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	Srinivasa Ramanujan - Wikipedia, the free encyclopedia
		Ramanujan was born in 1887 in Erode, Tamil Nadu, India, the place of residence of his maternal grandparents.
		Ramanujan was later appointed a Fellow of Trinity, and a Fellow of the Royal Society (FRS).
		That Ramanujan conjecture is an assertion on the size of the tau function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms.
	en.wikipedia.org /wiki/Ramanujan (3163 words)

	Srinivasa Aaiyangar Ramanujan
		Ramanujan mainly worked in analytical number theory and is famous for many amazingly deep and beautiful summation formulas involving constants such as π, prime numbers and partition function.
		As an orthodox Brahmin Ramanujan consulted the astrological data for his journey, because his mother was horrified that he would lose his caste by traveling to foreign shores.
		The Ramanujan Conjecture is an assertion on the size of the coefficients of the tau-function, a typical cusp form in the theory of modular forms[?].
	www.ebroadcast.com.au /lookup/encyclopedia/ra/Ramanujan.html (450 words)

	Srinivasa Ramanujan, , Legends, Srinivasa Ramanujan profile, Ramanujan was an astrologer of repute and a good speaker. ...
		Ramanujan was born at Erode in Tamil Nadu on December 22, 1887.
		Although Ramanujan secured a first class in mathematics in the matriculation examination and was awarded the Subramanyan Scholarship, he failed twice in his first year arts examination in college, as he neglected other subjects such as history, English and physiology.
		Ramanujan was elected Fellow of the Royal Society on February 28, 1918.
	www.4to40.com /legends/index.asp?article=legends_sramanujan (1291 words)

	Vidyapatha :: Indian Scientists : India's Largest Portal on Educational Information
		Mathematical ideas began to come in such a flood to his mind that he was not able to write all of them down.He used to do problems on loose sheets of paper or on a slate and to jot the results down in notebooks.
		But Ramanujan was ignorant of the work of the German mathematician, George F. Riemann, who had earlier arrived at the series, a rare achievement.Also included was Ramanujan's conjecture about the kind of equations called "modular".
		Ramanujan found himself a stranger at Cambridge.The cold was hard to bear and, being a Brahmin and a vegetarian, he had to cook his own food.
	www.vidyapatha.com /scientists/ramanujun.php (1226 words)

	Seminars & Colloquia - Boston College (Site not responding. Last check: 2007-10-29)
		A conjecture of Thurston predicts that whenever the fundamental group of M is infinite the manifold M admits a finite cover with non-trivial (rational) first homology.
		Ramanujan hypergraphs are higher dimensional analogs of Ramanujan graphs.
		Abstract: The Birch/Swinnerton-Dyer Conjecture (one of the Clay Millenium problems) predicts that the rank of the group of rational points on an elliptic curve is equal to the order of vanishing at s=1 of the L-function of the elliptic curve.
	www.bc.edu /schools/cas/math/seminars (720 words)

	Pure Group Publications
		Voevodsky's work establishes (at the prime $p=2$) the 1973 conjecture of Quillen and Lichtenbaum, which relates K-theory with mod $p^{n}$ coefficients to Grothendieck's \'{e}tale cohomology with mod $p^{n}$ coefficients.
		This is an extremely active area and the maojor open questions are: the Birch-Sinnerton-Dyer conjecture, the Brumer-Stark conjecture, the Lichtenbaum conjecture, the Quillen-Lichtenbaum conjecture, the Beilinson conjectures, the Kato conjecture, the Tate conjecture, the Hodge conjecture, the Coates-Sinnott conjecture and one of mine - the Chinburg-Snaith conjecture concerning the "Wiles unit".
		Snaith uses connective topological K-theory to study Chow groups in arithmetic-algebraic geometry as above and to study the famous problem of the existence/non-existence of framed manifolds of Arf-Kervaire invariant one (a problem that is the natural successor to JF Adams' "Hopf invariant one" work.
	www.maths.soton.ac.uk /pure/researchabstract.phtml?keyword=K-theory (422 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Ramanujan graphs are regular graphs, whose adjacency matrices, (or equivalently their Laplacians), have eigenvalues satisfying some natural up- per bounds.
		In fact it turns out that the Ramanujan property of these graphs is equivalent to the fact that the associated representations sat- isfy the Ramanujan-Petersson conjecture.
		In the case d = 2, the Ramanujan property can be shown by using the so called Jacquet- Langlands correspondence between automorphic repre- sentations of D?) (then D is a quaternion algebra) and automorphic rep- resentations of GL (2;AF), not given by multiplicative characters.
	www.uni-math.gwdg.de /asarveni/Short-versionI.txt (5292 words)

