In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers, invariant under the action of a discrete subgroup Γ ⊂ G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are automorphic forms defined over the groups SL or PSL with the discrete subgroup being the modular group, or one of its congruence subgroups. More one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group G, for an algebraic group G and an algebraic number field F, is a complex-valued function on G, left invariant under G and satisfies certain smoothness and growth conditions. Poincaré first discovered automorphic forms as generalizations of trigonometric and elliptic functions. Through the Langlands conjectures automorphic forms play an important role in modern number theory.
An automorphic form is a function F on G, subject to three kinds of conditions: to transform under translation by elements γ ∈ Γ according to the given factor of automorphy j. It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F with F for γ ∈ Γ. In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to'twist' them; the Casimir operator condition says. The third condition is to handle the case where G/Γ has cusps; the formulation requires the general notion of factor of automorphy j for Γ, a type of 1-cocycle in the language of group cohomology. The values of j may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms; the cocycle condition imposed on the factor of automorphy is something that can be checked, when j is derived from a Jacobian matrix, by means of the chain rule. Before this general setting was proposed, there had been substantial developments of automorphic forms other than modular forms.
The case of Γ a Fuchsian group had received attention before 1900. The Hilbert modular forms were proposed not long after that, though a full theory was long in coming; the Siegel modular forms, for which G is a symplectic group, arose from considering moduli spaces and theta functions. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by Ilya Piatetski-Shapiro, in the years around 1960, in creating such a theory; the theory of the Selberg trace formula, as applied by others, showed the considerable depth of the theory. Robert Langlands showed how the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms, he 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.
From the point of view of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan, as the heart of the matter. The subsequent notion of an "automorphic representation" has proved of great technical value when dealing with G an algebraic group, treated as an adelic algebraic group, it does not include the automorphic form idea introduced above, in that the adelic approach is a way of dealing with the whole family of congruence subgroups at once. Inside an L2 space for a quotient of the adelic form of G, an automorphic representation is a representation, an infinite tensor product of representations of p-adic groups, with specific enveloping algebra representations for the infinite prime. 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, it is this concept, basic to the formulation of the Langlands philosophy. One of Poincaré's first discoveries in mathematics, dating to the 1880s, was automorphic forms.
He named them Fuchsian functions, after the mathematician Lazarus Fuchs, because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one, analytic in its domain and is invariant under a discrete infinite group of linear fractional transformations. Automorphic functions generalize both trigonometric and elliptic functions. Poincaré explains how he discovered Fuchsian functions: For fifteen days I strove to prove
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, was one of the creators of the contemporary theory of linear algebraic groups, he studied at the ETH Zürich, where he came under the influence of the topologist Heinz Hopf and Lie-group theorist Eduard Stiefel. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Jean Leray and Henri Cartan, he collaborated with Jacques Tits in fundamental work on algebraic groups, with Harish-Chandra on their arithmetic subgroups. In an algebraic group G a Borel subgroup H is one minimal with respect to the property that the homogeneous space G/H is a projective variety. For example, if G is GLn we can take H to be the subgroup of upper triangular matrices. In this case it turns out that H is a maximal solvable subgroup, that the parabolic subgroups P between H and G have a combinatorial structure.
Both those aspects generalize, play a central role in the theory. The Borel−Moore homology theory applies to general locally compact spaces, is related to sheaf theory, he published a number of books, including a work on the history of Lie groups. In 1978 he received the Brouwer Medal and in 1992 he was awarded the Balzan Prize "For his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, for his indefatigable action in favour of high quality in mathematical research and the propagation of new ideas", he died in Princeton. He used to answer the question of whether he was related to Émile Borel alternately by saying he was a nephew, no relation. "I feel that what mathematics needs least are pundits who issue prescriptions or guidelines for less enlightened mortals." Borel–Weil–Bott theorem Borel conjecture Borel fixed-point theorem Borel's theorem Borel, Seminar on transformation groups, With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais.
