Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Moshe Jarden is an Israeli mathematician, specialist in field arithmetic. Moshe Jarden was born in 1942 in Tel Aviv, his father, Dr. Dov Jarden, was a mathematician and linguist, who transmitted him his love to mathematics. In 1970 he received his Ph. D in Mathematics from the Hebrew University of Jerusalem, with Hillel Furstenberg as his thesis advisor, he accomplished his post doctorate during the years 1971-1973 at the Institut of Mathematics, Heidelberg University, with Peter Roquette as his mentor, habilitated there in 1972. During these years in Heidelberg, he initiated an intense and long term cooperation with German mathematicians with Peter Roquette, Wulf-Dieter Geyer, Gerhard Frey, Juergen Ritter, his achievements in mathematics, as well as the foundation of this fruitful cooperation with German mathematicians, earned him the L. Meithner-A.v. Humboldt Prize by the Alexander von Humboldt Foundation in 2001. In 1974, Moshe Jarden returned to Israel, joined the School of Mathematics of Tel Aviv University.
He became a full professor in 1982, the incumbent of the Cissie and Aaron Beare chair in Algebra and Number Theory in 1998. One of his great achievements is the publication of the book "Field Arithmetic" in the series Ergebnisse der Mathematik und ihrer Grenzgebiete of Springer, which earned him the Landau Prize; the main contribution of Moshe Jarden in algebra, in mathematics in general, is his research on families of large algebraic extensions of Hilbertian fields, parametrized by the automorphisms of the absolute Galois group of the base field. Notable results in this domain are the zero theorem, the transfer theorem, the free generators theorem, the Frey-Jarden theorem about the rank of algebraic varieties over large algebraic fields, Geyer-Jarden theorem about torsion points on elliptic curves over large algebraic fields, the strong approximation theorem over such fields; the remarkable development of Galois theory over a class of large fields, called ample fields, is described in the second book of Jarden: Algebraic Patching.
In 1979, Moshe Jarden came to Irvine and met Michael Fried. This was during this visit that Fried suggested to write a joint book on the topics they had worked about so far, proposed the name "Field arithmetic"; this book, where Diophantine fields are explored through their absolute Galois groups, was influential in the domain of field theory. Owing to this work, the intense activity of Moshe Jarden, his influence on his colleagues, "Field Arithmetic" has become a recognized name of a branch of algebra, with its own classification number; the impact of this book can be measured by the number of open problems presented in the first and second editions, which were solved between the publication of the first and third editions. In 1987, Moshe Jarden won the Landau Prize for the publication of "Field Arithmetic"; some years after the release of the first edition, it was realized that the term "field arithmetic" had been coined by the mathematician Paulo Ribenboim somewhat earlier, who had published a book in French named "L'Arithmétique des Corps".
He was awarded the Landau prize for the book "Field Arithmetic" in 1987, the L. Meithner-A.v. Humboldt Prize in 2001 for his achievements in mathematics. Moshe Jarden, Elementary statements over large algebraic fields, Transactions of AMS 164, 67-91. Gerhard Frey and Moshe Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proceedings of the London Mathematical Society 28, 112-128. M. Jarden, Algebraic extensions of finite corank of Hilbertian fields, Israel Journal of Mathematics 18, 279-307. W.-D. Geyer and M. Jarden, Torsion points of elliptic curves over large algebraic extensions of finitely generated fields,Israel Journal of Mathematics 31, 157-197. M. Jarden and U. Kiehne, The elementary theory of algebraic fields of finite corank, Inventiones Mathematicae 30, 275-294. M. Fried, D. Haran, M. Jarden, Galois stratification over Frobenius fields, Advances of Mathematics, 51, 1-35. D. Haran and M. Jarden, The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik 383, 147-206.
W.-D. Geyer and M. Jarden, On stable fields in positive characteristic, Geometriae Dedicata 29, 335-375. M. Jarden, Large normal extensions of Hilbertian fields, Mathematische Zeitschrift 224, 555-565. D. Haran and M. Jarden, Regular split embedding problems over complete valued fields, Forum Mathematicum 10, 329-351. D. Haran, M. Jarden, F. Pop, The absolute Galois group of subfields of the field of S-adic numbers, Functiones et Approximatio Commentarii Mathematici, 46, 205-223. M. Jarden, Diamonds in torsion of Abelian varieties, Journal of the Institute of Mathematics Jussieu 9, 477-480. Absolute Galois group Biographic details about Moshe Jarden Some insights about "field arithmetic" Other details on the book The mathematics Genealogy Project Moshe Jarden homepage Publications of Dr. Dov Jarden
Professor Alexander Lubotzky is an Israeli mathematician and former politician, a professor at the Hebrew University of Jerusalem and an adjunct professor at Yale University. He served as a member of the Knesset for The Third Way party between 1996 and 1999. In 2018 he won the Israel Prize for his accomplishments in maths and computer science. Lubotzky was born in Tel Aviv to Holocaust survivors, his father, Iser Lubotzky was the legal advisor of Herut. After school, Lubotzky did his IDF national service as a captain officer in a special intelligence and communication unit, he studied mathematics at Bar-Ilan University during highschool, gaining a BA and continued to direct PhD. He worked as a professor of mathematics at the Hebrew University, he has been a visiting professor at the Institute for Advanced Study in Princeton and the University of Chicago, with visits at Columbia, Yale, NYU and ETH Zurich. Lubotzky holds a Maurice and Clara Weil Chair in mathematics at the Einstein Institute of Mathematics of the Hebrew University of Jerusalem.
