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
Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
A wiki is a website on which users collaboratively modify content and structure directly from the web browser. In a typical wiki, text is written using a simplified markup language and edited with the help of a rich-text editor. A wiki is run using wiki software, otherwise known as a wiki engine. A wiki engine is a type of content management system, but it differs from most other such systems, including blog software, in that the content is created without any defined owner or leader, wikis have little inherent structure, allowing structure to emerge according to the needs of the users. There are dozens of different wiki engines in use, both standalone and part of other software, such as bug tracking systems; some wiki engines are open source. Some permit control over different functions. Others may permit access without enforcing access control. Other rules may be imposed to organize content; the online encyclopedia project Wikipedia is the most popular wiki-based website, is one of the most viewed sites in the world, having been ranked in the top ten since 2007.
Wikipedia is not a single wiki but rather a collection of hundreds of wikis, with each one pertaining to a specific language. In addition to Wikipedia, there are tens of thousands of other wikis in use, both public and private, including wikis functioning as knowledge management resources, notetaking tools, community websites, intranets; the English-language Wikipedia has the largest collection of articles. Ward Cunningham, the developer of the first wiki software, WikiWikiWeb described wiki as "the simplest online database that could work". "Wiki" is a Hawaiian word meaning "quick". Ward Cunningham and co-author Bo Leuf, in their book The Wiki Way: Quick Collaboration on the Web, described the essence of the Wiki concept as follows: A wiki invites all users—not just experts—to edit any page or to create new pages within the wiki Web site, using only a standard "plain-vanilla" Web browser without any extra add-ons. Wiki promotes meaningful topic associations between different pages by making page link creation intuitively easy and showing whether an intended target page exists or not.
A wiki is not a crafted site created by experts and professional writers, designed for casual visitors. Instead, it seeks to involve the typical visitor/user in an ongoing process of creation and collaboration that changes the website landscape. A wiki enables communities of contributors to write documents collaboratively. All that people require to contribute is a computer, Internet access, a web browser, a basic understanding of a simple markup language. A single page in a wiki website is referred to as a "wiki page", while the entire collection of pages, which are well-interconnected by hyperlinks, is "the wiki". A wiki is a database for creating and searching through information. A wiki allows non-linear, evolving and networked text, while allowing for editor argument and interaction regarding the content and formatting. A defining characteristic of wiki technology is the ease with which pages can be created and updated. There is no review by a moderator or gatekeeper before modifications are accepted and thus lead to changes on the website.
Many wikis are open to alteration by the general public without requiring registration of user accounts. Many edits can be made in real-time and appear instantly online, but this feature facilitates abuse of the system. Private wiki servers require user authentication to edit pages, sometimes to read them. Maged N. Kamel Boulos, Cito Maramba, Steve Wheeler write that the open wikis produce a process of Social Darwinism. "'Unfit' sentences and sections are ruthlessly culled and replaced if they are not considered'fit', which results in the evolution of a higher quality and more relevant page. While such openness may invite'vandalism' and the posting of untrue information, this same openness makes it possible to correct or restore a'quality' wiki page." Some wikis have an Edit button or link directly on the page being viewed, if the user has permission to edit the page. This can lead to a text-based editing page where participants can structure and format wiki pages with a simplified markup language, sometimes known as Wikitext, Wiki markup or Wikicode.
An example of this is the VisualEditor on Wikipedia. WYSIWYG controls do not, always provide
European Mathematical Society
The European Mathematical Society is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians; the current president is Pavel Exner, Scientific Director of the Doppler Institute for Mathematical Physics and Applied Mathematics in Prague. The Society seeks to serve all kinds of mathematicians in universities, research institutes and other forms of higher education, its aims are to Promote mathematical research, both pure and applied and advise on problems of mathematical education, Concern itself with the broader relations of mathematics to society, Foster interaction between mathematicians of different countries, Establish a sense of identity amongst European mathematicians, Represent the mathematical community in supra-national institutions. The EMS is itself an Affiliate Member of the International Mathematical Union and an Associate Member of the International Council for Industrial and Applied Mathematics.
