1.
Lecce
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It is the main city of the Salentine Peninsula, a sub-peninsula at the heel of the Italian Peninsula and is over 2,000 years old. Because of the rich Baroque architectural monuments found in the city, the city also has a long traditional affinity with Greek culture going back to its foundation, the Messapii who founded the city are said to have been Cretans in Greek records. To this day, in the Grecìa Salentina, a group of towns not far from Lecce, in terms of industry, the Lecce stone—a particular kind of limestone—is one of the citys main exports, because it is very soft and workable, thus suitable for sculptures. Lecce is also an important agricultural centre, chiefly for its oil and wine production. According to legend, a city called Sybar existed at the time of the Trojan War and it was conquered by the Romans in the 3rd century BC, receiving the new name of Lupiae. Under the emperor Hadrian the city was moved 3 kilometres to the northeast, Lecce had a theater and an amphitheater and was connected to the Hadrian Port. Orontius of Lecce, locally called SantOronzo, is considered to have served as the citys first Christian bishop and is Lecces patron saint, after the fall of the Western Roman Empire, Lecce was sacked by the Ostrogoth king Totila in the Gothic Wars. It was restored to Roman rule in 549, and remained part of the Eastern Empire for five centuries, with brief conquests by Saracens, Lombards, Hungarians and Slavs. After the Norman conquest in the 11th century, Lecce regained commercial importance, flourishing in the subsequent Hohenstaufen, the County of Lecce was one of the largest and most important fiefs in the Kingdom of Sicily from 1053 to 1463, when it was annexed directly to the crown. From the 15th century, Lecce was one of the most important cities of southern Italy, to avert invasion by the Ottomans, a new line of walls and a castle were built by Charles V, in the first part of the 16th century. In 1656, a plague broke out in the city, killing a thousand inhabitants, in 1943, fighter aircraft based in Lecce helped support isolated Italian garrisons in the Aegean Sea during World War 2. Because they were delayed by the Allies, they couldnt prevent a defeat, church of the Holy Cross, Construction of the Chiesa di Santa Croce) was begun in 1353, but work halted until 1549, and it was completed only by 1695. The church has a richly decorated façade with animals, grotesque figures and vegetables, next to the church is the Government Palace, a former convent. San Niccolò and Cataldo The church is an example of Italo-Norman architecture and it was founded by Tancred of Sicily in 1180. In 1716 the façade was rebuilt, with the addition of numerous statues, the walls were frescoed during the 15th-17th centuries. Celestine Convent, Built in Baroque-style by Giuseppe Zimbalo, the courtyard was designed by Gabriele Riccardi. Santa Irene, This church was commissioned in 1591 by the Theatines and it has a large façade showing different styles in the upper and lower parts. Above the portal stands a statue of Ste Irene by Mauro Manieri, the interior is on the Latin cross plan and is rather sober
Lecce
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Top left:Church of Santa Croce, Top right:Lecce Teatro Romano, Bottom left:Lecce Porta Napoli in Universita Street, Bottom middle:Saint Giovanni Cathedral in Perroni area, Bottom right:Lecce Cathedral in Duomo Square
Lecce
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Piazza del Duomo
Lecce
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The Roman Amphitheatre
2.
Italy
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Italy, officially the Italian Republic, is a unitary parliamentary republic in Europe. Located in the heart of the Mediterranean Sea, Italy shares open land borders with France, Switzerland, Austria, Slovenia, San Marino, Italy covers an area of 301,338 km2 and has a largely temperate seasonal and Mediterranean climate. Due to its shape, it is referred to in Italy as lo Stivale. With 61 million inhabitants, it is the fourth most populous EU member state, the Italic tribe known as the Latins formed the Roman Kingdom, which eventually became a republic that conquered and assimilated other nearby civilisations. The legacy of the Roman Empire is widespread and can be observed in the distribution of civilian law, republican governments, Christianity. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, science, exploration, Italian culture flourished at this time, producing famous scholars, artists and polymaths such as Leonardo da Vinci, Galileo, Michelangelo and Machiavelli. The weakened sovereigns soon fell victim to conquest by European powers such as France, Spain and Austria. Despite being one of the victors in World War I, Italy entered a period of economic crisis and social turmoil. The subsequent participation in World War II on the Axis side ended in defeat, economic destruction. Today, Italy has the third largest economy in the Eurozone and it has a very high level of human development and is ranked sixth in the world for life expectancy. The country plays a prominent role in regional and global economic, military, cultural and diplomatic affairs, as a reflection of its cultural wealth, Italy is home to 51 World Heritage Sites, the most in the world, and is the fifth most visited country. The assumptions on the etymology of the name Italia are very numerous, according to one of the more common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú, meaning land of young cattle. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. The name Italia originally applied only to a part of what is now Southern Italy – according to Antiochus of Syracuse, but by his time Oenotria and Italy had become synonymous, and the name also applied to most of Lucania as well. The Greeks gradually came to apply the name Italia to a larger region, excavations throughout Italy revealed a Neanderthal presence dating back to the Palaeolithic period, some 200,000 years ago, modern Humans arrived about 40,000 years ago. Other ancient Italian peoples of undetermined language families but of possible origins include the Rhaetian people and Cammuni. Also the Phoenicians established colonies on the coasts of Sardinia and Sicily, the Roman legacy has deeply influenced the Western civilisation, shaping most of the modern world
Italy
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The Colosseum in Rome, built c. 70 – 80 AD, is considered one of the greatest works of architecture and engineering of ancient history.
Italy
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Flag
Italy
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The Iron Crown of Lombardy, for centuries symbol of the Kings of Italy.
Italy
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Castel del Monte, built by German Emperor Frederick II, UNESCO World Heritage site
3.
Partial differential equation
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In mathematics, a partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs, just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations, Partial differential equations are equations that involve rates of change with respect to continuous variables. The dynamics for the body take place in a finite-dimensional configuration space. This distinction usually makes PDEs much harder to solve ordinary differential equations. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, a partial differential equation for the function u is an equation of the form f =0. If f is a function of u and its derivatives. Common examples of linear PDEs include the equation, the wave equation, Laplaces equation, Helmholtz equation, Klein–Gordon equation. A relatively simple PDE is ∂ u ∂ x =0 and this relation implies that the function u is independent of x. However, the equation gives no information on the dependence on the variable y. Hence the general solution of equation is u = f. The analogous ordinary differential equation is d u d x =0, which has the solution u = c and these two examples illustrate that general solutions of ordinary differential equations involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique, additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the example above, the function f can be determined if u is specified on the line x =0. Even if the solution of a differential equation exists and is unique. The mathematical study of questions is usually in the more powerful context of weak solutions. The derivative of u with respect to y approaches 0 uniformly in x as n increases and this solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y
Partial differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
4.
Alma mater
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Alma mater is an allegorical Latin phrase for a university or college. In modern usage, it is a school or university which an individual has attended, the phrase is variously translated as nourishing mother, nursing mother, or fostering mother, suggesting that a school provides intellectual nourishment to its students. Before its modern usage, Alma mater was a title in Latin for various mother goddesses, especially Ceres or Cybele. The source of its current use is the motto, Alma Mater Studiorum, of the oldest university in continuous operation in the Western world and it is related to the term alumnus, denoting a university graduate, which literally means a nursling or one who is nourished. The phrase can also denote a song or hymn associated with a school, although alma was a common epithet for Ceres, Cybele, Venus, and other mother goddesses, it was not frequently used in conjunction with mater in classical Latin. Alma Redemptoris Mater is a well-known 11th century antiphon devoted to Mary, the earliest documented English use of the term to refer to a university is in 1600, when University of Cambridge printer John Legate began using an emblem for the universitys press. In English etymological reference works, the first university-related usage is often cited in 1710, many historic European universities have adopted Alma Mater as part of the Latin translation of their official name. The University of Bologna Latin name, Alma Mater Studiorum, refers to its status as the oldest continuously operating university in the world. At least one, the Alma Mater Europaea in Salzburg, Austria, the College of William & Mary in Williamsburg, Virginia, has been called the Alma Mater of the Nation because of its ties to the founding of the United States. At Queens University in Kingston, Ontario, and the University of British Columbia in Vancouver, British Columbia, the ancient Roman world had many statues of the Alma Mater, some still extant. Modern sculptures are found in prominent locations on several American university campuses, outside the United States, there is an Alma Mater sculpture on the steps of the monumental entrance to the Universidad de La Habana, in Havana, Cuba. Media related to Alma mater at Wikimedia Commons The dictionary definition of alma mater at Wiktionary Alma Mater Europaea website
Alma mater
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The Alma Mater statue by Mario Korbel, at the entrance of the University of Havana in Cuba.
Alma mater
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John Legate's Alma Mater for Cambridge in 1600
Alma mater
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Alma Mater (1929, Lorado Taft), University of Illinois at Urbana–Champaign
5.
Luigi Ambrosio
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Luigi Ambrosio is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory, Ambrosio entered the Scuola Normale Superiore di Pisa in 1981. He obtained his degree under the guidance of Ennio de Giorgi in 1985 at University of Pisa, and he obtained his PhD in 1988. He is currently professor at the Scuola Normale, having taught previously at the University of Rome Tor Vergata, the University of Pisa, and the University of Pavia. Ambrosio also taught and conducted research at the Massachusetts Institute of Technology, the ETH in Zurich, and he is the Managing Editor of the scientific journal Calculus of Variations and Partial Differential Equations, and member of the editorial boards of scientific journals. In 1998 Ambrosio won the Caccioppoli Prize of the Italian Mathematical Union, in 2002 he was plenary speaker at the International Congress of Mathematicians in Beijing and in 2003 he has been awarded with the Fermat Prize. From 2005 he is a member of Accademia Nazionale dei Lincei. Ambrosio is listed as an ISI highly cited researcher, a compactness theorem for a new class of functions of bounded variation. New functionals in the calculus of variations, existence theory for a new class of variational problems. Ambrosio, Luigi, Fusco, Nicola, Pallara, Diego, functions of bounded variation and free discontinuity problems. The Clarendon Press, Oxford University Press, New York, currents in metric spaces, Acta Math. Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe, gradient flows in metric spaces and in the space of probability measures. Luigi Ambrosio, Edward Norman Dancer, Giuseppe Buttazzo, A. Marino, M. K. Venkatesha Murthy, Calculus of variations and partial differential equations. CS1 maint, Uses editors parameter Site of Caccioppoli Prize Luigi Ambrosio at the Mathematics Genealogy Project
Luigi Ambrosio
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Luigi Ambrosio
6.
