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.
Alexander Grothendieck was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics, he is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was raised and lived in France. For much of his working life, however, he was, in effect, stateless; as he spelled his first name "Alexander" rather than "Alexandre" and his surname, taken from his mother, was the Dutch-like Low German "Grothendieck", he was sometimes mistakenly believed to be of Dutch origin. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques and remained there until 1970, driven by personal and political convictions, he left following a dispute over military funding.
He became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political causes. Soon after his formal retirement in 1988, he moved to the Pyrenees, where he lived in isolation until his death in 2014. Grothendieck was born in Berlin to anarchist parents, his father, Alexander "Sascha" Schapiro, had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a Protestant family in Hamburg and worked as a journalist. Both had broken away from their early backgrounds in their teens. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and his birthname was recorded as "Alexander Raddatz." The marriage was dissolved in 1929 and Schapiro/Tanaroff acknowledged his paternity, but never married Hanka. Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism, followed soon thereafter by his mother.
They left Grothendieck in the care of a Lutheran pastor and teacher in Hamburg. During this time, his parents took part in the Spanish Civil War, according to Winfried Scharlau, as non-combatant auxiliaries, though others state that Sascha fought in the anarchist militia. In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterwards his father was interned in Le Vernet, he and his mother were interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". The first was the Rieucros Camp, where his mother contracted the tuberculosis which caused her death and where Alexander managed to attend the local school, at Mende. Once Alexander managed to escape from the camp, intending to assassinate Hitler, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II. Alexander was permitted to live, separated from his mother, in the village of Le Chambon-sur-Lignon and hidden in local boarding houses or pensions, though he had to seek refuge in the woods during Nazis raids, surviving at times without food or water for several days.
His father was arrested under the Vichy anti-Jewish legislation, sent to the Drancy, handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Chambon, Grothendieck attended the Collège Cévenol, a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Chambon attended Cévenol, it was at this school that Grothendieck first became fascinated with mathematics. After the war, the young Grothendieck studied mathematics in France at the University of Montpellier where he did not perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of independent studies there, he went to continue his studies in Paris in 1948. Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz and Jean Dieudonné on functional analysis, from 1950 to 1953.
At this time he was a leading expert in the theory of topological vector spaces. From 1953 to 1955 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he refused to take French Nationality. By 1957, he set this subject aside in order to work in algebraic homological algebra; the same year he was invited to visit Harvard by Oscar Zariski, but the offer fell through when he refused to sign a pledge promising not to work to overthrow the United States government, a position that, he was warned, might have landed him in prison. The prospect did not worry him. Comparing Grothendieck during his Nancy years to the École Normale Supérieure trained students at that time: Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, Bernard Malgrange, Leila Schneps says: He was so unknown to this group and to their professors, came from such a deprived and chaotic background, was, compared to them, so ignorant at the start of his research career
Raoul Bott was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, the Borel–Bott–Weil theorem. Bott was born in Budapest, the son of Margit Kovács and Rudolph Bott, his father was of Austrian descent, his mother was of Hungarian Jewish descent. Bott spent his working life in the United States, his family emigrated to Canada in 1938, subsequently he served in the Canadian Army in Europe during World War II. Bott went to college at McGill University in Montreal, where he studied electrical engineering, he earned a Ph. D. in mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled Electrical Network Theory, was written under the direction of Richard Duffin. Afterward, he began teaching at the University of Michigan in Ann Arbor. Bott continued his study at the Institute for Advanced Study in Princeton, he was a professor at Harvard University from 1959 to 1999.
In 2005 Bott died of cancer in San Diego. With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive-real functions. In 1949 they proved a fundamental theorem of filter synthesis. Duffin and Bott extended earlier work by Otto Brune that requisite functions of complex frequency s could be realized by a passive network of inductors and capacitors; the proof, relying on induction on the sum of the degrees of the polynomials in the numerator and denominator of the rational function, was published in Journal of Applied Physics, volume 20, page 816. In his 2000 interview with Allyn Jackson of the American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with network synthesis and electronic filter topology introduced him to algebraic topology. Bott met Arnold S. Shapiro at the IAS and they worked together, he studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem.
