty pictures of overlapping spheres.
		Group theoretic mathematics for determining whether a polygon formed out of hexagons can be dissected into three-hexagon triangles, or whether a polygon formed out of squares can be dissected into restricted-orientation triominoes.
	www.ics.uci.edu /~eppstein/junkyard/sym.html (1155 words)

	Category.org - The Online Shopping Center: Books - Group Theory (Site not responding. Last check: 2007-10-07)
		Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index.
		As well as determining the range of possible 'growth types', for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure.
		Throughout the emphasis is on Cayley maps: imbeddings of Cayley graphs for finite groups as (possibly branched) covering projections of surface imbeddings of loop graphs with one vertex.
	www.category.org /browse/books/13940/index.html (5868 words)

	PlanetMath: deck transformation (Site not responding. Last check: 2007-10-07)
		It is straightforward to check that the set of deck transformations is closed under compositions and the operation of taking inverses.
		It is worth noting that an alternative name for the group of deck transformations is the Galois group of the covering.
		This terminology arises from an analogy with the fundamental theorem of Galois theory which gives the inclusion-reversing identification addressed in the classification of covering spaces.
	planetmath.org /encyclopedia/DeckTransformation.html (357 words)

	[No title]
		The monodromy m.i of branchpoint b.i is the permutation of L one obtains by applying analytic continuation on L following a path from x0 to b.i, going around b.i counter clockwise, and returning to x0.
		If the optional argument group is given then the output will be the monodromy group G, the permutation group generated by the m.i.
		The homology is characterized by a canonical basis of cycles on the Riemann surface, i.e., these cycles split apart in two groups, the elements of the first group are denoted a.i, the elements of the second group are denoted b.i, with i=1..g, with g the genus of the Riemann surface.
	www.math.fsu.edu /~hoeij/periodmatrix/help_pages (1351 words)

	Computation of the Monodromy of the Generalized Hypergeometric Function ... - of, for, hypergeometric, Kyushu, ...
		Ohara, Computation of the monodromy for the generalized hypergeometric function p F p\Gamma1 (a 1 ; : : : ; a p ; b 2 ; : : : ; b p ; z), Kyushu Journal of Mathematics 51 (1997), 101--124.
		4 the structure of cohomology groups attached to the integral..
		2 Monodromy of the hypergeometric differential equation of typ..
	citeseer.ist.psu.edu /2272.html (453 words)

	Monodromy - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-07)
		Monodromy Groups Groups of Isolated Singularities of Complete Intersections
		Moments, Monodromy, and Perversity : A Diophantine Perspective.
		Special Values of Dirichlet Series, Monodromy, and the Periods of Automorphic Forms (Memoirs of the American Mathematical Society, 299)
	encyclopedia.worldsearch.com /monodromy.htm (533 words)

	Citations: Schur covers and Carlitz's conjecture - Fried, Guralnick, Saxl (ResearchIndex)
		The bulk of their effort was group theoretic: they used the above conditions on the monodromy groups, together with another condition reflecting....
		....that the Galois group is typically an affine group (that is, a group of invertible affine transformations of a vector space) except for certain unexpected possibilities over fields of characteristic two and three.
		One is of affine groups (x5.1) V Theta s H with H acting irreducibly on the vector space V.
	citeseer.ist.psu.edu /context/135879/0 (1426 words)

	tilings.org -- publications
		As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified.
		Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short, simple, closed geodesics on this surface.
		The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.
	www.tilings.org /publications.html (2346 words)

	MMJ: Vol.3 (2003), N.2 - Abstracts (Site not responding. Last check: 2007-10-07)
		The correspondence is based on an isomorphism of the discriminants and on the description of a relevant monodromy group of the determinantal curve.
		We prove that under certain spectral assumptions on the monodromy group, solutions of {Fuchsian systems} of linear equations on the Riemann sphere admit explicit global uniform bounds on the number of their isolated zeros in a way remotely resembling algebraic functions of one variable.
		Deformations of polynomials, singularities at infinity, monodromy, boundary singularities.
	www.ams.org /journals/distribution/mmj/vol3-2-2003/abst3-2-2003.html (2004 words)