Minimal surface
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In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature, the term minimal surface is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a solution, forming a soap film. However the term is used for general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas, Minimal surfaces can be defined in several equivalent ways in R3. Local least area definition, A surface M ⊂ R3 is minimal if, note that this property is local, there might exist other surfaces that minimize area better with the same global boundary. Variational definition, A surface M ⊂ R3 is minimal if and this definition makes minimal surfaces a 2-dimensional analogue to geodesics. Note that spherical bubbles are not minimal surfaces as per this definition, while they minimize total area subject to a constraint on internal volume, mean curvature definition, A surface M ⊂ R3 is minimal if and only if its mean curvature vanishes identically. A direct implication of this definition is that point on the surface is a saddle point with equal. This definition ties minimal surfaces to harmonic functions and potential theory, harmonic definition, If X =, M → R3 is an isometric immersion of a Riemann surface into 3-space, then X is said to be minimal whenever xi is a harmonic function on M for each i. A direct implication of this definition and the principle for harmonic functions is that there are no compact complete minimal surfaces in R3. This definition uses that the curvature is half of the trace of the shape operator. If the projected Gauss map obeys the Cauchy–Riemann equations then either the trace vanishes or every point of M is umbilic, mean curvature flow definition, Minimal surfaces are the critical points for the mean curvature flow. The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3, Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z of least area stretched across a given closed contour. He derived the Euler–Lagrange equation for the solution d d x + d d y =0 He did not succeed in finding any solution beyond the plane. By expanding Lagranges equation to z y y −2 z x z y z x y + z x x =0 Gaspard Monge, while these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface, progress had been fairly slow until the middle of the century, when the Björling problem was solved using complex methods. The first golden age of minimal surfaces began, schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 using complex methods
Minimal surface
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A helicoid minimal surface formed by a soap film on a helical frame
7.
House of Giorgi
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The family is listed in the Almanach de Gotha. The family was founded in 1169 and its main branch still survives in Italy, tradition links the Zorzi to the origins of the city of Venice. One tradition is that the Giorgi came to Dubrovnik from Rome, the House of Giorgi officially entered the Golden Book of the Republic of Genoa in 1370 after helping the Republic of Genoa winning an important naval battle over the Republic of Venice. The family established strong ties with the House of Grimaldi, the family of Monaco. The family was also the most loyal ally of the House of Hunyadi, damiano de Giorgi served Matthias Corvinus, King of Hungary and Croatia, receiving the award of large estates and the right to insert the royal crow in the family coat of arms. The island of Curzola has been a fiefdom of the family since 1254, over the centuries, the Giorgi were divided into several branches in Italy and abroad, merging with other noble families of Dubrovnik and continental Europe. A branch of the family joined his name and arms to those of the House of Bona, between 1440 and 1640 the House of Giorgi had 109 members of the Great Council, representing 4. 95% of the total. In the two hundred years, they count for 203 senators,163 Rectors of the Republic,173 representatives in the Minor Council and 41 Guardians of Justice. The Almanach de Gotha enumerates them among the eleven oldest native families of the Sovereign Republic of Ragusa still residing in the city in mid-nineteenth century
House of Giorgi
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House of Giorgi De Giorgi, Đurđević
8.
Accademia Nazionale dei Lincei
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The Accademia dei Lincei is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy. Founded in 1603 by Federico Cesi, it was one of the first academies of science to exist in the World as a locus for the incipient scientific revolution, the academy was named after the lynx, an animal whose sharp vision symbolizes the observational prowess that science requires. The Lincei did not long survive the death in 1630 of Cesi, its founder and patron and it was revived in the 1870s to become the national academy of Italy, encompassing both literature and science among its concerns. The Pontifical Academy of Science also claims a heritage descending from the first two incarnations of the Academy, by way of the Accademia Pontificia dei Nuovi Lincei, founded in 1847. The first Accademia dei Lincei was founded in 1603 by Federico Cesi, Cesis father disapproved of the research career that Federico was pursuing. His mother, Olimpia Orsini, supported him financially and morally. The Academy struggled due to this disapproval, but after the death of Fredericos father he had money to allow the academy to flourish. Cesi founded the Accademia dei Lincei with three friends, the Dutch physician Johannes Van Heeck and two fellow Umbrians, mathematician Francesco Stelluti and polymath Anastasio de Filiis, at the time of the Accademias founding Cesi was only 18, and the others only 8 years older. Cesi and his friends aimed to understand all of the natural sciences, the literary and antiquarian emphasis set the Lincei apart from the host of sixteenth and seventeenth century Italian Academies. While originally an association, the Academy became a semi-public establishment during the Napoleonic domination of Rome. This shift allowed local scientific elite to carve out a place for themselves in larger scientific networks, however, as a semi-public establishment, the Academys focus was directed by Napoleonic politics. This focus directed the efforts towards stimulating industry, turning public opinion in favor of the French regime. Accademia dei Linceis symbols were both a lynx and an eagle, animals with, or reputed to have, keen sight, the academys motto, chosen by Cesi, was, Take care of small things if you want to obtain the greatest results. When Cesi visited Naples, he met many scientists in fields of interest to him including the botanist, Fabio Colonna, the natural history writer, Ferrante Imperato. Della Porta encouraged Cesi to continue with his endeavours, giambattista della Porta joined Cesis academy in 1610. While in Naples, Cesi also met with Nardo Antonio Recchi to negotiate the acquisition of a collection of material describing Aztec plants and this collection of material would eventually become the Tesoro Messicano. Galileo was inducted to the exclusive Academy on April 25,1611, Galileo clearly felt honoured by his association with the Academy for he adopted Galileo Galilei Linceo as his signature. The Academy published his works and supported him during his disputes with the Roman Inquisition, with this publication, the first, most famous phase of the Lincei was concluded
Accademia Nazionale dei Lincei
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Palazzo Corsini
Accademia Nazionale dei Lincei
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Federico Cesi
9.
International Standard Serial Number
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An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication. The ISSN is especially helpful in distinguishing between serials with the same title, ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature. The ISSN system was first drafted as an International Organization for Standardization international standard in 1971, ISO subcommittee TC 46/SC9 is responsible for maintaining the standard. When a serial with the content is published in more than one media type. For example, many serials are published both in print and electronic media, the ISSN system refers to these types as print ISSN and electronic ISSN, respectively. 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 form of the ISSN code can be expressed as follows, NNNN-NNNC where N is in the set, a digit character. The ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, 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, the modulus 11 of the sum must be 0. There is an online ISSN checker that can validate an ISSN, ISSN codes are assigned by a network of ISSN National Centres, usually located at national libraries and coordinated by the ISSN International Centre based in Paris. The International Centre is an organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, at the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept, where ISBNs are assigned to individual books, 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 identifier associated with a serial title. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change, separate ISSNs are needed for serials in different media. Thus, the print and electronic versions of a serial need separate ISSNs. Also, 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
International Standard Serial Number
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ISSN encoded in an EAN-13 barcode with sequence variant 0 and issue number 5
10.
Springer-Verlag
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Springer also hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages. Book publications include major works, textbooks, monographs and book series. Springer has major offices in Berlin, Heidelberg, Dordrecht, on 15 January 2015, Holtzbrinck Publishing Group / Nature Publishing Group and Springer Science+Business Media announced a merger. In 1964, Springer expanded its business internationally, opening an office in New York City, offices in Tokyo, Paris, Milan, Hong Kong, and Delhi soon followed. The academic publishing company BertelsmannSpringer was formed after Bertelsmann bought a majority stake in Springer-Verlag in 1999, the British investment groups Cinven and Candover bought BertelsmannSpringer from Bertelsmann in 2003. They merged the company in 2004 with the Dutch publisher Kluwer Academic Publishers which they bought from Wolters Kluwer in 2002, Springer acquired the open-access publisher BioMed Central in October 2008 for an undisclosed amount. In 2009, Cinven and Candover sold Springer to two private equity firms, EQT Partners and Government of Singapore Investment Corporation, the closing of the sale was confirmed in February 2010 after the competition authorities in the USA and in Europe approved the transfer. In 2011, Springer acquired Pharma Marketing and Publishing Services from Wolters Kluwer, in 2013, the London-based private equity firm BC Partners acquired a majority stake in Springer from EQT and GIC for $4.4 billion. In 2014, it was revealed that Springer had published 16 fake papers in its journals that had been computer-generated using SCIgen, Springer subsequently removed all the papers from these journals. IEEE had also done the thing by removing more than 100 fake papers from its conference proceedings. In 2015, Springer retracted 64 of the papers it had published after it was found that they had gone through a fraudulent peer review process, Springer provides its electronic book and journal content on its SpringerLink site, which launched in 1996. SpringerProtocols is home to a collection of protocols, recipes which provide step-by-step instructions for conducting experiments in research labs, SpringerImages was launched in 2008 and offers a collection of currently 1.8 million images spanning science, technology, and medicine. SpringerMaterials was launched in 2009 and is a platform for accessing the Landolt-Börnstein database of research and information on materials, authorMapper is a free online tool for visualizing scientific research that enables document discovery based on author locations and geographic maps. The tool helps users explore patterns in scientific research, identify trends, discover collaborative relationships. While open-access publishing typically requires the author to pay a fee for copyright retention, for example, a national institution in Poland allows authors to publish in open-access journals without incurring any personal cost - but using public funds. Springer is a member of the Open Access Scholarly Publishers Association, the Academic Publishing Industry, A Story of Merger and Acquisition – via Northern Illinois University
Springer-Verlag
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Springer Science+Business Media
11.
Scuola Normale Superiore
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The Scuola Normale Superiore di Pisa is a public higher learning institution in Pisa, Italy. The Scuola Normale, together with the University of Pisa and SantAnna School of Advanced Studies, is a part of the Pisa University System. It is one of the three officially sanctioned special-statute public universities in Italy, being part of the process of Superior Graduate School in Italy, or Scuola Superiore Universitaria. According to the World University Ranking 2016 made by the Times, Normale is considered to be the best university in Italy and one among the best 50 in Europe. The Scuola Normale Superiore was founded in 1810 by Napoleonic decree, as twin institution of the École Normale Supérieure in Paris, ces écoles doivent être en effet le type et la règle de toutes les autres. Napoleon I rethought the project of an école normale in 1808, by establishing a hall of residence in Paris to house young students, when, in 1814, Ferdinand III, Grand Duke of Tuscany returned to Tuscany, the project of a Scuola Normale in Pisa ceased. Only at the beginning of the 1840s, in connection with the university reform of 1839-1841, was the project resumed. The question was combined with the proposals of resumption of the activities of the ancient Order of Saint Stephen. There was to be a division into two Faculties, of Arts and Sciences. In 1863, was appointed a new Director of the Scuola Normale, the new regulations, issued by Minister Michele Coppino in 1877, reviewed and simplified the internal study regulations and equalized, from an organizational point of view. The philosopher Giovanni Gentile was placed at the head of the Scuola Normale as commissioner in 1928, the new colleges were later merged in the Collegio Medico-Giuridico, which continued to operate under the jurisdiction of the Scuola Normale Superiore di Pisa. During the post-war period, there were practical difficulties, however. The new institution, while committed to the model established by the Scuola Normale Superiore di Pisa, was administered by the University of Pisa. Over time, the Scuola Normale has increasingly opened up to society, the educational programs at the Scuola Normale are divided into two levels, Undergraduate and Doctoral. The undergraduate program corresponds to the 1st-cycle and 2nd-cycle programs provided by Italian universities, the Scuola Normale is located in its original historical building, called Palazzo della Carovana, in Piazza dei Cavalieri, in the medieval centre of Pisa. The Scuola Normale offers classes in humanities and sciences. There are only sixty candidates admitted out of nearly 1000 applicants on average every year, the exam comprises questions covering the entire chosen field of study. The normalisti receive free housing, free lunches and dinners, students live in halls of residence, Collegio Domenico Timpano, Collegio Alessandro dAncona, Collegio Enrico Fermi, Collegio Giosue Carducci and Collegio Alessandro Faedo
Scuola Normale Superiore
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A detail of the main building, Palazzo della Carovana
Scuola Normale Superiore
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Scuola Normale of Pisa
Scuola Normale Superiore
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The first statute of the Scuola Normale Superiore
Scuola Normale Superiore
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Palazzo della Carovana, Scuola Normale's main building
12.