In the course of this work, he introduced Morse–Bott functions, an important generalization of Morse functions. This led to his role as collaborator over many years with Michael Atiyah via the part played by periodicity in K-theory. Bott made important contributions towards the index theorem in formulating related fixed-point theorems, in particular the so-called'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem; the major Atiyah–Bott papers on what is now the Atiyah–Bott fixed-point theorem were written in the years up to 1968. In the 1980s, Atiyah and Bott investigated gauge theory, using the Yang–Mills equations on a Riemann surface to obtain topological information about the moduli spaces of stable bundles on Riemann surfaces. In 1983 he spoke to the Canadian Mathematical Society in a talk he called "A topologist marvels at Physics", he is well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups.
He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds and the Bott cannibalistic class. In 1964, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. In 1987, he was awarded the National Medal of Science. In 2000, he received the Wolf Prize. In 2005, he was elected an Overseas Fellow of the Royal Society of London. Bott had 35 Ph. D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, András Szenes, Kevin Corlette. 1995: Collected Papers. Vol. 4. Mathematics Related to Physics. Edited by Robert MacPherson. Contemporary Mathematicians. Birkhäuser Boston, xx+485 pp. ISBN 0-8176-3648-X MR1321890 1995: Collected Papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxii+610 pp. ISBN 0-8176-3647-1 MR1321886 1994: Collected Papers.
Vol. 2. Differential Operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxiv+802 pp. ISBN 0-8176-3646-3 MR1290361 1994: Collected Papers. Vol. 1. Topology and Lie Groups. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xii+584 pp. ISBN 0-8176-3613-7 MR1280032 1982: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics #82. Springer-Verlag, New York-Berlin. Xiv+331 pp. ISBN 0-387-90613-4 doi:10.1007/978-1-4757-3951-0 MR0658304 1969: Lectures on K. Mathematics Lecture Note New York-Amsterdam x +203 pp. MR0258020 Raoul Bott at the Mathematics Genealogy Project Commemorative website at Harvard Math Department "The Life and Works of Raoul Bott", by Loring Tu. "Raoul Bott, an Innovator in Mathematics, Dies at 82", The New York Times, January 8, 2006
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
Frequency is the number of occurrences of a repeating event per unit of time. It is referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency; the period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals, radio waves, light. For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics and radio, frequency is denoted by a Latin letter f or by the Greek letter ν or ν; the relation between the frequency and the period T of a repeating event or oscillation is given by f = 1 T.
The SI derived unit of frequency is the hertz, named after the German physicist Heinrich Hertz. One hertz means. If a TV has a refresh rate of 1 hertz the TV's screen will change its picture once a second. A previous name for this unit was cycles per second; the SI unit for period is the second. A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. 60 rpm equals one hertz. As a matter of convenience and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are described by their frequency instead of period; these used conversions are listed below: Angular frequency denoted by the Greek letter ω, is defined as the rate of change of angular displacement, θ, or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument to the sine function: y = sin = sin = sin d θ d t = ω = 2 π f Angular frequency is measured in radians per second but, for discrete-time signals, can be expressed as radians per sampling interval, a dimensionless quantity.
Angular frequency is larger than regular frequency by a factor of 2π. Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.: y = sin = sin d θ d x = k Wavenumber, k, is the spatial frequency analogue of angular temporal frequency and is measured in radians per meter. In the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has an inverse relationship to the wavelength, λ. In dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ. In the special case of electromagnetic waves moving through a vacuum v = c, where c is the speed of light in a vacuum, this expression becomes: f = c λ; when waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can done in the following ways, Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period dividing the count by the length of the time period.
For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz If the number of counts is not large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count; this is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. A contractible space is one with the homotopy type of a point, it follows. Therefore any space with a nontrivial homotopy group cannot be contractible. Since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a topological space X the following are all equivalent: X is contractible. X is homotopy equivalent to a one-point space. X deformation retracts onto a point. For any space Y, any two maps f,g: Y → X are homotopic. For any space Y, any map f: Y → X is null-homotopic; the cone on a space X is always contractible. Therefore any space can be embedded in a contractible one. Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X; every contractible space is path connected and connected.
Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0. A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractible spaces are not locally contractible nor vice versa. For example, the comb space is contractible but not locally contractible. Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally connected, locally path connected, locally connected. Any Euclidean space is contractible; the Whitehead manifold is contractible. Spheres of any finite dimension are not contractible; the unit sphere in an infinite-dimensional Hilbert space is contractible. The house with two rooms is standard example of a space, contractible, but not intuitively so; the Dunce hat is not collapsible. The cone on a Hawaiian earring is contractible, but not locally contractible or locally connected. All manifolds and CW complexes are in general not contractible; the Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting and.
It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not contractible
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