The precursor to the EMS, the European Mathematical Council was founded in 1978 at the International Congress of Mathematicians in Helsinki. This informal federation of mathematical societies was chaired by Sir Michael Atiyah; the European Mathematical Society was founded on 28 October 1990 in Mądralin near Warsaw, with Friedrich Hirzebruch as founding President. The EMS had 27 member societies; the first European Congress of Mathematics was held at the Sorbonne and Panthéon-Sorbonne universities in Paris in 1992, is now held every 4 years at different locations around Europe, organised by the EMS. The next ECM will be in 2020 in Portoroz in Slovenia. Friedrich Hirzebruch, 1990 - 1994 Jean-Pierre Bourguignon, 1995 - 1998 Rolf Jeltsch, 1999 - 2002 John Kingman, 2003 - 2006 Ari Laptev, 2007 - 2010 Marta Sanz-Solé, 2011 - 2014 Pavel Exner, 2015 - 2018 Volker Mehrmann, 2019 - 2023 The governing body of the EMS is its Council, which comprises delegates representing all of the societies which are themselves members of the EMS, along with delegates representing the institutional and individual EMS members.
The Council meets every 2 years, appoints the President and Executive Committee who are responsible for the running of the society. Besides the Executive Committee, the EMS has standing committees on: Applied Mathematics, Developing Countries, Mathematical Education, ERCOM, European Solidarity, Meetings and Electronic Dissemination, Raising Public Awareness of Mathematics, Women in Mathematics; the EMS's rules are set down in its Bylaws. The EMS is headquartered at the University of Helsinki; the European Congress of Mathematics is held every four years under the Society's auspices, at which ten EMS Prizes are awarded to "recognize excellent contributions in Mathematics by young researchers not older than 35 years". Since 2000, the Felix Klein Prize has been awarded to "a young scientist or a small group of young scientists for using sophisticated methods to give an outstanding solution, which meets with the complete satisfaction of industry, to a concrete and difficult industrial problem."
Since 2012, the Otto Neugebauer Prize has been awarded to a researcher or group of researchers'"for original and influential work in the field of history of mathematics that enhances our understanding of either the development of mathematics or a particular mathematical subject in any period and in any geographical region". Here are the awardees so far. EMS Prizes: Richard Borcherds F – Jens Franke – Alexander Goncharov – Maxim Kontsevich F – François Labourie – Tomasz Łuczak – Stefan Müller – Vladimír Šverák – Gábor Tardos – Claire Voisin EMS Prizes: Alexis Bonnet – Timothy Gowers F – Annette Huber-Klawitter – Aise Johan de Jong – Dmitry Kramkov – Jiří Matoušek – Loïc Merel – Grigori Perelman F, declined – Ricardo Pérez-Marco – Leonid Polterovich EMS Prizes: Semyon Alesker – Raphaël Cerf – Dennis Gaitsgory – Emmanuel Grenier – Dominic Joyce – Vincent Lafforgue – Michael McQuillan – Stefan Nemirovski – Paul Seidel – Wendelin Werner FFelix Klein Prize: David C. Dobson EMS Prizes: Franck Barthe – Stefano Bianchini – Paul Biran – Elon Lindenstrauss F – Andrei Okounkov F – Sylvia Serfaty – Stanislav Smirnov F – Xavier Tolsa – Warwick Tucker – Otmar Venjakob Felix Klein Prize: Not Awarded EMS Prizes: Artur Avila F – Alexei Borodin – Ben J. Green – Olga Holtz – Boáz Klartag – Alexander Kuznetsov – Assaf Naor – Laure Saint-Raymond – Agata Smoktunowicz – Cédric Villani FFelix Klein Prize: Josselin Garnier EMS Prizes: Simon Brendle - Emmanuel Breuillard - Alessio Figalli F - Adrian Ioana - Mathieu Lewin - Ciprian Manolescu - Grégory Miermont - Sophie Morel - Tom Sanders - Corinna Ulcigrai - Felix Klein Prize: Emmanuel Trélat Otto Neugebauer Prize: Jan P. Hogendijk EMS Prizes: Sara Zahedi - Mark Braverman - Vincent Calvez - Guido de Philippis - Peter Scholze F - Péter Varjú