He is known for contributions to geometric group theory, the study of lattices in Lie groups, representation theory of discrete groups and Kazhdan's property, the study of subgroup growth and applications of group theory to combinatorics and computer science and error correcting codes. Lubotzky received the Erdős Prize in 1990. in the years 1994–1996 Lubotzky was the chairman of Einstein Institute of Mathematics at the Hebrew University of Jerusalem. In 1992 Lubotzky was a recipient of the Sunyer i Balaguer Prize from the Institut d'Estudis Catalans for his book "Discrete Groups Expanding Graphs and Invariant Measures" and again in 2002 with Dan Segal for their book "Subgroup Growth". In 2002 he has received the Rothschild Prize in mathematics. Lubotzky is listed as an ISI cited researcher in mathematics since 2003. Lubotzky was elected a foreign member of the American Academy of Arts and Sciences in 2005. In 2005-6 He led in the Institute for Advanced Study in Princeton a year long program on "Pro-finite groups and the congruence subgroup problem".
In 2006, he got an honorary degree from the University of Chicago for his contribution to Modern mathematics. In 2008 Lubotzky received the European Research Council advanced grant for exceptional established research leaders. In 2011 Lubotzky was chosen to be the keynote speaker at the joint meeting of the American Mathematical Society and the Mathematical Association of America in New Orleans. Lubotzky's keynote address in front of the conference's 6,000 attendees marked the first time that an Israeli was the keynote speaker at one of these conferences. In 2012 he was a visiting researcher at Microsoft Research Center. In 2014, he was elected to the Israel Academy of Humanities. In 2015 Lubotzky received the European Research Council advanced grant for exceptional established research leaders, becoming one of the only researchers receiving the grant twice. In honor of Lubotzky's sixtieth birthday The Israel Institute for Advanced Studies hosted a conference from 5 November through 11 November 2016, with scholars from around the world convening to celebrate his work and collaborations.
In 2018 Lubotzky received the Israel Prize, for mathematics. Lubotzky gave a Plenary lecture in the 2018 International Congress of Mathematicians at Rio de Janeiro, Brazil. A founding member of The Third Way in March 1996, he chaired its secretariat and was elected to the Knesset in the May 1996 elections, he served as a member of the Knesset's Foreign Defense Committee. As an MK, Prof. Lubotzky was known for his compromise proposals on religious issues and pluralism, he was involved in setting up a solution to the conversion bill crisis via the Ne'eman Commission, working to avoid a conflict between Israel and the Jewish Diaspora. Prof. Lubotzky co-drafted a comprehensive proposal for a new covenant for religion-state affairs in Israel with MK Yossi Beilin. Lubotzky married Yardenna, a lecturer in Art History and English, in 1980; the couple had six children. 1991: Erdős Prize for the best Israeli mathematician/computer scientist under the age of 40 1993: Ferran Sunyer i Balaguer Prize for the book Discrete groups, Expanding Graphs and Invariant Reassures 2002: The Rothschild Prize 2002: Ferran Sunyer i Balaguer Prize for the book Subgroup growth 2003: ISI list of Highly Cited Researchers 2005: Elected Foreign Honorary Member of the American Academy of Arts and Sciences 2006: Honorary doctoral degree from the University of Chicago 2014: Elected to the Israel Academy of Sciences and Humanities 2018: Israel Prize for mathematics Varieties of Representations of finitely generated groups, 1985 Discrete Groups, Expanding Graphs and Invariant Measures, 1994 Tree Lattices, 2000 Subgroup Growth, 2001 מבנים אלגבריים: חבורות, חוגים ושדות, 2018 Alexander Lubotzky on the Knesset website Alexander Lubotzky's webpage, Hebrew University Alexander Lubotzky at the Mathematics Genealogy Project Alexander Lubotzky at the Mathematical Reviews author summary
Lou van den Dries
Laurentius Petrus Dignus "Lou" van den Dries is a Dutch mathematician working in model theory. He is a professor of mathematics at the University of Illinois at Urbana–Champaign, he completed his PhD at Utrecht University in 1978 under the supervision of Dirk van Dalen with a dissertation entitled Model Theory of Fields. Van den Dries is most known for his seminal work in o-minimality, he has made contributions to the model theory of p-adic fields, valued fields, finite fields, to the study of transseries. With Alex Wilkie, he improved Gromov's theorem on groups of polynomial growth using nonstandard methods. Van den Dries has been a corresponding member of the Royal Netherlands Academy of Arts and Sciences since 1993, he was an invited speaker at the 1990 International Congress of Mathematicians in Kyoto and delivered the Tarski Lectures at the University of California, Berkeley in 2017. He was awarded the Shoenfield Prize from the Association for Symbolic Logic in 2016 for his chapter "Lectures on the Model Theory of Valued Fields" in Model Theory in Algebra and Arithmetic, edited by Dugald Macpherson and Carlo Toffalori.
