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Topic: Langlands philosophy

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	Langlands program (Site not responding. Last check: 2007-10-21)
		Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group over the adele ring of Q.
		Langlands attached L-functions to these automorphic representations, and conjectured that every L-function arising from finite-dimensional representations of the Galois group is equal to one arising from an automorphic cuspidal representation.
		Langlands then formulated a much more general "Functoriality Principle", which relates automorphic representations of different groups (not just the general linear group) over the adele ring of Q, in a way which is compatible with their L-functions.
	www.sciencedaily.com /encyclopedia/langlands_program (731 words)

	Knowledge King - Robert Langlands (Site not responding. Last check: 2007-10-21)
		Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory.
		Langlands is the author of the Langlands program, a deep web of conjectures connecting number theory and representation theory.
		Langlands understood that the theory of automorphic forms gives a generalization of class field theory, a central topic in algebraic number theory.
	www.knowledgeking.net /encyclopedia/r/ro/robert_langlands.html (333 words)

	Langlands program - Encyclopedia Glossary Meaning Explanation Langlands program (Site not responding. Last check: 2007-10-21)
		It was proposed by Robert Langlands beginning in 1967.
		Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group GL n
		Langlands received the Wolf Prize in 1996 for his work on these conjectures.
	www.encyclopedia-glossary.com /en/Langlands-program.html (887 words)

	Unifying theories in mathematics - Encyclopedia, History, Geography and Biography
		In fact the Langlands philosophy is much more like a web of unifying conjectures; it really does postulate that the general theory of automorphic forms is regulated by the L-groups introduced by Robert Langlands.
		Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine (connections between elliptic modular functions as Fourier series, and the group representations of the Monster group and other sporadic groups).
		The Langlands philosophy neither foreshadowed nor was able to include this line of research.
	www.arikah.net /encyclopedia/Unifying_conjecture (770 words)

	Langlands program -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Langlands then generalized these to (Click link for more info and facts about automorphic cuspidal representation) automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the (Click link for more info and facts about general linear group) general linear group GL n
		For example, in the work of Harish-Chandra one finds the principle that what can be done for one semisimple (or (Click link for more info and facts about reductive) reductive) (Click link for more info and facts about Lie group) Lie group, should be done for all.
		Langlands received the (Click link for more info and facts about Wolf Prize) Wolf Prize in 1996 for his work on these conjectures.
	www.absoluteastronomy.com /encyclopedia/L/La/Langlands_program.htm (831 words)

	Automorphic form - Wikpedia (Site not responding. Last check: 2007-10-21)
		Langlands showed how (in generality, many cases being known) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion.
		One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis, though not so obviously for the number theory.
		It is this concept that is basic to the formulation of the Langlands philosophy.
	bostoncoop.net /~tpryor/wiki/index.php?title=Automorphic_representation (693 words)

	EPSRC Web Site - Grants on the Web
		In the 1970s, Langlands outlined a profound series of conjectures and ideas which have since become known as "the Langlands philosophy".
		Langlands' conjectures related representations of Galois groups and related groups to automorphic forms, objects which are typically defined analytically.
		Langlands' conjectures explain many phenomena in this area, but they do not seem to shed too much light on the "extra" p-adic objects that may show up in this more algebraic setting (for example, modular forms of non-integral weight studied by Katz and Hida).
	gow.epsrc.ac.uk /ViewGrant.aspx?Grant=GR/T02256/01 (205 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		One way to construct them is via induction from compact open subgroups, and this has led to the philosophy of studying admissible representations of a reductive group by restricting them to appropriate compact open subgroups.
		Bushnell and Kutzko have applied this philosophy with great success to the general or special linear groups; this has led them to formalize it via the very general notion of S-types.
		Th e Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9623288.txt (272 words)

	Scotsman.com News - Big city survey: Dundee - Small city, Big world (Site not responding. Last check: 2007-10-21)
		Langlands is soft spoken and always measured in his statements.
		Langlands’ opposite number (in more ways than one) is the outspoken head of the University of Abertay since 1992, Bernard King.
		King argued that while Scotland needed a highly educated workforce itwould not be achieved by helping "rich and exclusive institutions to cream off a few of the very brightest state school pupils".
	news.scotsman.com /topics.cfm?tid=575&id=1084722002 (1580 words)

	H I S T O R Y O F S T G E O R G E
		Langlands adding the extras which included a magnificent conversion from the touchline.
		Under Langlands, it was another successful year as Saints came within one match of the grand final.
		Already classed as one the game's greatest players, Graeme Langlands was elevated to the position of captain-coach of Australia thereby becoming the last player to be selected as captain and coach of an Australian touring side.
	www.showroom.com.au /dragons/dragonshistory/history_stgeorge5b.htm (2799 words)

