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Thomas Callister Hales
Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof of the fundamental lemma, due to Ngô Bảo Châu). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings, the honeycomb conjecture, and the dodecahedral conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture. Biography He received his Ph.D. from Princeton University in 1986 with a dissertation titled ''The Subregular Germ of Orbital Integrals''. Hales taught at Harvard University and the University of Chicago, and from 1993 and 2002 he worked at the University of Michigan. In 1998, Hales submitted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
San Antonio, Texas
San Antonio ( ; Spanish for "Anthony of Padua, Saint Anthony") is a city in the U.S. state of Texas and the most populous city in Greater San Antonio. San Antonio is the List of Texas metropolitan areas, third-largest metropolitan area in Texas and the List of Metropolitan Statistical Areas, 24th-largest metropolitan area in the United States at 2.6 million people in the 2020 United States census. It is the most populous city in and the county seat of Bexar County. San Antonio is the List of United States cities by population, seventh-most populous city in the United States, and the second-most populous in the Southern United States List of municipalities in Texas, and Texas, after Houston. Founded as a Spanish colonization of the Americas, Spanish mission and colonial outpost in 1718, the city in 1731 became the first chartered civil settlement in what is now present-day Texas. The area was then part of the Spanish Empire. From 1821 to 1836, it was part of the Mexico, Mexican ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lester R
Lester is an ancient Anglo-Saxon surname and given name. People Given name * Lester Bangs (1948–1982), American music critic * Lester Oliver Bankhead (1912–1997), American architect * Lester W. Bentley (1908–1972), American artist from Wisconsin * Lester Bird (1938–2021), second prime minister of Antigua and Barbuda (1994–2004) * Lester D. Boronda (1886–1953), American painter, furniture designer, sculptor * Lester Cotton (born 1996), American football player * Lester del Rey (1915–1993), American science fiction author and editor * Lester Ellis (born 1965), Australian former professional boxer * Lester Flatt (1914–1979), American bluegrass musician * Lester Gillis (1908–1934), better known as Baby Face Nelson, American gangster * Les Gold (born 1950), American pawnbroker and reality TV star * Lester Holt (born 1959), American television journalist * Lester Charles King (1907–1989), English geomorphologist * Lester Lanin (1907–2004), American j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to Non-Euclidean geometry, non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the ''packing density'' of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ngô Bảo Châu
Ngô Bảo Châu (, born June 28, 1972) is a Vietnamese-French mathematician at the University of Chicago, best known for proving the fundamental lemma for automorphic forms (proposed by Robert Langlands and Diana Shelstad). He is the first Vietnamese national to have received the Fields Medal. Early life Ngô Bảo Châu was born in 1972, the son of an intellectual family in Hanoi, North Vietnam. His father, professor Ngô Huy Cẩn, is full professor of physics at the Vietnam National Institute of Mechanics. His mother, Trần Lưu Vân Hiền, is a physician and associate professor at an herbal medicine hospital in Hanoi. The beginning of Châu's schooling was at an experimental elementary school that had been founded by the revolutionary pedagogue Hồ Ngọc Đại, but when his father returned from the Soviet Union with his doctoral degree, he decided that Châu would learn more in traditional schools and enrolled him in the "chuyên toán" (special classes for gifted st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name " symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Lemma (Langlands Program)
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. ''Time'' magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009". In 2010, Ngô was awarded the Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ... for this proof. Motivation and history Langlands outlined a strategy for pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Program
In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number theory to automorphic forms and, more generally, the representation theory of algebraic groups over local fields and adeles. It was described by Edward Frenkel as the " grand unified theory of mathematics." Background The Langlands program is built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and , the work and Harish-Chandra's approach on semisimple Lie groups, and in technical terms the trace formula of Selberg and others. What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called functoriality). Harish-Chandra's work exploited the principle that what can be d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Verification
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods. It represents an important dimension of analysis and verification in electronic design automation and is one approach to software verification. The use of formal verification enables the highest Evaluation Assurance Level ( EAL7) in the framework of common criteria for computer security certification. Formal verification can be helpful in proving the correctness of systems such as: cryptographic protocols, combinational circuits, digital circuits with internal memory, and software expressed as source code in a programming language. Prominent examples of verified software systems include the CompCert verified C compiler and the seL ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
