|
List Of Unsolved Problems In Mathematics
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance. Lists of unsolved problems in math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Problems
A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox. Real-world problems Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems, they are used in mathematics education to teach students to connect real-world situations to the abstract language of mathematics. In general, to use mathematics for solving a real-world problem, the first ste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Problems
In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist. Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum operators. Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential. In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems. Among these was the problem of proving that the set of energy levels of one particular abstract quantum system was in fact the Cantor set, a challenge known as the "Ten Martini Problem" after the reward that Mark Kac offered for solving it. The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984. Context Background definitions for the "Coulomb energies" problems (N nonr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's scientific activities are managed from the President's office in Oxford, United Kingdom. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 through the sponsorship of Boston businessman Landon T. Clay. Harvard mathematician Arthur Jaffe was the first president of CMI. While the institute is best known for its Millennium Prize Problems, it carries out a wide range of activities, including a postdoctoral program (ten Clay Research Fellows are supported currently), conferences, workshops, and summer schools. Governance The institute is run according to a standard structure comprising a scientific advisory committee that decides on grant-awarding and research proposals, and a board of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis. Education and career Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smale's Problems
Smale's problems are a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century. Table of problems In later versions, Smale also listed three additional problems, "that don't seem important enough to merit a place on our main list, but it would still be nice to solve them:" # Mean value problem #Is the three-sphere a minimal set ( Gottschalk's conjecture)? #Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks? See also * Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems sel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yutaka Taniyama
was a Japanese mathematician known for the Taniyama–Shimura conjecture. Contribution Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura. The names Taniyama, Shimura and Weil have all been attached to this conjecture, but the idea is essentially due to Taniyama. “Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long. In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality. Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905. At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau's Problems
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows: # Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes? # Twin prime conjecture: Are there infinitely many primes ''p'' such that ''p'' + 2 is prime? # Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares? # Are there infinitely many primes ''p'' such that ''p'' − 1 is a perfect square? In other words: Are there infinitely many primes of the form ''n''2 + 1? , all four problems are unresolved. Progress toward solutions Goldbach's conjecture Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Mathematical Society
The Canadian Mathematical Society (CMS) (french: Société mathématique du Canada) is an association of professional mathematicians dedicated to the interests of mathematical research, outreach, scholarship and education in Canada. It serves the national community through the publication of academic journals, community bulletins, and the administration of mathematical competitions. It was originally conceived in June 1945 as the Canadian Mathematical Congress. A name change was debated for many years; ultimately, a new name was adopted in 1979, upon its incorporation as a non-profit charitable organization. The society is also affiliated with various national and international mathematical societies, including the Canadian Applied and Industrial Mathematics Society and the Society for Industrial and Applied Mathematics. The society is also a member of the International Mathematical Union and the International Council for Industrial and Applied Mathematics. History The Canadian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