	Fedor Bogomolov Lecture, NYU (Site not responding. Last check: 2007-10-07)
		with a given monodromy group and fibers whose Jacobians are nonisotrivial.
		The monodromy group is always a subgroup of finite index in SL(2,Z) and there is a simple combinatorial description of these groups.
		Also, there is a finite (but much bigger!) number of irreducible varieties parametrizing surfaces with a given monodromy group.
	www.math.uci.edu /~mfried/htmlfiles/bogomolov.html (148 words)

	Project-Team-Café (Site not responding. Last check: 2007-10-07)
		The dimension of the solution space of that linear differential system is the dimension of the Lie group and can be determined by the tools described in Section 3.1.
		For a given group of transformations on a set of independent and dependent variables there exist invariant derivations and a finite set of differential invariants that generate all the differential invariants [73].
		Many properties of its solutions, in particular the existence of closed-form solutions, are then equivalent to group-theoretic properties of the associated Galois group [70].
	www.inria.fr /rapportsactivite/RA2004/cafe2004/uid9.html (990 words)

	[No title]
		The Galois group $G$ of the system is, by definition, the group of all automorphisms $g$ of the field $L$ such that $g$ fixes all elements of $F$ and $[g,d/dz]=0$.
		The properties of group $G$ are the subject of Picard-Vessiot (or differential Galois) theory, and are described in \cite{Ko,R,Ka}.
		The Galois group of $L$ over $F$, $G=\text{Gal}(L/F)$ is, by definition, the group of all automorphisms of $L$ fixing all elements of $F$ and commuting with $T$.
	www.univie.ac.at /EMIS/journals/ERA-AMS/1995-01-001/1995-01-001.tex.html (1449 words)

	Monodromy Groups of Systems of Total Differential Equations of Two Variables (Site not responding. Last check: 2007-10-07)
		Monodromy Groups of Systems of Total Differential Equations of Two Variables: SIAM Journal on Mathematical Analysis Vol.
		This paper presents monodromy groups of the four systems of total differential equations.
		The theory for computing monodromy groups of the systems developed by Yokoyama is applied.
	epubs.siam.org /sam-bin/dbq/article/29687 (169 words)

	Bronek Wajnryb -- Selected Publications (Site not responding. Last check: 2007-10-07)
		B.Wajnryb, On the monodromy group of plane curve singularities, Math.
		Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math 45, (1983), 157-174.
		B.Wajnryb, Mapping class group of a surface is generated by two elements, Topology 35 (1996), 377-383.
	www.math.technion.ac.il /people/wajnryb/index_3.html (102 words)

	[No title]
		The monodromy group of a function on a general curve.
		On the lattice automorphisms of the finite Chevalley groups.
		Flag-transitive subgroups of Chevalley groups with abelian stabilizers.
	www.math.ufl.edu /fac/facmr/Voelklein.html (496 words)

	TMNA - Volume 19 Number 2
		We discuss the calculation of critical groups for jumping nonlinearities as the resonance set is crossed.
		For a ramified covering between Riemann surfaces the groups Deck of deck transformations and Mon of monodromy permutations are introduced.
		We associate with them groups of automorphisms of certain extensions of function fields.
	www.mat.uni.torun.pl /%7Etmna/htmls/archives/vol-19-2.html (506 words)

	Katz, N.M.: Moments, Monodromy, and Perversity: A Diophantine Perspective. (AM-159).
		It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions).
		The first is the theory of perverse sheaves, pioneered by Goresky and MacPherson in the topological setting and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber.
		These new techniques, which are of great interest in their own right, are first developed and then used to calculate the geometric monodromy groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields.
	pup.princeton.edu /titles/8118.html (368 words)

	Department of Mathematics - University of Georgia
		Diophantine geometry and arithmetic, local and global heights on algebraic varieties, uniform distribution on locally compact groups, applications of harmonic analysis to number theory, the Mahler measure of polynomials.
		Our number theory group is complemented by a large group in algebraic geometry, including Valery Alexeev, William Graham, Elham Izadi, Roy Smith, and Robert Varley.
		Members of the group are: Robert Brice, Sungkon Chang, Jerry Hower, Jacob Keenum, Nausheen Lotia, Daeshik Park, Clay Petsche, Charles Pooh, Dong Hoon Shin, and Juhyung Yi.
	www.math.uga.edu /research/number_theory.html (607 words)

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