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
David Kazhdan or Každan, Kazhdan named Dmitry Aleksandrovich Kazhdan, is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan was born on 20 June 1946 in Moscow, USSR, his father is Alexander Kazhdan. He earned a doctorate under Alexandre Kirillov in 1969 and was a member of Israel Gelfand's school of mathematics, he is Jewish, emigrated from the Soviet Union to take a position at Harvard University in 1975. He became an Orthodox Jew around that time. In 2002 he immigrated to Israel and is now a professor at the Hebrew University of Jerusalem as well as a professor emeritus at Harvard. On October 6, 2013, Kazhdan was critically injured in a car accident while riding a bicycle in Jerusalem. Kazhdan has four children, his son, Eli Kazhdan, was general director of Natan Sharansky's Yisrael BaAliyah political party. He is known for collaboration with Israel Gelfand, Victor Kac, George Lusztig, with Grigory Margulis, with Yuval Flicker and S. J. Patterson on the representations of metaplectic groups.
Kazhdan's property is used in representation theory. Kazhdan held a MacArthur Fellowship from 1990 to 1995, he was the doctoral advisor of Vladimir Voevodsky, a recipient of the Fields Medal, one of the highest awards in mathematics. Kazhdan has been a member of United States National Academy of Sciences since 1990, of the Israel Academy of Sciences since 2006, of the American Academy of Arts and Sciences since 2008. In 2012, he was awarded the Israel Prize, the country's highest academic honor, for mathematics and computer science. Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison and Edward Witten. American Mathematical Society, Providence, RI. Vol. 1: xxii+723 pp.. ISBN 0-8218-1198-3, 81-06 American Academy of Arts and Sciences, Class of 2008 Official Harvard home page Official Hebrew University home page David Kazhdan at the Mathematics Genealogy Project
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity taken to be the ring of real numbers or the ring of integers; the continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems in the context of the Stone–von Neumann theorem. More one can consider Heisenberg groups associated to n-dimensional systems, most to any symplectic vector space. In the three-dimensional case, the product of two Heisenberg matrices is given by: =. Since the multiplication is not commutative, the group is non-abelian; the neutral element of the Heisenberg group is the identity matrix, inverses are given by − 1 =. The group is a subgroup of the 2-dimensional affine group Aff: acting on corresponds to the affine transform x → +. There are several prominent examples of the three-dimensional case.
If a, b, c, are real numbers one has the continuous Heisenberg group H3. It is a nilpotent real Lie group of dimension 3. In addition to the representation as real 3x3 matrices, the continuous Heisenberg group has several different representations in terms of function spaces. By Stone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character; this representation has models. In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions. In the theta representation, it acts on the space of holomorphic functions on the upper half-plane. If a, b, c, are integers one has the discrete Heisenberg group H3, it is a non-abelian nilpotent group. It has two generators, x =, y = and relations z = x y x − 1 y − 1, x z = z x, y z = z y,where z =
André Weil was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was the de facto early leader of the mathematical Bourbaki group; the philosopher Simone Weil was his sister. André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71; the famous philosopher Simone Weil was Weil's only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he taught himself Sanskrit in 1920. After teaching for one year in Aix-Marseille University, he taught for six years in Strasbourg, he married Éveline in 1937. Weil was in Finland, his wife Éveline returned to France without him.
Weil was mistakenly arrested in Finland at the outbreak of the Winter War on suspicion of spying. Weil returned to France via Sweden and the United Kingdom, was detained at Le Havre in January 1940, he was charged with failure to report for duty, was imprisoned in Le Havre and Rouen. It was in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that Weil completed the work that made his reputation, he was tried on 3 May 1940. Sentenced to five years, he requested to be attached to a military unit instead, was given the chance to join a regiment in Cherbourg. After the fall of France, he met up with his family in Marseille, he went to Clermont-Ferrand, where he managed to join his wife Éveline, living in German-occupied France. In January 1941, Weil and his family sailed from Marseille to New York, he spent the remainder of the war in the United States, where he was supported by the Rockefeller Foundation and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated and poorly paid, although he didn't have to worry about being drafted, unlike his American students.