	Search Results for conjecture
		This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves.
		The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces.
		Aleksandrov and Urysohn had made a conjecture in 1923 concerning necessary and sufficient conditions for a Hausdorff space to be compact and this was not proved until 1935 when M H Stone gave an exceedingly complicated proof using representation theory of Boolean algebras.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=conjecture&CONTEXT=1 (7287 words)

	Ramanujan.htm
		Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
		Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic.
		MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions.
	www.cse.ohio-state.edu /~brinkmei/math/Ramanujan.htm (184 words)

	Mathematica Evidence that Ramanujan Kills Baker-Gammel-Wills -- from Mathematica Information Center
		A 1961 conjecture of Baker, Gammel and Wills asserts that if a function f is meromorphic in the unit ball, and analytic at zero, then a subsequence of its diagonal Padé approximants converges uniformly in compact subsets omitting poles.
		Inasmuch as the denominators of the Padé approximants are complex orthogonal polynomials, and the convergence of sequences of Padé approximants is determined largely by the behaviour of their poles, the conjecture deals with distribution of zeros of complex orthogonal polynomials.
		In this paper, we present numerical evidence derived using the Mathematica package, that Ramanujan's continued fraction H_q(z) provides a counterexample, provided q is appropriately chosen on the unit circle.
	library.wolfram.com /infocenter/Articles/2492 (119 words)

	List of conjectures - Wikipedia, the free encyclopedia
		Erdős conjecture, which lists conjectures of Paul Erdős and his collaborators
		Blattner's conjecture (now often known as the Blattner formula)
		Epsilon conjecture (an intermediate on the way to Fermat's last theorem)
	en.wikipedia.org /wiki/List_of_conjectures (126 words)

	Search Results for conjecture*
		In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all g(x) which have a maximum value and decrease monotonically on either side of this maximum.
		Burnside conjectured that every finite group of odd order is soluble and it is not surprising that he failed to prove this result as it was not proved until 1962 when W Feit and J C Thompson proved the result in a 300 page paper.
		E Winter conjectured (in 1933) - without proof - that these folios constitute meagre fragments of Bolzano's work 'Anti-Euclid' which - according to Bolzano's own report - was lost (it is perhaps possible that the lost 'Anti-Euclid' was written "according to such a detailed plan"), and that this work contained the concept of non-Euclidean geometry.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=conjecture*&CONTEXT=1 (10119 words)

	Pi - Wikipedia, the free encyclopedia
		Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π.
		Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of special right triangles that are either isosceles or halves of equilateral triangles.
		Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π.
	en.wikipedia.org /wiki/Pi (4630 words)

	The abc conjecture
		Assuming the Birch and Swinnerton-Dyer conjecture, it is shown in [Go-Sz] that this conjecture is equivalent to the Szpiro conjecture for modular elliptic curves.
		The pseudo-null conjecture of Greenberg (1976) asserts that these two invariants vanish for all p and all K. Assuming the truth of the abc conjecture for quadratic number fields for which the norm of a fundamental unit is -1, Ichimura [Ic] proved that there exist infinitely many primes p such that
		The Wieferich criterion, the ABC conjecture and Shimura's correspondence, Satya Mohit, M.Sc.
	www.math.unicaen.fr /~nitaj/abc.html (4274 words)