Heidelberg
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Heidelberg is a college town situated on the river Neckar in south-west Germany. At the 2015 census, its population was 156,257, located about 78 km south of Frankfurt, Heidelberg is the fifth-largest city in the German state of Baden-Württemberg. Heidelberg is part of the densely populated Rhine-Neckar Metropolitan Region, founded in 1386, Heidelberg University is Germanys oldest and one of Europes most reputable universities. A scientific hub in Germany, the city of Heidelberg is home to internationally renowned research facilities adjacent to its university. Heidelberg is in the Rhine Rift Valley, on the bank of the lower part of the Neckar in a steep valley in the Odenwald. It is bordered by the Königsstuhl and the Gaisberg mountains, the Neckar here flows in an east-west direction. On the right bank of the river, the Heiligenberg mountain rises to a height of 445 meters, the Neckar flows into the Rhine approximately 22 kilometres north-west in Mannheim. Villages incorporated during the 20th century stretch from the Neckar Valley along the Bergstraße, Heidelberg is on European walking route E1. Alongside the Philosophenweg on the side of the Old Town. There is a population of African rose-ringed parakeets, and a wild population of Siberian swan geese. Heidelberg is an authority within the Regierungsbezirk Karlsruhe. The Rhein-Neckar-Kreis rural district surrounds it and has its seat in the town, Heidelberg is a part of the Rhine-Neckar Metropolitan Region, often referred to as the Rhein-Neckar Triangle. The Rhein-Neckar Triangle became a European metropolitan area in 2005, Heidelberg consists of 15 districts distributed in six sectors of the town. The new district will have approximately 5, 000–6,000 residents, Heidelberg has an oceanic climate, defined by the protected valley between the Pfälzerwald and the Odenwald. Year-round, the temperatures are determined by maritime air masses coming from the west. In contrast to the nearby Upper Rhine Plain, Heidelbergs position in the leads to more frequent easterly winds than average. The hillsides of the Odenwald favour clouding and precipitation, the warmest month is July, the coldest is January. Temperatures often rise beyond 30 °C in midsummer, according to the German Meteorological Service, Heidelberg was the warmest place in Germany in 2009
Heidelberg
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Heidelberg, with Heidelberg Castle on the hill and the Old Bridge over river Neckar
Heidelberg
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Heidelberg with suburbs
Heidelberg
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The Altstadt from the Castle
Heidelberg
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The River Neckar at night
13.
Jacques-Louis Lions
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Jacques-Louis Lions was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAMs John von Neumann prize in 1986 and numerous other distinctions, Lions is listed as an ISI highly cited researcher. After being part of the French Résistance in 1943 and 1944, Lions entered the École Normale Supérieure in 1947. He was a professor of mathematics at the Université of Nancy, the Faculty of Sciences of Paris, and he joined the prestigious Collège de France as well as the French Academy of Sciences in 1973. Throughout his career, Lions insisted on the use of mathematics in industry, with an involvement in the French space program, as well as in domains such as energy. This eventually led him to be appointed director of the Centre National dEtudes Spatiales from 1984 to 1992, Lions was elected President of the International Mathematical Union in 1991 and also received the Japan Prize and the Harvey Prize that same year. In 1992, the University of Houston awarded him a doctoral degree. He was elected president of the French Academy of Sciences in 1996 and was also a Foreign Member of the Royal Society and his son Pierre-Louis Lions is also a well-known mathematician who was awarded a Fields Medal in 1994. In fact both Father and Son have also received recognition in the form of Honorary Doctorates from Heriot-Watt University in 1986 and 1995 respectively. With Enrico Magenes, Problèmes aux limites non homogènes et applications,1968,1970 Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. 1968 with L. Cesari, Quelques méthodes de résolution des problèmes aux limites non linéaires,1969 with Roger Dautray, Mathematical analysis and numerical methods for science and technology. 1984/5 with Philippe Ciarlet, Handbook of numerical analysis,7 vols. with Alain Bensoussan, Papanicolaou, Asymptotic analysis of periodic structures. Jacques-Louis Lions at the Mathematics Genealogy Project
Jacques-Louis Lions
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Jacques-Louis Lions
14.
Boston
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Boston is the capital and most populous city of the Commonwealth of Massachusetts in the United States. Boston is also the seat of Suffolk County, although the county government was disbanded on July 1,1999. The city proper covers 48 square miles with a population of 667,137 in 2015, making it the largest city in New England. Alternately, as a Combined Statistical Area, this wider commuting region is home to some 8.1 million people, One of the oldest cities in the United States, Boston was founded on the Shawmut Peninsula in 1630 by Puritan settlers from England. It was the scene of several key events of the American Revolution, such as the Boston Massacre, the Boston Tea Party, the Battle of Bunker Hill, and the Siege of Boston. Upon U. S. independence from Great Britain, it continued to be an important port and manufacturing hub as well as a center for education, through land reclamation and municipal annexation, Boston has expanded beyond the original peninsula. Its rich history attracts many tourists, with Faneuil Hall alone drawing over 20 million visitors per year, Bostons many firsts include the United States first public school, Boston Latin School, first subway system, the Tremont Street Subway, and first public park, Boston Common. Bostons economic base also includes finance, professional and business services, biotechnology, information technology, the city has one of the highest costs of living in the United States as it has undergone gentrification, though it remains high on world livability rankings. Bostons early European settlers had first called the area Trimountaine but later renamed it Boston after Boston, Lincolnshire, England, the renaming on September 7,1630 was by Puritan colonists from England who had moved over from Charlestown earlier that year in quest of fresh water. Their settlement was limited to the Shawmut Peninsula, at that time surrounded by the Massachusetts Bay and Charles River. The peninsula is thought to have been inhabited as early as 5000 BC, in 1629, the Massachusetts Bay Colonys first governor John Winthrop led the signing of the Cambridge Agreement, a key founding document of the city. Puritan ethics and their focus on education influenced its early history, over the next 130 years, the city participated in four French and Indian Wars, until the British defeated the French and their Indian allies in North America. Boston was the largest town in British America until Philadelphia grew larger in the mid-18th century, Bostons harbor activity was significantly curtailed by the Embargo Act of 1807 and the War of 1812. Foreign trade returned after these hostilities, but Bostons merchants had found alternatives for their investments in the interim. Manufacturing became an important component of the economy, and the citys industrial manufacturing overtook international trade in economic importance by the mid-19th century. Boston remained one of the nations largest manufacturing centers until the early 20th century, a network of small rivers bordering the city and connecting it to the surrounding region facilitated shipment of goods and led to a proliferation of mills and factories. Later, a network of railroads furthered the regions industry. Boston was a port of the Atlantic triangular slave trade in the New England colonies
Boston
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From top to bottom, left to right: the Boston skyline viewed from the Bunker Hill Monument; the Museum of Fine Arts; Faneuil Hall; Massachusetts State House; The First Church of Christ, Scientist; Boston Public Library; the John F. Kennedy Presidential Library and Museum; South Station; Boston University and the Charles River; Arnold Arboretum; Fenway Park; and the Boston Common
Boston
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State Street, 1801
Boston
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View of Boston from Dorchester Heights, 1841
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Scollay Square in the 1880s
15.
Stuttgart
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Stuttgart is the capital and largest city of the German state of Baden-Württemberg. It is located on the Neckar river in a fertile valley known as the Stuttgart Cauldron an hour from the Swabian Jura. Stuttgarts urban area has a population of 623,738, making it the sixth largest city in Germany. 2.7 million people live in the administrative region and another 5.3 million people in its metropolitan area. Since the 6th millennium BC, the Stuttgart area has been an important agricultural area and has been host to a number of cultures seeking to utilize the rich soil of the Neckar valley. The Roman Empire conquered the area in 83 AD and built a massive Castrum near Bad Cannstatt, Stuttgarts roots were truly laid in the 10th century with its founding by Liudolf, Duke of Swabia as a stud farm for his warhorses. Overshadowed by nearby Cannstatt, the town grew steadily and was granted a charter in 1320, the fortunes of Stuttgart turned with those of the House of Württemberg, and they made it the capital of their County, Duchy, and Kingdom from the 15th Century to 1918. Stuttgart prospered despite setbacks in the forms of the Thirty Years War and devastating air raids by the Allies on the city, however, by 1952, the city had bounced back and became the major economic, industrial, tourism and publishing center it is today. Stuttgart is also an important transport junction, and possesses the sixth largest airport in Germany. Such companies as Porsche, Mercedes-Benz, Daimler AG, Dinkelacker, Stuttgart is unusual in the scheme of German cities. It is spread across a variety of hills, valleys and parks and this is often a source of surprise to visitors who associate the city with its reputation as the Cradle of the Automobile. The citys tourism slogan is Stuttgart offers more, under current plans to improve transport links to the international infrastructure, the city unveiled a new logo and slogan in March 2008 describing itself as Das neue Herz Europas. For business, it describes itself as Where business meets the future, in July 2010, Stuttgart unveiled a new city logo, designed to entice more business people to stay in the city and enjoy breaks in the area. Stuttgart is a city of mostly immigrants, according to Dorling Kindersley Publishings Eyewitness Travel Guide to Germany, In the city of Stuttgart, every third inhabitant is a foreigner. 40% of Stuttgarts residents, and 64% of the population below the age of five are of immigrant background, the reason for this being that the city was founded in 950 AD by Duke Liudolf of Swabia to breed warhorses. Originally, the most important location in the Neckar river valley as the rim of the Stuttgart basin at what is today Bad Cannstatt. As with many military installations, a settlement sprang up nearby, when they did, the town was left in the capable hands of a local brickworks that produced sophisticated architectural ceramics and pottery. When the Romans were driven back past the Rhine and Danube rivers in the 3rd Century by the Alamanni, in 700, Duke Gotfrid mentions a Chan Stada in a document regarding property
Stuttgart
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Clockwise from top left: Staatstheater, Cannstatter Volksfest in Bad Cannstatt, fountain at Schlossplatz, Fruchtkasten façade and the statue of Friedrich Schiller at Schillerplatz, New Palace, and Old Castle at Schillerplatz.
Stuttgart
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Panorama of Stuttgart looking southeast. From the Neckar valley on the left the city rises to the city center, backdropped by high woods to the south (television tower). Stuttgart South and Stuttgart West are to the right.
Stuttgart
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Stuttgart at night, looking northwest
Stuttgart
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Schlossplatz
16.
