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C. P. Ramanujam
Chakravarthi Padmanabhan Ramanujam (9 January 1938 – 27 October 1974) was an Indian mathematician who worked in the fields of number theory and algebraic geometry. He was elected a fellow of the Indian Academy of Sciences in 1973. Like his namesake Srinivasa Ramanujan, Ramanujam also had a very short life. As David Mumford put it, Ramanujam felt that the spirit of mathematics demanded of him not merely routine developments but the right theorem on any given topic. "He wanted mathematics to be beautiful and to be clear and simple. He was sometimes tormented by the difficulty of these high standards, but in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuinely original stamp". Early life and education Ramanujam was born to a Tamil family on 9 January 1938 in Madras (now Chennai), India, as the eldest of seven, to Chakravarthi Srinivasa Padmanabhan. He finished his schooling in Town Higher Seconda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chennai
Chennai (, ), formerly known as Madras ( the official name until 1996), is the capital city of Tamil Nadu, the southernmost Indian state. The largest city of the state in area and population, Chennai is located on the Coromandel Coast of the Bay of Bengal. According to the 2011 Indian census, Chennai is the sixth-most populous city in the country and forms the fourth-most populous urban agglomeration. The Greater Chennai Corporation is the civic body responsible for the city; it is the oldest city corporation of India, established in 1688—the second oldest in the world after London. The city of Chennai is coterminous with Chennai district, which together with the adjoining suburbs constitutes the Chennai Metropolitan Area, the 36th-largest urban area in the world by population and one of the largest metropolitan economies of India. The traditional and de facto gateway of South India, Chennai is among the most-visited Indian cities by foreign tourists. It was ranked the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mumbai
Mumbai (, ; also known as Bombay — List of renamed Indian cities and states#Maharashtra, the official name until 1995) is the capital city of the Indian States and union territories of India, state of Maharashtra and the ''de facto'' financial centre of India. According to the United Nations, as of 2018, Mumbai is the List of cities in India by population, second-most populous city in India after Delhi and the List of largest cities, eighth-most populous city in the world with a population of roughly 20 million (2 crore). As per the Indian government population census of 2011, Mumbai was the list of cities in India by population, most populous city in India with an estimated city proper population of 12.5 million (1.25 crore) living under the Municipal Corporation of Greater Mumbai, Brihanmumbai Municipal Corporation. Mumbai is the centre of the Mumbai Metropolitan Region, the sixth most populous metropolitan area in the world with a population of over 23 million ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were (controversially) described as anti-semitic. Mathematics From his early years, Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups. Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waring's Problem
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants". Relationship with Lagrange's four-square theorem Long before Waring posed his problem, Diophantus had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by Claude Gaspard Bachet de Méziriac, and it was solved by Joseph-Louis Lag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Ludwig Siegel
Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, Siegel's method, Siegel's lemma and the Siegel mass formula for quadratic forms. He was named as one of the most important mathematicians of the 20th century.Pérez, R. A. (2011''A brief but historic article of Siegel'' NAMS 58(4), 558–566. André Weil, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. Atle Selberg said of Siegel and his work: Biography Siegel was born in Berlin, where he enrolled at the Humboldt University in Berlin in 1915 as a student in mathematics, astronomy, and physics. Amongst his teachers were Max Planck and Ferdinand Georg Frobenius, whose influence made the young Siegel abandon astronomy and turn towards number theory instead. His best-known student was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doctoral Advisor
A doctoral advisor (also dissertation director, dissertation advisor; or doctoral supervisor) is a member of a university faculty whose role is to guide graduate students who are candidates for a doctorate, helping them select coursework, as well as shaping, refining and directing the students' choice of sub- discipline in which they will be examined or on which they will write a dissertation. Students generally choose advisors based on their areas of interest within their discipline, their desire to work closely with particular graduate faculty, and the willingness and availability of those faculty to work with them. In some countries, the student's advisor serves as the chair of the dissertation committee or the examination committee. In some cases, though, the person who serves those roles may be different from the faculty member who has most closely advised the student. For instance, in the Dutch academic system, only full professors (''hoogleraren'') and associate profess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. * Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. * Additive num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