	Automorphic form (Site not responding. Last check: 2007-10-21)
		Langlands showed how (in generality, many casesbeing known) the Riemann-Roch theorem could be applied tothe calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion.
		Healso produced the general theory of Eisenstein series, which corresponds to what in spectral theory terms would be the 'continuous spectrum' for this problem, leaving the cusp form or discrete part to investigate.
		One way to express the shift in emphasis is that the Hecke operators are here in effect put on the same level as the Casimir operators; which is naturalfrom the point of view of functional analysis, though not soobviously for the number theory.
	www.therfcc.org /automorphic-form-79316.html (671 words)

	Langlands' philosophy and Koszul duality (Site not responding. Last check: 2007-10-21)
		I am trying to extend work of Adams-Barbasch-Vogan and some joint work of myself with Beilinson and Ginzburg.
		All this should be regarded as a contribution to (the local case of) Langlands' philosophy.
		This is however very different from the relation to be investigated in this article: Whereas localization leads to geometry on the group itself (or its flag manifold), the results of Adams-Barbasch-Vogan to be extended in this article lead to geometry on the Langlands dual group.
	home.mathematik.uni-freiburg.de /soergel/PReprints/langlands.html (109 words)

	CPANV.com (Site not responding. Last check: 2007-10-21)
		At Langlands and Grossa LLP, "professionalism" means the commitment to withstand all pressures, competitive and otherwise, to compromise our principles, standards and quality.
		As a matter of policy, Langlands and Grossa, LLP will only undertake engagements we believe we can perform with competence; that are useful to our clients; that do not impair our independence in fact or appearance; and that attract the top-notch professionals we need to serve the community with distinction.
		The adequacy of Langlands and Grossa, LLP quality control system for accounting and auditing is examined every three years through a peer review conducted by the American Institute of Certified Public Accountants (AICPA).
	www.cpanv.com /Values/Values.htm (293 words)

	[No title]
		L-functions have been used for over a century in number theory; Langlands isolated the correct analogue from representation theory (so-called "automorphic" L-functions) and was the first to understand the general picture.
		Standard references for the Langlands program are: Stephen Gelbart, "An Elementary Introduction to the Langlands Program", Bulletin of the AMS v.10 no. 2 April 1984.
		This is the proceedings of a conference on the Langlands conjectures, including expository articles on the Langlands program and some of the background.
	www.math.niu.edu /~rusin/known-math/94/langlands (908 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Title : Langlands' Lifting from Orthogonal and Symplectic Groups to GL(n) Abstract : Piatetski-Shapiro 9704997 The objective of the research funded under this award is to investigate the lifting of generic cuspidal automorphic representations of orthogonal and symplectic groups G to automorphic representations of GL(n).
		This proposal is in the part of mathematics known as the Langlands program.
		The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9704997.txt (199 words)

	[No title]
		There are a lot of more detailed sources of information on the Langlands program, but the problem for the beginner (me) is that the overall goal gets swamped in a mass of technicalities.
		First of all, it should be remarked that according to the Tannakian philosophy, one can reconstruct a group from the category of its finite-dimensional representations, equipped with the structure of the tensor product.
		More precisely, one asks that under the Langlands correspondence certain natural invariants attached to the Galois representations and to the automorphic representations be matched.
	math.ucr.edu /home/baez/twf_ascii/week221 (2641 words)

	The Philosophy of Real Mathematics - Ideas
		This regularity, at first blush so simple, is the germ of one of the major branches of the higher number theory and was the central theme of the development that began with Euler and Legendre in the eighteenth century, and continued down to our own time, with contributions by Gauss, Kummer, Takagi, and Artin.
		Garling brought up the case of three-dimensional topology, a subject to which Poincaré himself made very significant contributions, posing one of the central conjectures of the field that a three-dimensional manifold homotopic to the 3-sphere is homeomorphic to it.
		The philosophy of geometry has become a sub-branch of philosophy of physics.
	www.dcorfield.pwp.blueyonder.co.uk /ideas.html (4877 words)

	Artin conjecture (Site not responding. Last check: 2007-10-21)
		Therefore the Artin conjecture is concerned with zeroes of L-functions, just as the Riemann hypothesis family of conjectures is. It is believed that it would follow from strong enoughresults from the Langlands philosophy, relating to theL-functions associated to automorphicrepresentations for GL(n) for all n ≥ 1.
		In fact this is afolk-theorem; it certainly represents one of the major motivations for the generality present in Langlands' work.
		The second Artin conjecture relates to the density of the set of primes p modulo which a given integer a > 1 is a primitive root, when a is not a perfectsquare.
	www.therfcc.org /artin-conjecture-218255.html (466 words)