	Apostolos Doxiadis
		But "Uncle Petros and Goldbach's Conjecture" describes a passion wholly of the mind.
		Maybe I'd even have my Fields Medal, the Nobel Prize of Mathematics!' Even as he was speaking, in a flash of revelation, I guessed the awful truth," which readers may already have guessed themselves.
		Ramanujan's melancholy story echoes the novel's tone of bottomless longing.
	www.apostolosdoxiadis.com /page/?id=148&la=1 (1029 words)

	Ram Murty - Publications by Year
		On Ramanujan's tau function, Ramanujan Revisited, Proceedings of the Illinois conference on Ramanujan, (1987) 269-288.
		On Zagier's cusp form and the Ramanujan tau function, (with Ashwaq Hashim), Proceedings of the Indian Academy of Sciences, 104 (1) (1994) 93-98.
		The ABC conjecture and prime divisors of the Lucas and Lehmer sequences, (with Siman Wong), in Number Theory for the Millenium, Vol.
	www.mast.queensu.ca /~murty/index2.html (1878 words)

	Summer Seminar
		Abstract: In Ramanujan's 1916 letter to Hardy, he proposed some remarkable identities that were a special case of a general q-series continued fraction that he had discovered and investigated.
		Abstract: Selberg's orthogonality conjecture predicts that the coefficients of automorphic L-functions attached to different cuspidal representations are orthogonal.
		We conjecture that (i) There exists an N such that $\pi$ contains a vector invariant under the paramodular group of level N. (ii) If N is minimal with this property, then such a vector is unique up to multiples; we call it a local newform.
	www.math.wisc.edu /~thyang/Fall03.html (1724 words)

	References
		Hirschhorn, On the expansion of Ramanujan's continued fraction, The Ramanujan Journal, 2 (1998), 521-527.
		Ramanujan's contribution to continued fractions Wazir Hasan Abdi (ed), Toils and Triumphs of Srinivasa Ramanujan, the Man and the Mathematician.
		Hirschhorn, A generalisation of Winquist's identity and a conjecture of Ramanujan, J.
	web.maths.unsw.edu.au /~mikeh (935 words)

	Laurent Lafforgue and Nikita Nekrassov
		He gave the Peccot course at the Collège de France in 1996 and was an invited speaker at the 1998 International Congress of Mathematicians in Berlin.
		This conjecture relates arithmetic properties to properties of automorphic representations.
		In rank 2 and over number fields, the first significant confirmations of this conjecture were the proof of the Ramanujan conjecture by Pierre DELIGNE and the proof by LANGLANDS himself of the Artin conjecture, up to one case.
	www.ihes.fr /IHES-A/People/2nouveauxA.html (454 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		It was shown for classical holomorphic modular forms of weight 2 by Eichler, Shimura and Igusa, who reduced it to Weil's bounds on the number of points of curves over finite fields.
		The R-P conjecture for classical holomorphic cusp forms of any weight was then shown by Deligne to follow from the Weil conjectures, which he subsequently proved.
		One application of this is the so called Ramanujan property for ceratin graphs constructed by Lubotzky, Phillips and Sarnak.
	www.math.technion.ac.il /~techm/19991116093019991116liv (144 words)

	SASTRA PRIZE 2005
		This annual prize, being awarded for the first time, is for outstanding contributions by individuals not exceeding the age of 32 in areas of mathematics influenced by Ramanujan in a broad sense.
		The age limit was set at 32 because Ramanujan achieved so much in his brief life of 32 years.
		For an article describing the events leading to the launching of the SASTRA Ramanujan Prize, the prize ceremony, and the accomplishments of the winners, see Krishna Alladi's article The First SASTRA Ramanujan Prizes.
	www.math.ufl.edu /sastra-prize/2005.html (741 words)

	Mathematics Colloquium: Coble Memorial Lectures 2005
		One of the main themes in the study of the zeroes of these functions, is that of a family.
		In particular we highlight the role played by symmetry types (from random matrix theory) that are associated with a family.
		This will then be applied to the generalized Ramanujan conjectures, which we will introduce and discuss in some detail.
	www.math.uiuc.edu /Colloquia/05SP/sarnak_may03-05.html (117 words)

