Mathematics

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

Differentiable manifold

In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts known as an atlas. One may apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas.

The maps that relate the coordinates defined by the various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable tensor and vector fields. Differentiable manifolds are important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, Yang–Mills theory, it is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus; the study of calculus on differentiable manifolds is known as differential geometry.

The emergence of differential geometry as a distinct discipline is credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen, he motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, presciently described the role of coordinate systems and charts in subsequent formal developments: Having constructed the notion of a manifoldness of n dimensions, found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude... – B. RiemannThe works of physicists such as James Clerk Maxwell, mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one, invariant with respect to coordinate transformations; these ideas found a key application in Einstein's theory of general relativity and its underlying equivalence principle.

A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a topological manifold is a second countable Hausdorff space, locally homeomorphic to a vector space, by a collection of homeomorphisms called charts; the composition of one chart with the inverse of another chart is a function called a transition map, defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced contains the data of how it has been patched together. However, different atlases may produce "the same" manifold. Thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below. There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions.

Some common examples include the following: A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. More a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable. A smooth manifold or C∞-manifold is a differentiable manifold for which all the transition maps are smooth; that is, derivatives of all orders exist. An equivalence class of such atlases is said to be a smooth structure. An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is convergent and equals the function on some open ball. A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic. While there is a meaningful notion of a Ck atlas, there is no distinct notion of a Ck manifold other than C0 and C∞, because for every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure – a result of Whitney.

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Michael Atiyah

Sir Michael Francis Atiyah was a British-Lebanese mathematician specialising in geometry. Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at University of Oxford and University of Cambridge, in the United States at the Institute for Advanced Study, he was the President of the Royal Society, founding director of the Isaac Newton Institute, master of Trinity College, chancellor of the University of Leicester, the President of the Royal Society of Edinburgh. From 1997 until his death, he was an honorary professor at the University of Edinburgh. Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch and Isadore Singer, his students included Graeme Segal, Nigel Hitchin and Simon Donaldson. Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, informally speaking, describes ways in which spaces can be twisted, his best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations.

Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Atiyah was born on 22 April 1929 in Hampstead, England, the son of Jean and Edward Atiyah, his mother was Scottish and his father was a Lebanese Orthodox Christian. He had two brothers and Joe, a sister, Selma. Atiyah went to primary school at the Diocesan school in Khartoum, Sudan and to secondary school at Victoria College in Cairo and Alexandria, he returned to England and Manchester Grammar School for his HSC studies and did his national service with the Royal Electrical and Mechanical Engineers. His undergraduate and postgraduate studies took place at Cambridge, he was a doctoral student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry. During his time at Cambridge, he was president of The Archimedeans.

Atiyah spent the academic year 1955–1956 at the Institute for Advanced Study, Princeton returned to Cambridge University, where he was a research fellow and assistant lecturer a university lecturer and tutorial fellow at Pembroke College, Cambridge. In 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherine's College, he became Savilian Professor of Geometry and a professorial fellow of New College, from 1963 to 1969. He took up a three-year professorship at the Institute for Advanced Study in Princeton after which he returned to Oxford as a Royal Society Research Professor and professorial fellow of St Catherine's College, he was president of the London Mathematical Society from 1974 to 1976. Atiyah was president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002, he contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies, the European Mathematical Society.

Within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director. He was President of the Royal Society, Master of Trinity College, Chancellor of the University of Leicester, president of the Royal Society of Edinburgh. From 1997 until his death in 2019 he was an honorary professor in the University of Edinburgh, he was a Trustee of the James Clerk Maxwell Foundation. Atiyah collaborated with many mathematicians, his three main collaborations were with Raoul Bott on the Atiyah–Bott fixed-point theorem and many other topics, with Isadore M. Singer on the Atiyah–Singer index theorem, with Friedrich Hirzebruch on topological K-theory, all of whom he met at the Institute for Advanced Study in Princeton in 1955, his other collaborators included. Manin, Nick S. Manton, Vijay K. Patodi, A. N. Pressley, Elmer Rees, Wilfried Schmid, Graeme Segal, Alexander Shapiro, L. Smith, Paul Sutcliffe, David O. Tall, John A. Todd, Cumrun Vafa, Richard S. Ward and Edward Witten.