Web page
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A web page is a document that is suitable for the World Wide Web and web browsers. A web browser displays a web page on a monitor or mobile device, the web page is what displays, but the term also refers to a computer file, usually written in HTML or comparable markup language. Web browsers coordinate the various web resource elements for the web page, such as style sheets, scripts. Typical web pages provide hypertext that includes a bar or a sidebar menu to other web pages via hyperlinks. On a network, a web browser can retrieve a web page from a web server. On a higher level, the web server may restrict access to only a network such as a corporate intranet or it provides access to the World Wide Web. On a lower level, the web browser uses the Hypertext Transfer Protocol to make such requests, dynamic website pages help the browser to enhance the web page through user input to the server. Web pages usually include information as to the colors of text and backgrounds, an HTTP1.1 web server will maintain a connection with the browser until all related resources have been requested and provided. Web browsers usually render images along with the text and other material on the web page. These scripts may run on the client computer, if the user allows, a web browser can have a Graphical User Interface, like Internet Explorer, Mozilla Firefox, Google Chrome, and Opera, or can be text-based, like Lynx or Links. Web users with disabilities often use assistive technologies and adaptive strategies to access web pages, disabled and able-bodied users may disable the download and viewing of images and other media, to save time, network bandwidth or merely to simplify their browsing experience. Users of mobile devices often have restricted displays and bandwidth, anyone may prefer not to use the fonts, font sizes, styles and color schemes selected by the web page designer and may apply their own CSS styling to the page. The World Wide Web Consortium and Web Accessibility Initiative recommend that all web pages should be designed with all of options in mind. Non-textual information, Static images may be raster graphics, typically GIF, JPEG or PNG, Animated images typically Animated GIF and SVG, but also may be Flash, Shockwave, or Java applet. Audio, typically MP3, Ogg or various proprietary formats, video, WMV, RM, FLV, MPG, MOV Interactive information, see interactive media. For on page interaction, Interactive text, see DHTML, Interactive illustrations, ranging from click to play images to games, typically using script orchestration, Flash, Java applets, SVG, or Shockwave. Buttons, forms providing an interface, typically for use with script orchestration. For between pages interaction, Hyperlinks, standard change page reactivity, forms, providing more interaction with the server and server-side databases
Web page
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A screenshot of Wikimedia Commons, a web page
17.
Web site
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A website is a collection of related web pages, including multimedia content, typically identified with a common domain name, and published on at least one web server. A website may be accessible via a public Internet Protocol network, such as the Internet, or a local area network. Websites have many functions and can be used in various fashions, a website can be a website, a commercial website for a company. Websites are typically dedicated to a topic or purpose, ranging from entertainment and social networking to providing news. All publicly accessible websites collectively constitute the World Wide Web, while private websites, Web pages, which are the building blocks of websites, are documents, typically composed in plain text interspersed with formatting instructions of Hypertext Markup Language. They may incorporate elements from other websites with suitable markup anchors, Web pages are accessed and transported with the Hypertext Transfer Protocol, which may optionally employ encryption to provide security and privacy for the user. The users application, often a web browser, renders the page content according to its HTML markup instructions onto a display terminal. Hyperlinking between web pages conveys to the reader the site structure and guides the navigation of the site, Some websites require user registration or subscription to access content. As of 2016 end users can access websites on a range of devices, including desktop and laptop computers, tablet computers, smartphones, the World Wide Web was created in 1990 by the British CERN physicist Tim Berners-Lee. On 30 April 1993, CERN announced that the World Wide Web would be free to use for anyone, before the introduction of HTML and HTTP, other protocols such as File Transfer Protocol and the gopher protocol were used to retrieve individual files from a server. These protocols offer a directory structure which the user navigates and chooses files to download. Documents were most often presented as text files without formatting. Websites have many functions and can be used in various fashions, a website can be a website, a commercial website. Websites can be the work of an individual, a business or other organization, any website can contain a hyperlink to any other website, so the distinction between individual sites, as perceived by the user, can be blurred. Websites are written in, or converted to, HTML and are accessed using a software interface classified as a user agent. Web pages can be viewed or otherwise accessed from a range of computer-based and Internet-enabled devices of various sizes, including computers, laptops, PDAs. A website is hosted on a system known as a web server. These terms can refer to the software that runs on these systems which retrieves
Web site
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NASA.gov homepage as it appeared in April 2015
18.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
University of St Andrews
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College Hall, within the 16th century St Mary's College building
University of St Andrews
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University of St Andrews shield
University of St Andrews
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St Salvator's Chapel in 1843
University of St Andrews
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The "Gateway" building, built in 2000 and now used for the university's management department
19.
Israel Gelfand
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Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gelfand, or Izrail M. Gelfand was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and his legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, as well as his own son, Sergei Gelfand. A native of Kherson Governorate of the Russian Empire, Gelfand was born into a Jewish family in the small southern Ukrainian town of Okny, according to his own account, Gelfand was expelled from high school because his father had been a mill owner. Bypassing both high school and college, he proceeded to study at Moscow State University, where his advisor was the preeminent mathematician Andrei Kolmogorov. He nevertheless managed to attend lectures at the University and began study at the age of 19. The Gelfand–Tsetlin basis is a widely used tool in theoretical physics, Gelfand also published works on biology and medicine. For a long time he took an interest in cell biology and he worked extensively in mathematics education, particularly with correspondence education. In 1994, he was awarded a MacArthur Fellowship for this work, Gelfand was married to Zorya Shapiro, and their two sons, Sergei and Vladimir both live in the United States. A third son, Aleksandr, died of leukemia, following the divorce from his first wife, Gelfand married his second wife, Tatiana, together they had a daughter, Tatiana. The family also includes four grandchildren and three great-grandchildren, the memories about I. Gelfand are collected at the special site handled by his family. Gelfand held several degrees and was awarded the Order of Lenin three times for his research. In 1977 he was elected a Foreign Member of the Royal Society and he won the Wolf Prize in 1978, Kyoto Prize in 1989 and MacArthur Foundation Fellowship in 1994. Israel Gelfand died at the Robert Wood Johnson University Hospital near his home in Highland Park and he was less than five weeks past his 96th birthday. His death was first reported on the blog of his former collaborator Andrei Zelevinsky and confirmed a few hours later by an obituary in the Russian online newspaper Polit. ru. Gelfand, I. M. Lectures on linear algebra, Courier Dover Publications, ISBN 978-0-486-66082-0 Gelfand, I. M. Fomin, Sergei V. Silverman, Richard A. ed. Calculus of variations, Englewood Cliffs, ISBN 978-0-486-41448-5, MR0160139 Gelfand, I. Raikov, D. Shilov, G. Commutative normed rings, Translated from the Russian, with a chapter, New York. ISBN 978-0-8218-2022-3, MR0205105 Gelfand, I. M. Shilov, G. E. Generalized functions. Vol. I, Properties and operations, Translated by Eugene Saletan, Boston, MA, Academic Press, ISBN 978-0-12-279501-5, MR0166596 Gelfand, I. M. Shilov, G. E. Generalized functions
Israel Gelfand
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Israïl Moiseevich Gelfand
20.
Jean Leray
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Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology. He studied at École Normale Supérieure from 1926 to 1929 and he received his Ph. D. in 1933. Leray wrote an important paper that founded the study of solutions of the Navier–Stokes equations. From 1938 to 1939 he was professor at the University of Nancy and he did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was in a prisoner of war camp in Edelbach and he concealed his expertise on differential equations, fearing that its connections with applied mathematics could lead him to be asked to do war work. Lerays work of this period proved seminal to the development of spectral sequences and sheaves and these were subsequently developed by many others, each separately becoming an important tool in homological algebra. He returned to work on differential equations from about 1950. He was professor at the University of Paris from 1945 to 1947 and he was awarded the Malaxa Prize, the Grand Prix in mathematical sciences, the Feltrinelli Prize, the Wolf Prize in Mathematics, and the Lomonosov Gold Medal. Leray spectral sequence Leray cover Lerays theorem Leray–Hirsch theorem OConnor, John J. Robertson, Jean Leray, MacTutor History of Mathematics archive, University of St Andrews. Jean Leray at the Mathematics Genealogy Project Jean Leray, by Armand Borel, Gennadi M. Henkin, and Peter D. Lax, Notices of the American Mathematical Society, vol
Jean Leray
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Jean Leray in 1961
21.
Henri Cartan
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Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan and the brother of composer Jean Cartan, Cartan studied at the Lycée Hoche in Versailles, then at the École Normale Supérieure in Paris, receiving his doctorate in mathematics. Cartan is known for work in topology, in particular on cohomology operations, the method of killing homotopy groups. The number of his students was small, but includes Adrien Douady, Roger Godement, Max Karoubi, Jean-Louis Koszul, Jean-Pierre Serre. Cartan also was a member of the Bourbaki group and one of its most active participants. His book with Samuel Eilenberg Homological Algebra was an important text, Cartan used his influence to help obtain the release of some dissident mathematicians, including Leonid Plyushch and Jose Luis Massera. For his humanitarian efforts, he received the Pagels Award from the New York Academy of Sciences, the Cartan model in algebra is named after Cartan. Cartan died on 13 August 2008 at the age of 104 and his funeral took place the following Wednesday on 20 August in Die, Drome. Cartan received numerous honours and awards including the Wolf Prize in 1980 and he was an Invited Speaker at the ICM in 1932 in Zurich and a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts and in 1958 in Edinburgh. From 1974 until his death he had been a member of the French Academy of Sciences, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, thèse,1928 Sur les groupes de transformations analytiques,1935. Sur les classes de fonctions définies par des inégalités portant sur leurs dérivées successives,1940, cohomologie des groupes, suite spectrale, faisceaux, 1950-1951. Algèbres dEilenberg - Mac Lane et homotopie, 1954-1955, Homological Algebra, Princeton Univ Press,1956 ISBN 978-0-69104991-5 Séminaires de lÉcole normale supérieure, Secr. IHP, 1948-1964, New York, W. A. Benjamin ed.1967, théorie élémentaire des fonctions analytiques, Paris, Hermann,1961. Differential Forms, Dover 2006 Œuvres — Collected Works,3 vols, ed. Reinhold Remmert & Jean-Pierre Serre, Springer Verlag, Heidelberg,1967. Relations dordre en théorie des permutations des ensembles finis, Neuchâtel,1973, théorie élémentaire des fonctions analytiques dune ou plusieurs variables complexes, Paris, Hermann,1975. Elementary theory of functions of one or several complex variables, Dover 1995 Cours de calcul différentiel, Paris. Correspondance entre Henri Cartan et André Weil, Paris, SMF,2011, oConnor, John J. Robertson, Edmund F. Henri Cartan, MacTutor History of Mathematics archive, University of St Andrews. A17, retrieved 2008-08-25 Cartan, Henri, Eilenberg, Samuel, notices of the American Mathematical Society, Sept.2010, vol
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Henri Cartan
22.
Andrey Kolmogorov
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Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. Kolmogorova, died giving birth to him, Andrey was raised by two of his aunts in Tunoshna at the estate of his grandfather, a well-to-do nobleman. Little is known about Andreys father and he was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Nikolai had been exiled from St. Petersburg to the Yaroslavl province after his participation in the movement against the czars. He disappeared in 1919 and he was presumed to have killed in the Russian Civil War. Andrey Kolmogorov was educated in his aunt Veras village school, and his earliest literary efforts, Andrey was the editor of the mathematical section of this journal. In 1910, his aunt adopted him, and they moved to Moscow, later that same year, Kolmogorov began to study at the Moscow State University and at the same time Mendeleev Moscow Institute of Chemistry and Technology. Kolmogorov writes about this time, I arrived at Moscow University with a knowledge of mathematics. I knew in particular the beginning of set theory, I studied many questions in articles in the Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles. Kolmogorov gained a reputation for his wide-ranging erudition, during the same period, Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series. In 1922, Kolmogorov gained international recognition for constructing a Fourier series that diverges almost everywhere, around this time, he decided to devote his life to mathematics. In 1925, Kolmogorov graduated from the Moscow State University and began to study under the supervision of Nikolai Luzin, Kolmogorov became interested in probability theory. In 1929, Kolmogorov earned his Doctor of Philosophy degree, from Moscow State University, in 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen and Munich, and then to Paris. He had various contacts in Göttingen. His pioneering work, About the Analytical Methods of Probability Theory, was published in 1931, also in 1931, he became a professor at the Moscow State University. In 1935, Kolmogorov became the first chairman of the department of probability theory at the Moscow State University, around the same years Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator-prey systems. In 1936, Kolmogorov and Alexandrov were involved in the persecution of their common teacher Nikolai Luzin, in the so-called Luzin affair. In a 1938 paper, Kolmogorov established the basic theorems for smoothing and predicting stationary stochastic processes—a paper that had military applications during the Cold War
Andrey Kolmogorov
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Andrey Kolmogorov
Andrey Kolmogorov
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Kolmogorov (left) delivers a talk at a Soviet information theory symposium. (Tallinn, 1973).