Van den Dries was jointly awarded the 2018 Karp Prize with Matthias Aschenbrenner and Joris van der Hoeven "for their work in model theory on asymptotic differential algebra and the model theory of transseries."His doctoral students include Matthias Aschenbrenner. M. Aschenbrenner. Asymptotic Differential Algebra and Model Theory of Transseries. Annals of Mathematics Studies. 195. Princeton University Press. ArXiv:1509.02588. Doi:10.1515/9781400885411. ISBN 9781400885411. MR 3585498. Zbl 06684722. Z. Chatzidakis. "Definable sets over finite fields". J. Reine Angew. Math. 427: 107–135. Doi:10.1515/crll.1992.427.107. MR 1162433. Zbl 0759.11045. J. Denef. "p-adic and real subanalytic sets". Ann. of Math. Series 2. 128: 79–138. Doi:10.2307/1971463. JSTOR 1971463. MR 0951508. Zbl 0693.14012. L. van den Dries. "Gromov's theorem of groups of polynomial growth and elementary logic". J. Algebra. 89: 349–374. Doi:10.1016/0021-869390223-0. MR 0751150. Zbl 0552.20017. L. van den Dries. "The elementary theory of restricted analytic fields with exponentiation".
Ann. of Math. Series 2. 140: 183–205. Doi:10.2307/2118545. JSTOR 2118545. MR 1289495. Zbl 0837.12006. L. van den Dries. "Geometric categories and o-minimal structures". Duke Math. J. 84: 497–540. Doi:10.1215/S0012-7094-96-08416-1. MR 1404337. Zbl 0889.03025. L. van den Dries. Tame topology and o-minimal structures. London Mathematical Society Lecture Notes. 248. Cambridge University Press. Doi:10.1017/CBO9780511525919. ISBN 9780511525919. MR 1633348. Zbl 0953.03045. L. van den Dries, "Lectures on the Model Theory of Valued Fields", in H. Dugald Macpherson. Toffalori, Model Theory in Algebra and Arithmetic, Lecture Notes in Mathematics, 2111, Springer-Verlag, pp. 55–157, doi:10.1007/978-3-642-54936-6_4, ISBN 978-3-642-54935-9, MR 3330198, Zbl 1347.03074
Florian Pop is a Romanian mathematician, a professor of mathematics at the University of Pennsylvania. Pop received his Ph. D. in 1987 and his habilitation in 1991, both from the University of Heidelberg. He has been a member of the Institute for Advanced Study in Princeton, New Jersey, a professor at the University of Bonn prior to joining the U. Penn faculty. Pop's research concerns algebraic geometry, arithmetic geometry, anabelian geometry, Galois theory. Kuhlmann, Kuhlmann & Marshall call his habilitation thesis, concerning the characterization of certain fields by their absolute Galois groups, a "milestone". In 1996, Pop was awarded the Gay-Lussac–von Humboldt Prize for Mathematics, in 2003 he was awarded the Romanian Order of Merit. In 2012 he became a fellow of the American Mathematical Society. Pop, Florian, "On the Galois theory of function fields of one variable over number fields", Journal für die reine und angewandte Mathematik, 406: 200–218, doi:10.1515/crll.1990.406.200, MR 1048241 Pop, Florian, "On Grothendieck's conjecture of birational anabelian geometry", Annals of Mathematics, 139: 145–182, doi:10.2307/2946630, MR 1259367 Pop, Florian, "Embedding problems over large fields", Annals of Mathematics, 144: 1–34, doi:10.2307/2118581, MR 1405941 Tamás Szamuely, Groupes de Galois de corps de type fini, Astérisque, No.