His research on gauge field theories Yang–Mills theory, stimulated important interactions between geometry and physics, most notably in the work of Edward Witten. Atiyah's students included.

Grothendieck–Riemann–Roch theorem

In mathematics in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more their characteristic classes in homology or algebraic analogues thereof; the classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves; the theorem has been influential, not least for the development of the Atiyah–Singer index theorem.

Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript published. Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958. Grothendieck and his collaborators simplified and generalized the proof. Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group K 0 of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character as a functorial transformation ch: K 0 → A, where A d is the Chow group of cycles on X of dimension d modulo rational equivalence, tensored with the rational numbers. In case X is defined over the complex numbers, the latter group maps to the topological cohomology group H 2 d i m − 2 d. Now consider a proper morphism f: X → Y between smooth quasi-projective schemes and a bounded complex of sheaves F ∙ on X.

The Grothendieck–Riemann–Roch theorem relates the pushforward map f! = ∑ i R i f ∗: K 0 → K 0 and the pushforward f ∗: A → A, by the formula ch td = f ∗. Here td is the Todd genus of X, thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the Chern character and shows that the needed correction factors depend on X and Y only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck–Riemann–Roch formula as ch = f ∗, where Tf is the relative tangent sheaf of f, defined as the element TX − f*TY in K0. For example, when f is a smooth morphism, Tf is a vector bundle, known as the tangent bundle along the fibers of f. Using A1-homotopy theory, the Grothendieck–Riemann–Roch theorem has been extended by Navarro & Navarro to the situation where f is a proper map between two smooth schemes. Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination chtd and to the non-proper case by considering cohomology with compact support.

The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem to arithmetic schemes. The Hirzebruch–Riemann–Roch theorem is the special case where Y is a point and the field is the field of complex numbers; the version of Riemann-Roch theorem for oriented cohom

Friedrich Hirzebruch

Friedrich Ernst Peter Hirzebruch ForMemRS was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period." Hirzebruch was born in Hamm, Westphalia in 1927. His father of the same name was a math teacher. Hirzebruch studied at the University of Münster from 1945–1950, with one year at ETH Zürich. Hirzebruch held a position at Erlangen, followed by the years 1952–54 at the Institute for Advanced Study in Princeton, New Jersey. After one year at Princeton University 1955–56, he was made a professor at the University of Bonn, where he remained, becoming director of the Max-Planck-Institut für Mathematik in 1981. More than 300 people gathered in celebration of his 80th birthday in Bonn in 2007; the Hirzebruch–Riemann–Roch theorem for complex manifolds was a major advance and became part of the mainstream developments around the classical Riemann–Roch theorem.

Hirzebruch's book Neue topologische Methoden in der algebraischen Geometrie was a basic text for the'new methods' of sheaf theory, in complex algebraic geometry. He went on to write the foundational papers on topological K-theory with Michael Atiyah, collaborate with Armand Borel on the theory of characteristic classes. In his work he provided a detailed theory of Hilbert modular surfaces, working with Don Zagier. In March 1945, Hirzebruch became a soldier, in April, in the last weeks of Hitler's rule, he was taken prisoner by the British forces invading Germany from the west; when a British soldier found that he was studying mathematics, he drove him home and released him, told him to continue studying. Hirzebruch died at the age of 84 on 27 May 2012. Amongst many other honours, Hirzebruch was awarded a Wolf Prize in Mathematics in 1988 and a Lobachevsky Medal in 1989; the government of Japan awarded him the Order of the Sacred Treasure in 1996. Hirzebruch won an Einstein Medal in 1999, received the Cantor medal in 2004.

Hirzebruch was a foreign member of numerous academies and societies, including the United States National Academy of Sciences, the Russian Academy of Sciences, the Royal Society and the French Academy of Sciences. In 1980–81 he delivered the first Sackler Distinguished Lecture in Israel