Andrey Kolmogorov
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Kolmogorov works on his talk (Tallinn, 1973).
23.
Shiing-Shen Chern
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Shiing-Shen Chern was a Chinese-American mathematician. Shiing-Shen Chern co-founded the world-renowned Mathematical Sciences Research Institute at Berkeley in 1982, Chern was born in Xiushui County, Jiaxing, in Zhejiang province. The year after his birth, China changed its regime from the Qing Dynasty to the Republic of China and he graduated from Xiushui Middle School and subsequently moved to Tianjin in 1922 to accompany his father. In 1926, after spending four years in Tianjin, Chern graduated from Fulun High School, at age 15, Chern entered the Faculty of Sciences of the Nankai University in Tianjin, studied mathematics there, and graduated with a Bachelor of Science degree in 1930. At Nankai, Cherns mentor was Li-Fu Chiang, a Harvard-trained geometer, also at Nankai, he was heavily influenced by the physicist Rao Yutai. Rao is today considered to be one of the fathers of modern Chinese informatics. Chern went to Beiping to work at the Tsinghua University Department of Mathematics as a teaching assistant, at the same time he also registered at Tsinghua Graduate School as a student. He studied projective geometry under Prof. Sun Guangyuan, a University of Chicago-trained geometer and logician who was also from Zhejiang, Sun is another mentor of Chern who is considered a founder of modern Chinese mathematics. In 1932, Chern published his first research article in the Tsinghua University Journal, in the summer of 1934, Chern graduated from Tsinghua with a masters degree, the first ever masters degree in mathematics issued in China. Chen-Ning Yangs father — Yang Ko-Chuen, another Chicago-trained professor at Tsinghua, at the same time, Chern was Chen-Ning Yangs teacher of undergraduate maths at Tsinghua. At Tsinghua, Hua Luogeng, also a mathematician, was Cherns colleague, in 1932, Wilhelm Blaschke from the University of Hamburg visited Tsinghua and was impressed by Chern and his research. In 1934, co-funded by Tsinghua and the Chinese Foundation of Culture and Education, Chern studied at the University of Hamburg and worked under Blaschkes guidance first on the geometry of webs then on the Cartan-Kähler theory. Blaschke recommended Chern to study in Paris, in August 1936, Chern watched summer Olympics in Berlin together with Hua Luogeng who paid Chern a brief visit. During that time, Hua was studying at the University of Cambridge in Britain, in September 1936, Chern went to Paris and worked with Élie Cartan. Chern spent one year at the Sorbonne in Paris, in 1937, Chern accepted Tsinghuas invitation and was promoted to professor of mathematics at Tsinghua. However, at the time the Marco Polo Bridge Incident happened. Three universities including Peking University, Tsinghua, and Nankai formed the National Southwestern Associated University, in the same year, Hua Luogeng was promoted to professor of mathematics at Tsinghua
Shiing-Shen Chern
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Shiing-Shen Chern, 1976
24.
Kunihiko Kodaira
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Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese national to receive this honour and he graduated from the University of Tokyo in 1938 with a degree in mathematics and also graduated from the physics department at the University of Tokyo in 1941. During the war years he worked in isolation, but was able to master Hodge theory as it then stood and he obtained his Ph. D. from the University of Tokyo in 1949, with a thesis entitled Harmonic fields in Riemannian manifolds. He was involved in work from about 1944, while holding an academic post in Tokyo. In 1949 he travelled to the Institute for Advanced Study in Princeton, at this time the foundations of Hodge theory were being brought in line with contemporary technique in operator theory. Kodaira rapidly became involved in exploiting the tools it opened up in algebraic geometry and this work was particularly influential, for example on Hirzebruch. In a second phase, Kodaira wrote a long series of papers in collaboration with D. C. Spencer, founding the theory of complex structures on manifolds. This gave the possibility of constructions of moduli spaces, since in such structures depend continuously on parameters. This theory is still foundational, and also had an influence on the theory of Grothendieck. Spencer then continued work, applying the techniques to structures other than complex ones. In a third part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds. This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically and this work also included a characterisation of K3 surfaces as deformations of quartic surfaces in P4, and the theorem that they form a single diffeomorphism class. Again, this work has proved foundational, Kodaira left the Institute for Advanced Study in 1961, and briefly served as chair at the Johns Hopkins University and Stanford University. In 1967, returned to the University of Tokyo and he was awarded a Wolf Prize in 1984/5. He died in Kofu on 26 July 1997, ISBN 978-0-691-08158-8, MR0366598 Kodaira, Kunihiko, Baily, Walter L. ed. Kunihiko Kodaira, collected works, II, Iwanami Shoten, Publishers, Tokyo, Princeton University Press, Princeton, N. J. ISBN 978-0-691-08163-2, MR0366599 Kodaira, Kunihiko, Baily, Walter L. ed. Kunihiko Kodaira, collected works, III, Iwanami Shoten, Publishers, Tokyo, Princeton University Press, Princeton, N. J. Robertson, Edmund F. Kunihiko Kodaira, MacTutor History of Mathematics archive, spencer, Kunihiko Kodaira, Notices of the AMS,45, 388–389
Kunihiko Kodaira
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Kunihiko Kodaira
25.
Hans Lewy
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Hans Lewy was a German born American mathematician, known for his work on partial differential equations and on the theory of functions of several complex variables. Lewy was born in Breslau, Germany, on October 20,1904, at Göttingen, he studied both mathematics and physics, his teachers there included Max Born, Richard Courant, James Franck, David Hilbert, Edmund Landau, Emmy Noether, and Alexander Ostrowski. He earned his doctorate in 1926, at time he. After Hitlers election as chancellor in 1933, Lewy was advised by Herbert Busemann to leave Germany again and he was offered a position in Madrid, but declined it, fearing for the future there under Francisco Franco. At the end of term, in 1935, he moved to the University of California. During World War II, Lewy obtained a license. He married Helen Crosby in 1947, in 1950, Lewy was fired from Berkeley for refusing to sign a loyalty oath. He taught at Harvard University and Stanford University in 1952 and 1953 before being reinstated by the California Supreme Court case Tolman v. Underhill and he retired from Berkeley in 1972, and in 1973 became one of two Ordway Professors of Mathematics at the University of Minnesota. He died on August 23,1988, in Berkeley, Lewy was elected to the National Academy of Sciences in 1964, and was also a member of the American Academy of Arts and Sciences. He became a member of the Accademia dei Lincei in 1972. He was awarded a Leroy P. Steele Prize in 1979, in 1986, the University of Bonn gave him an honorary doctorate. A priori limitations for Monge-Ampère equations, on the non-vanishing of the Jacobian in certain one-to-one mappings. Proc Natl Acad Sci U S A.22, 377–381, a priori limitations for Monge-Ampère equations. On the existence of a convex surface realizing a given Riemannian metric. Proc Natl Acad Sci U S A.24, 104–106, on differential geometry in the large. Aspects of the Calculus of Variations, Berkeley, U. of California Press, notes by J. W. Green from lectures by Hans Lewy, vi+96 pp. Lewy, Hans. On the boundary behavior of minimal surfaces, proc Natl Acad Sci U S A.37, 103–110. A note on harmonic functions and a hydrodynamical application, on the reflection laws of second order differential equations in two independent variables
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Hans Lewy in 1975 (photo by George Bergman)
26.
Peter Lax
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Peter David Lax is a Hungarian-born American mathematician working in the areas of pure and applied mathematics. Lax is listed as an ISI highly cited researcher, Lax was born in Budapest, Hungary to a Jewish family. His parents Klara Kornfield and Henry Lax were both physicians, and his uncle, Albert Kornfeld, was a mathematician and a friend of Leó Szilárd, Lax began displaying an interest in mathematics at age twelve, and soon his parents hired Rózsa Péter as a tutor for him. The family left Hungary on November 15,1941, and traveled via Lisbon to the United States, as he was still 17 when he finished high school, he could avoid military service, and was able to study for three semesters at New York University. In a complex analysis class that he had begun in the role of a student, before being able to complete his studies, Lax was drafted into the U. S. Army. After basic training, the Army sent him to Texas A&M University for more studies, then Oak Ridge National Laboratory, at Los Alamos, he began working as a calculator operator, but eventually moved on to higher-level mathematics. Lax returned to NYU for the 1946-1947 academic year, and by pooling credits from the four universities at which he had studied, he graduated that year. He stayed at NYU for his studies, marrying Anneli in 1948. In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the Lax conjecture grew as mathematicians working in different areas recognized the importance of its implications in their field. Lax holds a faculty position in the Department of Mathematics, Courant Institute of Mathematical Sciences and he is a member of the Norwegian Academy of Science and Letters and the National Academy of Sciences, USA. He won a Lester R. Ford Award in 1966 and again in 1973 and he was awarded the National Medal of Science in 1986, the Wolf Prize in 1987, the Abel Prize in 2005 and the Lomonosov Gold Medal in 2013. The American Mathematical Society selected him as its Gibbs Lecturer for 2007, in 2012 he became a fellow of the American Mathematical Society. Some of the present, possibly members of the Weathermen, threatened to destroy the computer with incendiary devices. Complex Proofs of Real Theorems, with Lawrence Zalcman, University Lecture Series,2012,90 pp, softcover, Volume,58, ISBN 978-0-8218-7559-9 Functional Analysis, Wiley-Interscience, linear Algebra and Its Applications, 2nd ed. Wiley-Interscience, New York. Hyperbolic Partial Differential Equations, American Mathematical Society/Courant Institute of Mathematical Sciences, scattering Theory, with R. S. Phillips, Academic Press. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, with J. Glimm, American Mathematical Society. Recent Mathematical Methods in Nonlinear Wave Propagation, with G. Boillat, C. M. Dafermos, scattering Theory for Automorphic Functions with R. S. Phillips, Princeton Univ
Peter Lax
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Peter Lax in Tokyo, 1969
27.
Friedrich Hirzebruch
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Friedrich Ernst Peter Hirzebruch ForMemRS was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and 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. He studied at the University of Münster from 1945–1950, with one year at ETH Zürich, Hirzebruch then 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, more than 300 people gathered in celebration of his 80th birthday in Bonn in 2007. Hirzebruchs book Neue topologische Methoden in der algebraischen Geometrie was a text for the new methods of sheaf theory. He went on to write the foundational papers on topological K-theory with Michael Atiyah, in his later work he provided a detailed theory of Hilbert modular surfaces, working with Don Zagier. In March 1945, Hirzebruch became a soldier, and in April, in the last weeks of Hitlers rule, when a British soldier found that he was studying mathematics, he drove him home and released him, and 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, and received the Cantor medal in 2004. In 1980–81 he delivered the first Sackler Distinguished Lecture in Israel
Friedrich Hirzebruch
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Friedrich Hirzebruch in 1980 (picture courtesy MFO)
28.
