Neil Trudinger

Neil Sidney Trudinger is an Australian mathematician, known for his work in the field of nonlinear elliptic partial differential equations. After completing his B. Sc at the University of New England in 1962, he continued his graduate studies at Stanford University, he was awarded a Ph. D in 1966 for his thesis "Quasilinear Elliptical Partial Differential Equations in n Variables". After the award of his doctorate from Stanford University, Trudinger became a Courant Instructor at the Courant Institute of Mathematical Sciences of New York University during the academic year 1966-67, he returned to Australia where he was appointed as a lecturer at Macquarie University in 1967. In 1970, he moved to University of Queensland where he was first appointed as a Reader as Professor. In 1973 he moved to the Australian National University. In 2016 he moved to the University of Wollongong, where he is appointed as a Distinguished Professor. At the ANU Trudinger served as Head of the Department of Pure Mathematics, as Director of the Centre for Mathematical Analysis and as Director of the Centre for Mathematics and its Applications, before becoming Dean of the School of Mathematical Sciences in 1992.

He coordinates ANU's Applied and Nonlinear Analysis program. He is co-author, together with his thesis advisor, David Gilbarg, of the book Elliptic Partial Differential Equations of Second Order. 1978, elected as a fellow of the Australian Academy of Science. 1981, first recipient of the Australian Mathematical Society Medal. 1996, awarded the Hannan Medal of the Australian Academy of Science. 1997, elected as a Fellow of the Royal Society of London. 2008, awarded the Leroy P. Steele Prize for Mathematical Exposition by the American Mathematical Society. 2012, elected as a fellow of the American Mathematical Society. 2014, gave the Łojasiewicz Lecture at the Jagiellonian University in Kraków. Trudinger's theorem Yamabe problem MacTutor biography Neil Trudinger at the Mathematics Genealogy Project ANU Applied and Nonlinear Analysis program

Jürgen Moser

Jürgen Kurt Moser was an award-winning, German-American mathematician, honored for work spanning over 4 decades, including Hamiltonian dynamical systems and partial differential equations. Moser's mother Ilse Strehlke was a niece of composer Louis Spohr, his father was the neurologist Kurt E. Moser, born to the merchant Max Maync and Clara Moser; the latter descended from 17th century French Hugenot immigrants to Prussia. Jürgen Moser's parents lived in Königsberg, German empire and resettled in Stralsund, East Germany as a result of the second world war. Moser attended the Wilhelmsgymnasium in his hometown, a high school specializing in mathematics and natural sciences education, from which David Hilbert had graduated in 1880, his older brother Friedrich Robert Ernst Moser served in the German Army and died in Pillkallen/Schloßberg. Moser married the biologist Dr. Gertrude C. Courant on September 10, 1955 and took up permanent residence in New Rochelle, New York in 1960, commuting to work in New York City.

In 1980 he moved to Switzerland. He was a member of the Akademisches Orchester Zürich, he was survived by his younger brother, the photographic printer and processor Klaus T. Moser-Maync from Northport, New York, his wife, Gertrude Moser from Seattle, their daughters, the theater designer Nina Moser from Seattle and the mathematician Lucy I. Moser-Jauslin from Dijon, his stepson, the lawyer Richard D. Emery from New York City. Moser played the piano and the cello, performing chamber music since his childhood in the tradition of a musical family, where his father played the violin and his mother the piano, he was a lifelong amateur astronomer and took up paragliding in 1988 during a visit at IMPA in Rio de Janeiro. Moser completed his undergraduate education at and received his Ph. D. from the University of Göttingen in 1952, studying under Franz Rellich. After his thesis, he came under the influence of Carl Ludwig Siegel, with whom he coauthored the second and expanded English language edition of a monography on celestial mechanics.

Having spent the year 1953 at the Courant Institute of New York University as a Fulbright scholar, he emigrated to the United States in 1955 becoming a citizen in 1959. He became a professor at MIT and at New York University, he served as director of the Courant Institute of New York University in the period of 1967–1970. In 1970 he declined the offer of a chair at the Institute for Advanced Study in Princeton. After 1980 he was at ETH Zürich, becoming professor emeritus in 1995, he was director of the Forschungsinstitut für Mathematik at ETH Zürich in 1984 - 1995, where he succeeded Beno Eckmann. He led a rebuilding of the ETH Zürich mathematics faculty. Moser was president of the International Mathematical Union in 1983–1986. Among Moser's students were Mark Adler of Brandeis University, Ed Belbruno, Charles Conley, Howard Jacobowitz of Rutgers University, Paul Rabinowitz of University of Wisconsin. Moser won the first George David Birkhoff Prize in 1968 for contributions to the theory of Hamiltonian dynamical systems, the James Craig Watson Medal in 1969 for his contributions to dynamical astronomy, the Brouwer Medal of the Royal Dutch Mathematical Society in 1984, the Cantor Medal of the Deutsche Mathematiker-Vereinigung in 1992 and the Wolf Prize in 1995 for his work on stability in Hamiltonian systems and on nonlinear differential equations.

He was elected to membership of the National Academy of Sciences in 1973 and was corresponding member of numerous foreign academies such as the London Mathematical Society and the Akademie der Wissenschaften und Literatur, Mainz. At three occasions he was an invited speaker at the quadrennial International Congress of Mathematicians, namely in Stockholm in the section on Applied Mathematics, in Helsinki in the section on Complex Analysis, a plenary speaker in Berlin. In 1990 he was awarded honorary doctorates from University of Bochum and from Pierre and Marie Curie University in Paris; the Society for Industrial and Applied Mathematics established a lecture prize in his honor in 2000. Mather, John N.. "Jürgen K. Moser". Notices of the AMS. 4: 1392–1405. Retrieved 2007-08-20. J. J. O'Connor. "Jürgen Kurt Moser". Retrieved 2008-07-04. Sylvia Nasar. "Obituary, New York Times". The New York Times. Retrieved 2010-09-14. American Institute of Physics. "Professional biography Jürgen Moser". Retrieved 2010-12-05.

Vladimir Arnold. "Déclin des Mathématiques". Retrieved 2012-10-11. ETH. "Biography of Jürgen Moser, by ETH". ETH. Retrieved 2013-04-02. Guardian. "Obituary of Moser, by Guardian". Retrieved 2013-05-27. SIAM. "Moser Lecture, by SIAM". Retrieved 2013-11-16. Max Planck Institut Leipzig. "In memoriam Jürgen Moser". Moser Symposium, by MPI Leipzig. Retrieved 2013-11-16

Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.

In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.

During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.

In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.

In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set