294, ix, 403–431. MR2111651
David Harbater is an American mathematician at the University of Pennsylvania, well known for his work in Galois theory, algebraic geometry and arithmetic geometry. Harbater was born in New York City and attended Stuyvesant High School, where he was on the math team. After graduating in 1970, he entered Harvard University. After graduating summa cum laude in 1974, Harbater earned a master's degree from Brandeis University and a Ph. D. in 1978 from MIT, where he wrote a dissertation under the direction of Michael Artin. In 1995, Harbater was awarded the Cole Prize for his solution, with Michel Raynaud, of the long outstanding Abhyankar conjecture, he has solved the inverse Galois problem over Q p, made many other significant contributions to the field of Galois theory. In 2012 he became a fellow of the American Mathematical Society. Harbater's recent work on patching over fields, together with Julia Hartmann and Daniel Krashen, has had applications in such varied fields as quadratic forms, central simple algebras and local-global principles.
Harbater, D.. "Abhyankar's Conjecture on Galois Groups Over Curves". Invent. Math. 117: 1–25. Doi:10.1007/BF01232232. Recollections of Arthur Rothstein, Cole Prize citation for David Harbater David Harbater at the Mathematics Genealogy Project Harbater's home page at Penn
Providence, Rhode Island
Providence is the capital and most populous city of the U. S. is one of the oldest cities in the United States. It was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony, he named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city is situated at the mouth of the Providence River at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool and silverware industries. Today, the city of Providence is home to eight hospitals and seven institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturing activity; the city is the third most populous city in New England after Worcester, Massachusetts. Providence was one of the original Thirteen Colonies. Williams and his company were compelled to leave Massachusetts Bay Colony, Providence became a refuge for persecuted religious dissenters, as Williams himself had been exiled from Massachusetts.
The city was burned to the ground in March 1676 by the Narragansetts during King Philip's War, despite the good relations between Williams and the sachems with whom the United Colonies of New England were waging war. In the year, the Rhode Island legislature formally rebuked the other colonies for provoking the war. Providence residents were among the first Patriots to spill blood in the lead-up to the American Revolutionary War during the Gaspée Affair of 1772, Rhode Island was the first of the Thirteen Colonies to renounce its allegiance to the British Crown on May 4, 1776, it was the last of the Thirteen Colonies to ratify the United States Constitution on May 29, 1790, once assurances were made that a Bill of Rights would become part of the Constitution. Following the war, Providence was the country's ninth-largest city with 7,614 people; the economy shifted from maritime endeavors to manufacturing, in particular machinery, silverware and textiles. By the start of the 20th century, Providence hosted some of the largest manufacturing plants in the country, including Brown & Sharpe, Nicholson File, Gorham Manufacturing Company.
Providence residents ratified a city charter in 1831 as the population passed 17,000. The seat of city government was located in the Market House in Market Square from 1832 to 1878, the geographic and social center of the city; the city offices outgrew this building, the City Council resolved to create a permanent municipal building in 1845. The city offices moved into the Providence City Hall in 1878. During the American Civil War, local politics split over slavery as many had ties to Southern cotton and the slave trade. Despite ambivalence concerning the war, the number of military volunteers exceeded quota, the city's manufacturing proved invaluable to the Union. Providence thrived after the war, waves of immigrants brought the population from 54,595 in 1865 to 175,597 by 1900. By the early 1900s, Providence was one of the wealthiest cities in the United States. Immigrant labor powered one of the nation's largest industrial manufacturing centers. Providence was a major manufacturer of industrial products, from steam engines to precision tools to silverware and textiles.
Giant companies were based in or near Providence, such as Brown & Sharpe, the Corliss Steam Engine Company, Babcock & Wilcox, the Grinnell Corporation, the Gorham Manufacturing Company, Nicholson File, the Fruit of the Loom textile company. From 1975 until 1982, $606 million of local and national community development funds were invested throughout the city. In the 1990s, the city pushed for revitalization, realigning the north-south railroad tracks, removing the huge rail viaduct that separated downtown from the capitol building and moving the rivers to create Waterplace Park and river walks along the rivers' banks, constructing the Fleet Skating Rink and the Providence Place Mall. Despite new investment, poverty remains an entrenched problem. 27.9 percent of the city population is living below the poverty line. Recent increases in real estate values further exacerbate problems for those at marginal income levels, as Providence had the highest rise in median housing price of any city in the United States from 2004 to 2005.
The Providence city limits enclose a small geographical region with a total area of 20.5 square miles. Providence is located at the head of Narragansett Bay, with the Providence River running into the bay through the center of the city, formed by the confluence of the Moshassuck and Woonasquatucket Rivers; the Waterplace Park amphitheater and riverwalks line the river's banks through downtown. Providence is one of many cities claimed to be founded on seven hills like Rome; the more prominent hills are: Constitution Hill, College Hill, Federal Hill. The other four are: Tockwotten Hill at Fox Point, Smith Hill, Christian Hill at Hoyle Square, Weybosset Hill at the lower end of Weybosset Street, leveled in the early 1880s. Providence has 25 official neighborhoods, though these neighborhoods are grouped together and referred to