Ilya Piatetski-Shapiro
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Ilya Piatetski-Shapiro was a Soviet-born Israeli mathematician. During a career that spanned 60 years he made contributions to applied science as well as theoretical mathematics. In the last forty years his research focused on mathematics, in particular, analytic number theory, group representations. His main contribution and impact was in the area of automorphic forms, for the last 30 years of his life he suffered from Parkinsons disease. However, with the help of his wife Edith, he was able to continue to work and do mathematics at the highest level, even when he was able to walk. Ilya was born in 1929 in Moscow, Soviet Union, both his father, Iosif Grigorevich, and mother, Sofia Arkadievna, were from traditional Jewish families, but which had become assimilated. His father was from Berdichev, a city in the Ukraine. His mother was from Gomel, a small city in Belorussia. Both parents families were middle-class, but they sank into poverty after the October revolution of 1917, in 1952, Piatetski-Shapiro won the Moscow Mathematical Society Prize for a Young Mathematician for work done while still an undergraduate at Moscow University. His winning paper contained a solution to the problem of the French analyst Raphaël Salem on sets of uniqueness of trigonometric series, the award was especially remarkable because of the atmosphere of strong anti-Semitism in Soviet Union at that time. Ilya was ultimately admitted to the Moscow Pedagogical Institute, where he received his Ph. D. in 1954 under the direction of Alexander Buchstab and his early work was in classical analytic number theory. His contact with Shafarevich, who was a professor at the Steklov Institute, broadened Ilyas mathematical outlook and directed his attention to modern number theory and this led, after a while, to the influential joint paper in which they proved a Torelli theorem for K3 surfaces. Ilyas career was on the rise, and in 1958 he was made a professor of mathematics at the Moscow Institute of Applied Mathematics, by the 1960s, he was recognized as a star mathematician. In 1965 he was appointed to a professorship at the prestigious Moscow State University. He conducted seminars for advanced students, among them Grigory Margulis and he was invited to attend 1962 International Congress of Mathematicians in Stockholm, but was not allowed to go by Soviet authorities. In 1966, Ilya was again invited to ICM in Moscow where he presented a 1-hour lecture on Automorphic Functions, but despite his fame, Ilya was not allowed to travel abroad to attend meetings or visit colleagues except for one short trip to Hungary. The Soviet authorities insisted on one a condition, become a party member, Ilya gave his famous answer, “The membership in the Communist Party will distract me from my work. ”During the span of his career Piatetski-Shapiro was influenced greatly by Israel Gelfand. The aim of their collaboration was to introduce novel representation theory into classical modular forms, together with Graev, they wrote the classic “Automorphic Forms and Representations” book
Ilya Piatetski-Shapiro
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Ilya Piatetski-Shapiro
29.
Lennart Carleson
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Lennart Axel Edvard Carleson is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most famous achievements is his proof of Lusins conjecture and he was a student of Arne Beurling and received his Ph. D. from Uppsala University in 1950. Between 1978 and 1982 he served as president of the International Mathematical Union, Carleson married Butte Jonsson in 1953, and they had two children, Caspar and Beatrice. His work has included the solution of some outstanding problems, using techniques from combinatorics, in the theory of Hardy spaces, Carlesons contributions include the corona theorem and establishing the almost everywhere convergence of Fourier series for square-integrable functions. He is also known for the theory of Carleson measures, in the theory of dynamical systems, Carleson has worked in complex dynamics. He is a member of the Norwegian Academy of Science and Letters, in 2012 he became a fellow of the American Mathematical Society. Selected Problems on Exceptional Sets, Van Nostrand,1967 Matematik för vår tid, Prisma 1968 with T. W. Gamelin, Complex Dynamics, Springer,1993
Lennart Carleson
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Lennart Carleson in May 2006.
30.
Mikhail Leonidovich Gromov
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Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University, Gromov has won several prizes, including the Abel Prize in 2009 for his revolutionary contributions to geometry. Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union and his father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. Gromov was born during World War II, and his mother, when Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a degree in 1965. His thesis advisor was Vladimir Rokhlin, in 1970, invited to give a presentation at the International Congress of Mathematicians in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings, disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel and he changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook. In 1981 he left Stony Brook to join the faculty of University of Paris VI, at the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996. He adopted French citizenship in 1992, Gromovs style of geometry often features a coarse or soft viewpoint, analyzing asymptotic or large-scale properties. In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two metric spaces. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, Gromov was also the first to study the space of all possible Riemannian structures on a given manifold. Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs, in 1981 he proved Gromovs theorem on groups of polynomial growth, a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof uses the Gromov–Hausdorff metric mentioned above, along with Eliyahu Rips he introduced the notion of hyperbolic groups. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves and this led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem. Gromov is also interested in biology, the structure of the brain and the thinking process. Member of the French Academy of Sciences Gromov, M. Hyperbolic manifolds, groups, riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, pp. 183–213, Ann. of Math
Mikhail Leonidovich Gromov
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Mikhail Gromov
31.
Jacques Tits
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Jacques Tits is a Belgium-born French mathematician who works on group theory and incidence geometry, and who introduced Tits buildings, the Tits alternative, and the Tits group. Tits was born in Uccle to Léon Tits, a professor, Jacques attended the Athénée of Uccle and the Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation Généralisation des groupes projectifs basés sur la notion de transitivité. His academic career includes professorships at the Free University of Brussels, the University of Bonn and he changed his citizenship to French in 1974 in order to teach at the Collège de France, which at that point required French citizenship. Because Belgian nationality law did not allow dual nationality at the time and he has been a member of the French Academy of Sciences since then. Tits received the Wolf Prize in Mathematics in 1993, the Cantor Medal from the Deutsche Mathematiker-Vereinigung in 1996, and the German distinction Pour le Mérite. In 2008 he was awarded the Abel Prize, along with John Griggs Thompson, “for their profound achievements in algebra and he is a member of the Norwegian Academy of Science and Letters. He became a member of the Royal Netherlands Academy of Arts. He introduced the theory of buildings, which are structures on which groups act. The related theory of pairs is a tool in the theory of groups of Lie type. Of particular importance is his classification of all buildings of spherical type. In the rank-2 case spherical building are generalized n-gons, and in joint work with Richard Weiss he classified these when they admit a group of symmetries. In collaboration with François Bruhat he developed the theory of affine buildings, the Tits group and the Tits–Koecher construction are named after him. Buildings of spherical type and finite BN-pairs, lecture Notes in Mathematics, Vol.386. MR0470099 Tits, Jacques, Weiss, Richard M. Moufang polygons, MR1938841 J. Tits, Oeuvres - Collected Works,4 vol. J. Tits, Résumés des cours au Collège de France, Jacques Tits at the Mathematics Genealogy Project OConnor, John J. Robertson, Edmund F. Jacques Tits, MacTutor History of Mathematics archive, University of St Andrews. Biography at the Abel Prize site List of publications at the Université libre de Bruxelles
Jacques Tits
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Jacques Tits in May 2008
32.
Robert Langlands
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Robert Phelan Langlands is a Canadian mathematician. He is best known as the founder of the Langlands program and he is an emeritus professor and occupies Albert Einsteins office at the Institute for Advanced Study in Princeton. Langlands received a degree from the University of British Columbia in 1957. He then went to Yale University where he received a Ph. D. in 1960 and his academic positions since then include the years 1960–67 at Princeton University, ending up as Associate Professor, and the years 1967–72 at Yale University. He was a Miller Research Fellow at the University of California Berkeley from 1964-65 and he was appointed Hermann Weyl Professor at the Institute for Advanced Study in 1972, becoming Professor Emeritus in January 2007. His Ph. D. thesis was on the theory of semigroups. His first accomplishment in this field was a formula for the dimension of certain spaces of automorphic forms, as a first application, he proved the Weil conjecture on Tamagawa numbers for the large class of arbitrary simply connected Chevalley groups defined over the rational numbers. Previously this had been only in a few isolated cases. As a second application of work, he was able to show meromorphic continuation for a large class of L-functions arising in the theory of automorphic forms. These occurred in the constant terms of Eisenstein series, and meromorphicity as well as a functional equation were a consequence of functional equations for Eisenstein series. This work led in turn, in the winter of 1966–67 and these conjectures were first posed in relatively complete form in a famous letter to Weil, written in January 1967. It was in this letter that he introduced what has become known as the L-group and along with it. Langlandss introduction of these notions broke up large and to some extent intractable problems into smaller, for example, they made the infinite-dimensional representation theory of reductive groups into a major field of mathematical activity. Functoriality is the conjecture that automorphic forms on different groups should be related in terms of their L-groups, in its application to Artins conjecture, functoriality associated to every N-dimensional representation of a Galois group an automorphic representation of the adelic group of GL. In the theory of Shimura varieties it associates automorphic representations of groups to certain l-adic Galois representations as well. This book applied the trace formula for GL and quaternion algebras to do this. Subsequently James Arthur, a student of Langlands while he was at Yale, the functoriality conjecture is far from proved, but a special case was the starting point of Andrew Wiles attack on the Taniyama–Shimura conjecture and Fermats last theorem. In the mid-1980s Langlands turned his attention to physics, particularly the problems of percolation, in recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls beyond endoscopy
Robert Langlands
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Robert Langlands
33.
Andrew Wiles
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Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is most notable for proving Fermats Last Theorem, for which he received the 2016 Abel Prize, Wiles has received numerous other honours. Wiles was born in 1953 in Cambridge, England, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford and his father worked as the Chaplain at Ridley Hall, Cambridge, for the years 1952–55. Wiles attended Kings College School, Cambridge, and The Leys School, Wiles states that he came across Fermats Last Theorem on his way home from school when he was 10 years old. He stopped by his local library where he found a book about the theorem. Fascinated by the existence of a theorem that was so easy to state that he, a ten-year-old, could understand it, Wiles earned his bachelors degree in mathematics in 1974 at Merton College, Oxford, and a PhD in 1980 at Clare College, Cambridge. After a stay at the Institute for Advanced Study in New Jersey in 1981, in 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford and he rejoined Oxford in 2011 as Royal Society Research Professor. Wiless graduate research was guided by John Coates beginning in the summer of 1975, together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory. He further worked with Barry Mazur on the conjecture of Iwasawa theory over the rational numbers. The modularity theorem involved elliptic curves, which was also Wiless own specialist area, the conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove. Despite this, Wiles, with his fascination with Fermats Last Theorem, decided to undertake the challenge of proving the conjecture. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge and he gave a lecture a day on Monday, Tuesday and Wednesday with the title Modular Forms, Elliptic Curves and Galois Representations. There was no hint in the title that Fermats last theorem would be discussed, finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermats last theorem was true, in August 1993, it was discovered that the proof contained a flaw in one area. Wiles tried and failed for over a year to repair his proof, according to Wiles, the crucial idea for circumventing, rather than closing this area, came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a paper which circumvented the problem. Both papers were published in May 1995 in a volume of the Annals of Mathematics
Andrew Wiles
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Wiles at the 61st Birthday conference for P. Deligne (Institute for Advanced Study, 2005).
Andrew Wiles
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Andrew Wiles before the statue of Pierre de Fermat in Beaumont-de-Lomagne (October 1995)
34.