But, he hated Lehigh much for their heavy teaching workload and he swore that he would never talk about "Lehigh" any more. He quit the job at Lehigh, he moved to Brazil and taught at the Universidade de São Paulo from 1945 to 1947, where he worked with Oscar Zariski, he returned to the United States and taught at the University of Chicago from 1947 to 1958, before moving to the Institute for Advanced Study, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts, in 1954 in Amsterdam, in 1978 in Helsinki. In 1979, Weil shared the second Wolf Prize in Mathematics with Jean Leray. Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between algebraic geometry and number theory; this began in his doctoral work leading to the Mordell–Weil theorem. Mordell's theorem had an ad hoc proof. Both aspects of Weil's work have developed into substantial theories. Among his major accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields, his subsequent laying of proper foundations for algebraic geometry to support that result.
The so-called Weil conjectures were hugely influential from around 1950. Weil introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, gave a proof of the Riemann–Roch theorem with them. His'matrix divisor' Riemann–Roch theorem from 1938 was a early anticipation of ideas such as moduli spaces of bundles; the Weil conjecture on Tamagawa numbers proved resistant for many years. The adelic approach became basic in automorphic representation theory, he picked up another credited Weil conjecture, around 1967, which under pressure from Serge Lang became known as the Taniyama–Shimura conjecture based on a formulated question of Taniyama at the 1955 Nikkō conference. His attitude towards conjectures was that one should not dignify a guess as a conjecture and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s. Other significant results were on Pontryagin differential geometry, he introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki.
His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s, reprinted in his collected papers, proved most influential. He created the ∅, he discovered that the so-called Weil representation introduced in quantum mechanics by Irving Segal an
In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, of quadratic forms, they have been applied to soliton theory. When generalized to a Grassmann algebra, they appear in quantum field theory; the most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables, a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this comes from a line bundle condition of descent. There are several related functions called Jacobi theta functions, many different and incompatible systems of notation for them. One Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has positive imaginary part.
It is given by the formula ϑ = ∑ n = − ∞ ∞ exp = 1 + 2 ∑ n = 1 ∞ n 2 cos = ∑ n = − ∞ ∞ q n 2 η n where q = exp is the nome and η = exp. It is a Jacobi form. If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; the function behaves regularly with respect to its quasi-period τ and satisfies the functional equation ϑ = exp ϑ where a and b are integers. The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript: ϑ 00 = ϑ The auxiliary functions are defined by ϑ 01 = ϑ ϑ 10 = exp ϑ ϑ 11 = exp ϑ (
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups G is an extension of Q by N if there is a short exact sequence 1 → N → G → Q → 1. If G is an extension of Q by N G is a group, N is a normal subgroup of G and the quotient group G/N is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups; this fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center of G. One extension, the direct product, is obvious. If one requires G and Q to be abelian groups the set of isomorphism classes of extensions of Q by a given group N is in fact a group, isomorphic to Ext Z 1 .
Several other general classes of extensions are known but no theory exists which treats all the possible extensions at one time. Group extension is described as a hard problem. To consider some examples, if G = K × H G is an extension of both H and K. More if G is a semidirect product of K and H, written as G = K ⋊ H G is an extension of H by K, so such products as the wreath product provide further examples of extensions; the question of what groups G are extensions of H by N is called the extension problem, has been studied since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups, where each Ai+1 is an extension of Ai by some simple group; the classification of finite simple groups gives us a complete list of finite simple groups. Solving the extension problem amounts to classifying all extensions of H by K. In general, this problem is hard, all the most useful results classify extensions that satisfy some additional condition.
It is important to know when two extensions are congruent. We say that the extensions 1 → K → i G → π H → 1 and 1 → K → i ′ G ′ → π ′ H → 1 are equivalent if there exists a group isomorphism T: G → G ′ making commutative the diagram of Figure 1. In fact it is sufficient to have a group homomorphism, it may happen that the extensions 1 → K → G → H → 1 and 1 → K → G ′ → H → 1 are inequivalent but G and G' are isomorphic as groups. For instance, there are 8 inequivalent extensions of the Klein four-group by Z / 2 Z, but there are, up to group isomorphism, only four groups of order 8 containing a normal subgroup of order 2 with quotient group isomorphic to the Klein four-group. A trivial extension is an extension 1 → K → G → H → 1, equivalent to the extension 1 → K → K × H → H → 1 where the left and right arrows are the inclusion and the projection of each factor of: K × H. A split extension is an extension 1 → K → G → H → 1 with a homomorphism s: H → G such that going from H to G by s and back to H by the quotient map of the short exact sequence induces the identity map on H i.e. π ∘ s = i d H.
In this situation, it is said that s splits th
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