Annals of Mathematics Studies, No. 46, Princeton University Press, MR 0116341 Borel, Cohomologie des espaces localement compacts d'après J. Leray. Exposés faits au séminaire de Topologie algébrique de l'École Polytechnique Fédérale, printemps 1951, Lecture Notes in Mathematics, 2, New York: Springer-Verlag, doi:10.1007/BFb0097851, MR 0174045 Borel, Halpern, Edward, ed. Topics in the homology theory of fibre bundles, Lecture Notes in Mathematics, 36, New York: Springer-Verlag, doi:10.1007/BFb0096867, MR 0221507 Borel, Introduction aux groupes arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Paris: Hermann, MR 0244260 Borel, Représentations de groupes localement compacts, Lecture Notes in Mathematics, 276, New York: Springer-Verlag, doi:10.1007/BFb0058407, MR 0414779 Borel, Linear algebraic groups, Graduate Texts in Mathematics, 126, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012 Borel, Intersection cohomology, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4764-3, MR 0788171 Borel, Armand.
I.: American Mathematical Society, ISBN 978-0-8218-0851-1, MR 1721403 Borel, Essays in the History of Lie Groups and Algebraic Groups, Providence, R. I.: American Mathematical Society, ISBN 978-0-8218-0288-5, MR 1847105 Borel, Armand, Œuvres: collected papers, I, II, III, New York: Springer-Verlag, ISBN 978-3-540-12126-8, MR 0725852 Borel, Armand, Œuvres: collected papers, IV, New York: Springer-Verlag, ISBN 978-3-540-67640-9, MR 1829820 Borel, Armand. Springer, Tonny A. "Armand Borel's work in the theory of linear algebraic groups", Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Mat
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Roger Evans Howe
Roger Evans Howe is William R. Kenan, Jr. Professor Emeritus of Mathematics at Yale University, Curtis D. Robert Endowed Chair in Mathematics Education at Texas A&M University, he is well known for his contributions to representation theory, in particular for the notion of a reductive dual pair, sometimes known as a Howe pair, the Howe correspondence. His contributions to mathematics education is well-documented, he attended Ithaca High School Harvard University as an undergraduate, winning the William Lowell Putnam Mathematical Competition in 1964. He obtained his Ph. D. from University of California, Berkeley in 1969. His thesis, titled On representations of nilpotent groups, was written under the supervision of Calvin Moore. Between 1969 and 1974, Howe taught at the State University of New York in Stony Brook before joining the Yale faculty in 1974, his doctoral students include Ju-Lee Kim, Jian-Shu Li, Zeev Rudnick, Eng-Chye Tan, Chen-Bo Zhu. He moved to Texas A&M University in 2015, he has been a member of the American Academy of Arts and Sciences since 1993, a member of the National Academy of Sciences since 1994.
Howe received a Lester R. Ford Award in 1984. In 2006 he was awarded the American Mathematical Society Distinguished Public Service Award in recognition of his "multifaceted contributions to mathematics and to mathematics education." In 2012 he became a fellow of the American Mathematical Society. In 2015 he received the inaugural Award for Excellence in Mathematics Education. Roger Howe, "Tamely ramified supercuspidal representations of Gln", Pacific Journal of Mathematics 73, no. 2, 437–460. Roger Howe and Calvin C. Moore, "Asymptotic properties of unitary representations", Journal of Functional Analysis 32, no. 1, 72–96. Roger Howe, "θ-series and invariant theory", in Automorphic forms, representations and L-functions, pp. 275–285. Roger Howe, "Wave front sets of representations of Lie groups". Automorphic forms, representation theory and arithmetic, pp. Tata Inst. Fund. Res. Studies in Math. 10, Tata Inst. Fundamental Res. Bombay, 1981. Roger Howe, "On a notion of rank for unitary representations of the classical groups".
Harmonic analysis and group representations, 223–331, Naples, 1982. Howe, Roger, "Remarks on classical invariant theory", Transactions of the American Mathematical Society, 313: 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR 2001418, MR 0986027 Howe, Roger, "Transcending classical invariant theory", Journal of the American Mathematical Society, 2: 535–552, doi:10.1090/S0894-0347-1989-0985172-6 Roger Howe, "Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond". The Schur lectures, 1–182, Israel Math. Conf. Proc. 8, Bar-Ilan Univ. Ramat Gan, 1995. Roger Howe & Eng-Chye Tan, "Nonabelian harmonic analysis. Applications of SL". Universitext. Springer-Verlag, New York, 1992. Xvi+257 pp. ISBN 0-387-97768-6. Roger Howe & William Barker Continuous Symmetry: From Euclid to Klein, American Mathematical Society, ISBN 978-0-8218-3900-3. Robin Hartshorne Review of Continuous Symmetry, American Mathematical Monthly 118:565–8. Oscillator semigroup Official website Roger Evans Howe at the Mathematics Genealogy Project