Elias M. Stein
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Elias Menachem Stein is a mathematician. He is a figure in the field of harmonic analysis. He is an emeritus of Mathematics at Princeton University. Stein was born to Elkan Stein and Chana Goldman, Ashkenazi Jews from Belgium, after the German invasion in 1940, the Stein family fled to the United States, first arriving in New York City. He graduated from Stuyvesant High School in 1949, where he was classmates with future Fields Medalist Paul Cohen, in 1955, Stein earned a Ph. D. from the University of Chicago under the direction of Antoni Zygmund. He began teaching in MIT in 1955, moved to the University of Chicago in 1958 as an assistant professor, and in 1963 became a professor at Princeton. Stein has worked primarily in the field of analysis, and has made contributions in both extending and clarifying Calderón–Zygmund theory. He has written books on harmonic analysis, which are often cited as the standard references on the subject. His Princeton Lectures in Analysis series were penned for his sequence of courses on analysis at Princeton. Stein is also noted as having trained a number of graduate students. They include two Fields medalists, Charles Fefferman and Terence Tao and his honors include the Steele Prize, the Schock Prize in Mathematics, the Wolf Prize in Mathematics, and the National Medal of Science. In addition, he has fellowships to National Science Foundation, Sloan Foundation, Guggenheim Foundation, in 2005, Stein was awarded the Stefan Bergman prize in recognition of his contributions in real, complex, and harmonic analysis. In 2012 he became a fellow of the American Mathematical Society, in 1959, he married Elly Intrator, a former Jewish refugee during World War II. They had two children, Karen Stein and Jeremy C, Stein, and grandchildren named Alison, Jason, and Carolyn. Singular Integrals and Differentiability Properties of Functions, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Introduction to Fourier Analysis on Euclidean Spaces, Lectures on Pseudo-differential Operators, Regularity Theorems and Applications to Non-elliptic Problems. Harmonic Analysis, Real-variable Methods, Orthogonality and Oscillatory Integrals, Stein, Elias, Shakarchi, R. Fourier Analysis, An Introduction. Stein, Elias, Shakarchi, R. Complex Analysis, real Analysis, Measure Theory, Integration, and Hilbert Spaces
Elias M. Stein
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Elias M. Stein
35.
Raoul Bott
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Raoul Bott, ForMemRS 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 context. Bott was born in Budapest, Hungary, the son of Margit Kovács and his father was of Austrian descent, and his mother was of Hungarian Jewish descent, Bott was raised a Catholic by his mother and stepfather. Bott grew up in Czechoslovakia and spent his life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Army in Europe during World War II, Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a Ph. D. in mathematics from Carnegie Mellon University in Pittsburgh in 1949 and 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 network of inductors and capacitors. Bott met Arnold S. Shapiro at the IAS and they worked together and he studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem. In the course of work, he introduced Morse–Bott functions. This led to his role as collaborator over many years with Michael Atiyah and he is also well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups, and for work on foliations. He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds, 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 26 Ph. D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, András Szenes, birkhäuser Boston, xx+485 pp. ISBN 0-8176-3648-X MR13218901995, Collected Papers. Birkhäuser, xxxii+610 pp. ISBN 0-8176-3647-1 MR13218861994, Collected Papers, birkhäuser, xxxiv+802 pp. ISBN 0-8176-3646-3 MR12903611994, Collected Papers
Raoul Bott
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Raoul Bott in 1986
36.
Jean-Pierre Serre
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Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000, born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951, from 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France and his wife, Professor Josiane Heulot-Serre, was a chemist, she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil, the French mathematician Denis Serre is his nephew. Serres thesis concerned the Leray–Serre spectral sequence associated to a fibration, together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, Serre subsequently changed his research focus. However, Weyls perception that the place of classical analysis had been challenged has subsequently been justified. In the 1950s and 1960s, a collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldnt capture as much topology as singular cohomology with integer coefficients, amongst Serres early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important and this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA5, from 1959 onward Serres interests turned towards group theory, number theory, in particular Galois representations and modular forms. In his paper FAC, Serre asked whether a finitely generated module over a polynomial ring is free. This question led to a deal of activity in commutative algebra. This result is now known as the Quillen-Suslin theorem, Serre, at twenty-seven in 1954, is the youngest ever to be awarded the Fields Medal. He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000 and he has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre. He is a member of several scientific Academies and has received many honorary degrees
Jean-Pierre Serre
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Jean-Pierre Serre
Jean-Pierre Serre
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Serre
37.
Vladimir Arnold
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Vladimir Igorevich Arnold was a Soviet and Russian mathematician. Arnold was also known as a popularizer of mathematics, through his lectures, seminars, and as the author of several textbooks and popular mathematics books, he influenced many mathematicians and physicists. Many of his books were translated into English, Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Soviet Union. His father was Igor Vladimirovich Arnold, a mathematician and his mother was Nina Alexandrovna Arnold, an art historian. This is the Kolmogorov–Arnold representation theorem, after graduating from Moscow State University in 1959, he worked there until 1986, and then at Steklov Mathematical Institute. He became an academician of the Academy of Sciences of the Soviet Union in 1990, Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms, Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. As of 2006 he was reported to have the highest citation index among Russian scientists, to his students and colleagues Arnold was known also for his sense of humour. In accordance with this principle I shall formulate some problems. ”Arnold died of pancreatitis on 3 June 2010 in Paris. He was buried on June 15 in Moscow, at the Novodevichy Monastery, in a telegram to Arnolds family, Russian President Dmitry Medvedev stated, “The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics, teaching had a special place in Vladimir Arnolds life and he had great influence as an enlightened mentor who taught several generations of talented scientists. The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man. ”Arnold is well known for his writing style, combining mathematical rigour with physical intuition. His defense is that his books are meant to teach the subject to those who wish to understand it. Arnold was a critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. Arnold was very interested in the history of mathematics and he liked to study the classics, most notably the works of Huygens, Newton and Poincaré, and many times he reported to have found in their works ideas that had not been explored yet. Arnold worked on systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics. Moser and Arnold expanded the ideas of Kolmogorov and gave rise to what is now known as KAM Theory, KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are. In 1965, Arnold attended René Thoms seminar on catastrophe theory, after this event, singularity theory became one of the major interests of Arnold and his students
Vladimir Arnold
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Vladimir Arnold in 2008
38.
Grigory Margulis
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Gregori Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, in 1991, he joined the faculty of Yale University, where he is currently the Erastus L. DeForest Professor of Mathematics. Margulis was born in Moscow, Soviet Union and he received his PhD in 1970 from the Moscow State University, starting research in ergodic theory under the supervision of Yakov Sinai. Early work with David Kazhdan produced the Kazhdan–Margulis theorem, a result on discrete groups. His superrigidity theorem from 1975 clarified an area of classical conjectures about the characterisation of arithmetic groups amongst lattices in Lie groups and he was awarded the Fields Medal in 1978, but was not permitted to travel to Helsinki to accept it in person. In 1991, Margulis accepted a position at Yale University. Margulis was elected a member of the U. S. National Academy of Sciences in 2001, in 2012 he became a fellow of the American Mathematical Society. In 2005, Margulis received the Wolf Prize for his contributions to theory of lattices and applications to ergodic theory, representation theory, number theory, combinatorics, and measure theory. Marguliss early work dealt with Kazhdans property and the questions of rigidity and arithmeticity of lattices in semisimple algebraic groups of rank over a local field. It had been known since the 1950s that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices and it is analogous to considering the subgroup SL of the real special linear group SL that consists of matrices with integer entries. Margulis proved that under suitable assumptions on G, any lattice Γ in it is arithmetic, thus Γ is commensurable with the subgroup G of G, i. e. they agree on subgroups of finite index in both. Unlike general lattices, which are defined by their properties, arithmetic lattices are defined by a construction, therefore, these results of Margulis pave a way for classification of lattices. Arithmeticity turned out to be related to another remarkable property of lattices discovered by Margulis. Superrigidity for a lattice Γ in G roughly means that any homomorphism of Γ into the group of invertible n × n matrices extends to the whole G. While certain rigidity phenomena had already known, the approach of Margulis was at the same time novel, powerful. Margulis solved the Banach–Ruziewicz problem that asks whether the Lebesgue measure is the only normalized rotationally invariant finitely additive measure on the n-dimensional sphere, Margulis gave the first construction of expander graphs, which was later generalized in the theory of Ramanujan graphs. In 1986, Margulis gave a resolution of the Oppenheim conjecture on quadratic forms. He has formulated a program of research in the same direction
Grigory Margulis
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Grigory Margulis
39.
Hillel Furstenberg
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He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. Hillel Furstenberg was born in Germany, in 1935, and the family emigrated to the United States in 1939 and he attended Marsha Stern Talmudical Academy and then Yeshiva University, where he concluded his BA and MSc studies in 1955. He obtained his Ph. D. under Salomon Bochner at Princeton University in 1958, after several years at the University of Minnesota he became a Professor of Mathematics at the Hebrew University of Jerusalem in 1965. He gained attention at a stage in his career for producing an innovative topological proof of the infinitude of prime numbers. He proved unique ergodicity of horocycle flows on compact hyperbolic Riemann surfaces in the early 1970s, in 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédis theorem. The Furstenberg boundary and Furstenberg compactification of a symmetric space are named after him. 1993 – Furstenberg received the Israel Prize, for exact sciences,1993 – Furstenberg received the Harvey Prize from Technion. 2006/7 – He received the Wolf Prize in Mathematics, Furstenberg, Harry, Stationary processes and prediction theory, Princeton, N. J. Furstenberg, Harry, Recurrence in ergodic theory and combinatorial number theory, Princeton, compactification Ratners theorems List of Israel Prize recipients OConnor, John J. Robertson, Edmund F. Hillel Furstenberg, MacTutor History of Mathematics archive, University of St Andrews. Mathematics Genealogy page Press release Israel Academy of Sciences and Humanities
Hillel Furstenberg
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Hillel Furstenberg in 1992 (photo by George Bergman)
40.
Pierre Deligne
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Pierre René, Viscount Deligne is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a proof in 1973. He is the winner of the 2013 Abel Prize,2008 Wolf Prize and he was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles. In 1968, he worked with Jean-Pierre Serre, their work led to important results on the l-adic representations attached to modular forms. Delignes also focused on topics in Hodge theory and he introduced weights and tested them on objects in complex geometry. He also collaborated with David Mumford on a new description of the spaces for curves. Their work came to be seen as an introduction to one form of the theory of algebraic stacks, perhaps Delignes most famous contribution was his proof of the third and last of the Weil conjectures. This proof completed a programme initiated and largely developed by Alexander Grothendieck, as a corollary he proved the celebrated Ramanujan–Petersson conjecture for modular forms of weight greater than one, weight one was proved in his work with Serre. From 1970 until 1984, when he moved to the Institute for Advanced Study in Princeton, during this time he did much important work outside of his work on algebraic geometry. He received a Fields Medal in 1978 and this idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. All this is part of the yoga of weights, uniting Hodge theory, the Shimura variety theory is related, by the idea that such varieties should parametrize not just good families of Hodge structures, but actual motives. This theory is not yet a finished product – and more recent trends have used K-theory approaches and he was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, and the Abel Prize in 2013. In 2006 he was ennobled by the Belgian king as viscount, in 2009, Deligne was elected a foreign member of the Royal Swedish Academy of Sciences. He is a member of the Norwegian Academy of Science and Letters, Quantum fields and strings, a course for mathematicians. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, american Mathematical Society, Providence, RI, Institute for Advanced Study, Princeton, NJ,1999. Vol.1, xxii+723 pp. Vol.2, pp. i--xxiv, Deligne wrote multiple hand-written letters to other mathematicians in the 1970s. These include Delignes letter to Piatetskii-Shapiro and it was proved by Kontsevich–Soibelman, McClure–Smith and others
Pierre Deligne
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Pierre Deligne, March 2005
41.