	Laurent Lafforgue, Fields Medal 2002
		In rank 1, this conjecture is nothing other than the now traditional "class field theory" of Emil Artin.
		In rank 2 and for number fields, the first great confirmations of this conjecture were the proof of the conjecture of Ramanujan per Pierre Deligne and the proof by Langlands itself of the conjecture of Artin except for a case.
		For that purpose, he built varieties similar to modular curves and showed certain cases of the conjecture of Langlands in rank 2.
	www.ihes.fr /EVENTS/lafforgue/aboutLaf.html (490 words)

	David Bressoud (Site not responding. Last check: 2007-10-29)
		The Borwein conjecture and partitions with prescribed hook differences.
		On the proof of Andrews' q-Dyson conjecture, pp.
		Definite integral evaluation by enumeration, partial results on the Macdonald conjectures.
	www.macalester.edu /~bressoud/pub/biblio.html (281 words)

	[No title]
		O1/O2 Ramanujan, Twelve Lectures on Subjects Suggested by his Life and Work.
		xxxiii A generalization of Winquist?s identity and a conjecture of Ramanujan.
		O On the Expansion of Ramanujan?s Continued Fraction.
	www.geocities.com /furmend/Ha_Hz.txt (4137 words)

	Rutgers Graduate Number Theory Seminar - Fall 2005
		Abstract I will restate the Quantum Unique Ergodicity conjecture for SL(2,Z) in terms of shifted convolution sums and summarize some analytic methods that have been applied in the hopes of showing Quantum Unique Ergodicity.
		Abstract We will state the Ramanujan Conjectures and reformulate them in terms of Representation Theory.
		Abstract Basic facts about Elliptic Curves including conjectures and what is known about the rank of an Elliptic Curve.
	www.math.rutgers.edu /~romanh/seminar/fall2005.html (484 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		ABSTRACT: Ramanujan graphs are k-regular graphs with optimal bounds on their eigenvalues.
		Their construction is based on the work of Deligne and Drinfeld on the Ramanujan conjecture for GL(2).
		The recent work of Lafforge which settles the Ramanujan conjecture for GL(n) over function fields opens the door to study of Ramanujan complexes: these are higher dimensional analogues which are obtained as quotients of the Bruhat-Tits building of PGL(n) over local fields.
	www.math.technion.ac.il /~techm/20040511161020040511lub (162 words)

	Who Can Name the Bigger Number?
		Here ‘efficient’ means using an amount of time proportional to at most the problem size raised to some fixed power—for example, the number of cities cubed.
		Take Goldbach’s conjecture, that every even number 4 or higher is a sum of two prime numbers: 10=7+3, 18=13+5.
		What’s more, brain-imaging evidence showed that the subjects’ parietal lobes, involved in spatial reasoning, were more active during approximation problems; while the left inferior frontal lobes, involved in verbal reasoning, were more active during exact calculation problems.
	www.scottaaronson.com /writings/bignumbers.html (6372 words)

	Ramanujan, Srinivasa
		Ramanujan and The Cubic Equation 3^3+4^3+5^3 = 6^3
		Ramanujan and The Quartic Equation 2^4+2^4+3^4+4^4+4^4 = 5^4
		Euler's Extended Conjecture and a^k+b^k+c^k = d^k for k > 4
	www.geocities.com /titus_piezas/ramanujan.html (184 words)

	Cohomology of Drinfeld Modular Varieties (Cambridge Studies in Advanced Mathematics) by Gérard Laumon [ISBN: ...
		The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field.
		By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case.
		This second volume is concerned with the ArthurSHSelberg trace formula, and to the proof in some cases of the Ramanujan-Petersson conjecture and the global Langlands conjecture for function fields.
	www.gettextbooks.com /isbn_0521470617.html (216 words)

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