|
Thomas C. 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 (Langlands program), fundamental lemma over the group symplectic 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 and the honeycomb 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, his dissertation was 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
San Antonio, Texas
("Cradle of Freedom") , image_map = , mapsize = 220px , map_caption = Interactive map of San Antonio , subdivision_type = Country , subdivision_name = United States , subdivision_type1= State , subdivision_name1 = Texas , subdivision_type2 = Counties , subdivision_name2 = Bexar, Comal, Medina , established_title = Foundation , established_date = May 1, 1718 , established_title1 = Incorporated , established_date1 = June 5, 1837 , named_for = Saint Anthony of Padua , government_type = Council-Manager , governing_body = San Antonio City Council , leader_title = Mayor , leader_name = Ron Nirenberg ( I) , leader_title2 = City Manager , leader_name2 = Erik Walsh , leader_title3 = City Council , leader_name3 = , unit_pref = Imperial , area_total_sq_mi = 504.64 , area_total_km2 = 1307.00 , area_land_sq_mi = 498.85 , area_land_km2 = 1292.02 , area_water_sq_mi = 5.79 , area_water_km2 ... [...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 for this proof. Motivation and history Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be related in a particular way. This relationship takes the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isabelle (proof Assistant)
The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring yet supporting explicit proof objects. Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle ''Archive of Formal Proofs'' (Isabelle AFP) Isabelle was named by Lawrence Paulson after Gérard Huet's daughter. The Isabelle theorem prover is free software, released under the revised BSD license. Features Isabelle is generic: it provides a meta-logic (a weak type theory), which is used to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HOL Light
HOL Light is a member of the HOL theorem prover family. Like the other members, it is a proof assistant for classical higher order logic. Compared with other HOL systems, HOL Light is intended to have relatively simple foundations. HOL Light is authored and maintained by the mathematician and computer scientist John Harrison. HOL Light is released under the simplified BSD license. Logical foundations HOL Light is based on a formulation of type theory with equality as the only primitive notion. The primitive rules of inference are the following: This formulation of type theory is very close to the one described in section II.2 of . References * Further reading *{{Citation , author = Freek Wiedijk , title = Formal Proof — Getting Started , journal = Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined Ju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marcus Terentius Varro
Marcus Terentius Varro (; 116–27 BC) was a Roman polymath and a prolific author. He is regarded as ancient Rome's greatest scholar, and was described by Petrarch as "the third great light of Rome" (after Vergil and Cicero). He is sometimes called Varro Reatinus to distinguish him from his younger contemporary Varro Atacinus. Biography Varro was born in or near Reate (now Rieti) to a family thought to be of equestrian rank, and always remained close to his roots in the area, owning a large farm in the Reatine plain, reported as near Lago di Ripasottile, until his old age. He supported Pompey, reaching the office of praetor, after having been tribune of the people, ''quaestor'' and ''curule aedile''. It is probable that Varro was discontented with the course on which Pompey entered when the First Triumvirate was formed, and he may thus have lost his chance of rising to the consulate. He actually ridiculed the coalition in a work entitled the ''Three-Headed Monster'' ( in th ... [...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 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, the densest packing uses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a 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 which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which 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 further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the best universities in the world and it is among the most selective in the United States. The university is composed of an undergraduate college and five graduate research divisions, which contain all of the university's graduate programs and interdisciplinary committees. Chicago has eight professional schools: the Law School, the Booth School of Business, the Pritzker School of Medicine, the Crown Family School of Social Work, Policy, and Practice, the Harris School of Public Policy, the Divinity School, the Graham School of Continuing Liberal and Professional Studies, and the Pritzker School of Molecular Engineering. The university has additional campuses and centers in London, Paris, Beijing, Delhi, and Hong Kong, as well as in downtown ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher learning in the United States and one of the most prestigious and highly ranked universities in the world. The university is composed of ten academic faculties plus Harvard Radcliffe Institute. The Faculty of Arts and Sciences offers study in a wide range of undergraduate and graduate academic disciplines, and other faculties offer only graduate degrees, including professional degrees. Harvard has three main campuses: the Cambridge campus centered on Harvard Yard; an adjoining campus immediately across Charles River in the Allston neighborhood of Boston; and the medical campus in Boston's Longwood Medical Area. Harvard's endowment is valued at $50.9 billion, making it the wealthiest academic institution in the world. Endowment inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Honeycomb Conjecture
The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales. Theorem Let \Gamma be any system of smooth curves in \mathbb^2, subdividing the plane into regions (connected components of the complement of \Gamma) all of which are bounded and have unit area. Then, averaged over large disks in the plane, the average length of \Gamma per unit area is at least as large as for the hexagon tiling. The theorem applies even if the complement of \Gamma has additional components that are unbounded or whose area is not one; allowing these additional components cannot shorten \Gamma. Formally, let B(0,r) denote the disk of radius r centered at the origin, let L_r denote the total length of \Gamma\cap B(0,r), and let A_r denote the total area of B(0,r) covered by bounded unit-area components. (If these are the only components, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> San Antonio, Texas")
_(cropped).jpg "Click for more on -> Sphere Packing")