Phillip Griffiths
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Phillip Augustus Griffiths IV is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a developer in particular of the theory of variation of Hodge structure in Hodge theory. He received his B. S. from Wake Forest College in 1959, since then, he has held positions at Berkeley, Princeton, Harvard University, and Duke University. From 1991 to 2003 he was the Director of the Institute for Advanced Study at Princeton and he has published on algebraic geometry, differential geometry, geometric function theory, and the geometry of partial differential equations. Griffiths serves as the Chair of the Science Initiative Group and he is co-author, with Joe Harris, of Principles of Algebraic Geometry, a well-regarded textbook on complex algebraic geometry. In 2008 he was awarded the Wolf Prize and the Brouwer Medal, in 2012 he became a fellow of the American Mathematical Society. Moreover, in 2014 Griffiths was awarded the Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society, also in 2014, Griffiths was awarded the Chern Medal for lifetime devotion to mathematics and outstanding achievements. Proc Natl Acad Sci U S A.48, 780–783, some remarks on automorphisms, analytic bundles, and embeddings of complex algebraic varieties. Proc Natl Acad Sci U S A.49, 817–820, on the differential geometry of homogeneous vector bundles. The residue calculus and some results in algebraic geometry, I. Proc Natl Acad Sci U S A.55, 1303–1309, the residue calculus and some transcendental results in algebraic geometry, II. Proc Natl Acad Sci U S A.55, 1392–1395, some results on locally homogeneous complex manifolds. Proc Natl Acad Sci U S A.56, 413–416, a transcendental method in algebraic geometry. Periods of integrals on algebraic manifolds, with Joe Harris, A Poncelet theorem in space. With S. S. Chern, Abels Theorem and Webs, introduction to Algebraic Curves, American Mathematical Society, Providence, RI,1989, ISBN0821845306 Differential Systems and Isometric Embeddings, with Gary R
Phillip Griffiths
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Phillip Griffiths in 2008 (photo from MFO)
42.
David Mumford
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David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow, in 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University, Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started a school in Tanzania and worked for the then newly created United Nations. In high school, he was a finalist in the prestigious Westinghouse Science Talent Search, after attending the Phillips Exeter Academy, Mumford went to Harvard, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956 and he completed his Ph. D. in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. He met his first wife, Erika Jentsch, at Radcliffe College, after Erika died in 1988, he married his second wife, Jenifer Gordon. He and Erika had four children, Mumfords work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety and his books Abelian Varieties and Curves on an Algebraic Surface combined the old and new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique and they are now available as The Red Book of Varieties and Schemes. Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and this work on the equations defining abelian varieties appeared in 1966–7. He published some books of lectures on the theory. He also was one of the founders of the toroidal embedding theory and these pathologies fall into two types, bad behavior in characteristic p and bad behavior in moduli spaces. This second example is developed further in Mumfords third paper on classification of surfaces in characteristic p, worse pathologies related to p-torsion in crystalline cohomology were explored by Luc Illusie. Further such examples arise in Zariski surface theory and he also conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the third paper, he gives an example of a surface for which Kodaira vanishing fails. The first example of a surface for which Kodaira vanishing fails was given by Michel Raynaud in 1978. In the second Pathologies paper, Mumford finds that the Hilbert scheme parametrizing space curves of degree 14, in the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves
David Mumford
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David Mumford in 2010
David Mumford
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David Mumford in 1975
43.
Dennis Sullivan
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Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems and he holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University. He received his B. A. in 1963 from Rice University and his Ph. D. thesis, entitled Triangulating homotopy equivalences, was written under the supervision of William Browder, and was a contribution to surgery theory. He was a permanent member of the Institut des Hautes Études Scientifiques from 1974 to 1997, Sullivan is one of the founders of the surgery method of classifying high-dimensional manifolds, along with Browder, Sergei Novikov and C. T. C. In homotopy theory, Sullivan put forward the concept that spaces could directly be localised. This area has generated considerable further research, in 1985, he proved the No wandering domain theorem. The Parry–Sullivan invariant is named after him and the English mathematician Bill Parry, in 1987, he proved Thurstons conjecture about the approximation of the Riemann map by circle packings together with Burton Rodin. 47, 269–331, MR0646078 OConnor, John J. Robertson, Edmund F. Dennis Sullivan, MacTutor History of Mathematics archive, Dennis Sullivan at the Mathematics Genealogy Project Sullivans homepage at CUNY Sullivans homepage at SUNY Stony Brook Dennis Sullivan International Balzan Prize Foundation
Dennis Sullivan
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Dennis Parnell Sullivan
44.
Shing-Tung Yau
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Shing-Tung Yau is a Hong Kong and naturalised American mathematician. He was awarded the Fields Medal in 1982 and he is currently the William Caspar Graustein Professor of Mathematics at Harvard. Yaus work is mainly in differential geometry, especially in geometric analysis and his contributions have influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics. His proof of the energy theorem in general relativity demonstrated—sixty years after its discovery—that Einsteins theory is consistent. His proof of the Calabi conjecture allowed physicists to show, using Calabi–Yau compactification, Calabi–Yau manifolds are part of the standard toolkit for string theorists today. Yau was born in Shantou, Guangdong Province, China with Hakka ancestry in Jiaoling, Guangdong in a family of eight children. When he was only a few old, his family emigrated to Hong Kong. After graduating from Pui Ching Middle School, he studied mathematics at the Chinese University of Hong Kong from 1966 to 1969. Yau left for the University of California, Berkeley in the fall of 1969 and he spent a year as a member of the Institute for Advanced Study at Princeton before joining Stony Brook University in 1972 as an assistant professor. In 1974, he became a professor at Stanford University. Yau has held American citizenship since 1990, since 1987, he has been at Harvard University. He is also involved in the activities of research institutes in Hong Kong. Duong Hong Phong of Columbia University has commented on the influence of Yaus research in geometric analysis, Yaus solution of the Calabi conjecture, concerning the existence of an Einstein–Kähler metric, has far-reaching consequences. The existence of such a canonical unique metric allows one to give explicit representatives of characteristic classes, Calabi–Yau manifolds are now fundamental in string theory, where the Calabi conjecture provides an essential piece in the model. Yau also made a contribution in the case that the first Chern number c1 >0 and this has motivated the work of Simon Donaldson on scalar curvature and stability. Another important result of Donaldson–Uhlenbeck–Yau is that a vector bundle is stable if. Yau pioneered the method of using minimal surfaces to study geometry, by analyzing how minimal surfaces behave in space-time, Yau and Richard Schoen proved the long-standing conjecture that the total mass in general relativity is positive. This theorem implies that flat space-time is stable, an issue for the theory of general relativity
Shing-Tung Yau
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Image of mathematician Shing Tung Yau
Shing-Tung Yau
45.
Michael Aschbacher
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Michael George Aschbacher is an American mathematician best known for his work on finite groups. He was a figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of groups had not been finished. This gap was fixed by Aschbacher and Stephen D. Smith in 2004, Aschbacher is currently the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology. Aschbacher received his B. S. at the California Institute of Technology in 1966 and he joined the faculty of the California Institute of Technology in 1970 and became a full professor in 1976. He was a scholar at the Institute for Advanced Study in 1978-79. He was awarded the Cole Prize in 1980, and was elected to the National Academy of Sciences in 1990, in 1992, Aschbacher was elected a Fellow of the American Academy of Arts and Sciences. He was awarded the Rolf Schock Prize for Mathematics by the Royal Swedish Academy of Sciences in 2011, in 2012 he received the Leroy P. Steele Prize for Mathematical Exposition and the Wolf Prize in Mathematics, and became a fellow of the American Mathematical Society. In 1973, Aschbacher became a figure in the classification of finite simple groups. Interestingly, Aschbacher considered himself somewhat of an outsider in the world of conventional group theory, Aschbacher only became interested in finite simple groups as a postdoctorate. In particular, Daniel Gorenstein, another leader of the classification of simple groups. In fact, the rate of Aschbachers results proved so astounding that many mathematicians decided to leave the field to pursue other problems. Aschbacher was proving one major result after another and when he announced his progress at the Duluth conference and this conference represented a turning point for the problem as many mathematicians decided to leave the field to pursue other problems. However, Aschbachers entrance into the field did not come without difficulties, Aschbachers papers, beginning with the first he wrote in the field for publication, were very difficult to read. Some commented that his proofs lacked explanations of very sophisticated counting arguments, as Aschbachers proofs became longer, it became even more difficult for others to understand his proofs. Even some of his own coauthors had trouble reading their own papers, from that point on, researchers no longer read papers as independent documents, but rather ones that required the context of its author. As a result, responsibility of finding errors in the problem was up to the entire community of researchers rather than just peer-reviewers alone. That Aschbachers proofs were hard to read was not due to a lack of ability, I Structure of Strongly Quasithin K-groups, Mathematical Surveys and Monographs,111, Providence, R. I
Michael Aschbacher
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Michael Aschbacher
46.
Michael Artin
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Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. Artin was born in Hamburg, Germany, and brought up in Indiana and his parents were Natalia Naumovna Jasny and Emil Artin, preeminent algebraist of the 20th century. Artins parents had left Germany in 1937, because Michael Artins maternal grandfather was Jewish, in the early 1960s Artin spent time at the IHÉS in France, contributing to the SGA4 volumes of the Séminaire de géométrie algébrique, on topos theory and étale cohomology. His work on the problem of characterising the representable functors in the category of schemes has led to the Artin approximation theorem and this work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved very influential in moduli theory. Additionally, he has made contributions to the theory of algebraic varieties. Small, which prompted first foray into ring theory, in 2002, Artin won the American Mathematical Societys annual Steele Prize for Lifetime Achievement. In 2005, he was awarded the Harvard Centennial Medal, in 2013 he won the Wolf Prize in Mathematics, and in 2015 was awarded the National Medal of Science. Artin–Mazur zeta function Artin stacks Artin–Verdier duality Michael Artin at MIT Mathematics
Michael Artin
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Michael Artin (photo by George Bergman)
47.
Virtual International Authority File
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The Virtual International Authority File is an international authority file. It is a joint project of national libraries and operated by the Online Computer Library Center. The project was initiated by the US Library of Congress, the German National Library, the National Library of France joined the project on October 5,2007. The project transitions to a service of the OCLC on April 4,2012, the aim is to link the national authority files to a single virtual authority file. In this file, identical records from the different data sets are linked together, a VIAF record receives a standard data number, contains the primary see and see also records from the original records, and refers to the original authority records. The data are available online and are available for research and data exchange. Reciprocal updating uses the Open Archives Initiative Protocol for Metadata Harvesting protocol, the file numbers are also being added to Wikipedia biographical articles and are incorporated into Wikidata. VIAFs clustering algorithm is run every month, as more data are added from participating libraries, clusters of authority records may coalesce or split, leading to some fluctuation in the VIAF identifier of certain authority records
Virtual International Authority File
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Screenshot 2012
48.
Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly for documentation in libraries and increasingly also by archives, the GND is managed by the German National Library in cooperation with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license, the GND specification provides a hierarchy of high-level entities and sub-classes, useful in library classification, and an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format
Integrated Authority File
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GND screenshot