	Diary for badvogato
		Below are references to works of expository character which touch on topics related to the Langlands program.
		His philosophy is like to promote that of a thornbird.
		In philosophy, where shibumi emerges as wabi, it is spiritual tranquility that is not passive; it is being without the angst of becoming.
	www.advogato.org /person/badvogato/diary.html?start=138 (1528 words)

	Seminar abstract (Site not responding. Last check: 2007-10-21)
		Langlands' philosophy is a coherent collection of theorems and conjectures describing the duality between the spectral theoretic and the geometric aspect of mathematical theories.
		The philosophy gives rise to detailed conjectures-many of which still defy mathematical proof.
		Via Langlands philosophy we obtain a relation between the spectrum of the Bose gas and the geometry of conjugacy classes in complex linear simple algebraic groups.
	www.win.tue.nl /~amc/seminar/abstracts/opdam.html (186 words)

	provost - Professor James G. Arthur (Site not responding. Last check: 2007-10-21)
		Automorphic forms is the branch of representation theory that relates symmetry with arithmetic and number theory.
		According to a general philosophy of R. Langlands of Princeton, automorphic forms hold the key to unifying vast areas of mathematics, some of which date back several centuries.
		The Langlands programme is a stunning blueprint for relating arithmetic and algebra on the one hand, with analysis and spectral theory on the other.
	www.provost.utoronto.ca /English/Professor-James-G.html (564 words)

	David Joyner (Site not responding. Last check: 2007-10-21)
		The talks for the academic year 2004-2005 are held Thursdays in Preble 242 at 3:45 pm unless otherwise stated.
		Abstract: The "Langlands Program" started out as a grand conjectural generalization of Artin reciprocity to non-Abelian Galois extensions of number fields.
		It has evolved to include relationships with other fields, such as parts of the theory of harmonic analysis on reductive groups and arithmetic algebraic geometry.
	web.usna.navy.mil /~wdj/colloq/talk04_7.html (76 words)

	m759's Xanga Site (Site not responding. Last check: 2007-10-21)
		The typical affirmation of Exaggerated Realism, the most outspoken ever made, appears in Plato's philosophy; the real must possess the attributes of necessity, universality, unity, and immutability which are found in our intellectual representations.
		The idea is absolutely stable and exists by itself (ontos on; auta kath' auta), isolated from the phenomenal world, distinct from the Divine and human intellect....
		Atiyah's misleading remarks may appeal to believers in the contemptible religion of Scientism, but they have little to do with either historical reality or authentic philosophy.
	www.xanga.com /home.aspx?user=m759&nextdate=11%2f1%2f2005+23%3a59%3a59.999 (976 words)

	Mathematics Seminars for the week of February 16, 2004
		By some local-to-global philosophy this very often is undertaken first for local number fields.
		It proceeds to the local Langlands correspondence which approaches the full group by representation theoretic means and which was proved a few years ago.
		I will finish with explaining why this apparent climax in fact is, in the light of new developments, far from being the end of the story.
	www.math.umn.edu /~seminar/2004_February_16.html (585 words)

	AAG 2006 (Site not responding. Last check: 2007-10-21)
		The study of L-functions of algebraic varieties is a central domain of investigation in this area, with specific problems arising from Iwasawa theory and the Birch and Swinnerton-Dyer and Bloch-Kato conjectures.
		This theme is unified by the so-called "Langlands philosophy''.
		Some of its essential goals are: to investigate automorphic forms and representations, and in particular the L-functions attached to these objects; to carry out Langlands' program of functoriality relating representations of different groups; and to establish correspondences between automorphic and Galois representations.
	www.uam.es /otros/aag2006/wellcome.html (223 words)

	Another proof for Fermat's last theorem
		In fact it, as well as the conjecture by Serre, are ingredients of a wider program to unify various areas of mathematics, known as Langlands philosophy.
		The idea behind such unifying theories is that it should be possible to directly translate every concept in a given area of maths into all the other areas of maths.
		The Langlands philosophy was conceived by the mathematician Robert Langland in the 1960's and consists of a set of conjectures concerning the intimate relationship between number theory, geometry and algebra.
	plus.maths.org /latestnews/jan-apr05/serre/index-gifd.html (765 words)

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