3-manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer; this is made more precise in the definition below. A topological space X is a 3-manifold if it is a second-countable Hausdorff space and if every point in X has a neighbourhood, homeomorphic to Euclidean 3-space; the topological, piecewise-linear, smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, so there is a prevalence of specialized techniques that do not generalize to dimensions greater than three; this special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, partial differential equations.

3-manifold theory is considered a part of low-dimensional topology or geometric topology. A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful in the non-Haken case. Thurston's contributions to the theory allow one to consider, in many cases, the additional structure given by a particular Thurston model geometry; the most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is fruitful; the fundamental groups of 3-manifolds reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group topological methods. Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it.

This is just the standard 3-dimensional vector space over the real numbers. A 3-sphere is a higher-dimensional analogue of a sphere, it consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Real projective 3-space, or RP3, is the topological space of lines passing through the origin 0 in R4, it is a compact, smooth manifold of dimension 3, is a special case Gr of a Grassmannian space. RP3 is SO, hence admits a group structure; the 3-dimensional torus is the product of 3 circles. That is: T 3 = S 1 × S 1 × S 1; the 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer lattice Z3. Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.

A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is an example of a compact abelian Lie group; this follows from the fact. Group multiplication on the torus is defined by coordinate-wise multiplication. Hyperbolic space is a homogeneous space, it is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, models of elliptic geometry that have a constant positive curvature; when embedded to a Euclidean space, every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially; the Poincaré homology sphere is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere with a finite fundamental group, its fundamental group is known as the binary icosahedral group and has order 120.

This shows. In 2003, lack of structure on the largest scales in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. However, there is no strong support for the correctness of the model, as yet. In mathematics, Seifert–Weber space is a closed hyperbolic 3-manifold, it is known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space

Surface (topology)

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids. Other surfaces arise as graphs of functions of two variables. However, surfaces can be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis; the various mathematical notions of surface can be used to model surfaces in the physical world. In mathematics, a surface is a geometrical shape; the most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres. The exact definition of a surface may depend on the context. In algebraic geometry, a surface may cross itself, while, in topology and differential geometry, it may not.

A surface is a two-dimensional space. In other words, around every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, latitude and longitude provide two-dimensional coordinates on it; the concept of surface is used in physics, computer graphics, many other disciplines in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. A surface is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2; such a neighborhood, together with the corresponding homeomorphism, is known as a chart. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane; these coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean.

In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is nonempty, second countable, Hausdorff. It is often assumed that the surfaces under consideration are connected; the rest of this article will assume, unless specified otherwise, that a surface is nonempty, second countable, connected. More a surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H2 in C; these homeomorphisms are known as charts. The boundary of the upper half-plane is the x-axis. A point on the surface mapped via a chart to the x-axis is termed a boundary point; the collection of such points is known as the boundary of the surface, a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the x-axis is an interior point; the collection of interior points is the interior of the surface, always non-empty. The closed disk is a simple example of a surface with boundary.

The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary, compact is known as a'closed' surface; the two-dimensional sphere, the two-dimensional torus, the real projective plane are examples of closed surfaces. The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip. For example, the sphere and torus are orientable. In differential and algebraic geometry, extra structure is added upon the topology of the surface; this added structures can be a smoothness structure, a Riemannian metric, a complex structure, or an algebraic structure. Surfaces were defined as subspaces of Euclidean spaces; these surfaces were the locus of zeros of certain functions polynomial functions.

Such a definition considered the surface as part of a larger space, as such was termed extrinsic. In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean; this topological space is not considered a subspace of another space. In this sense, the definition given above, the definition that mathematicians use at present, is intrinsic. A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space, it may seem possible for some surfaces defined intrinsically to not be surfaces in the

Hellmuth Kneser

Hellmuth Kneser was a Baltic German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds, his proof originated the concept of normal surface, a fundamental cornerstone of the theory of 3-manifolds. He was died in Tübingen, Germany, he was the father of the mathematician Martin Kneser. He assisted Wilhelm Süss in the founding of the Mathematical Research Institute of Oberwolfach and served as the director of the institute from 1958 to 1959. Kneser had formulated the problem of non-integer iteration of functions and proved the existence of the entire Abel function of the exponential. Kneser was a student of David Hilbert, he was an advisor including Reinhold Baer. Hellmuth Kneser was a member of the NSDAP and the SA. In July 1934 he wrote to Ludwig Bieberbach a short note supporting his anti-semitic views and stating: "May God grant German science a unitary and continued political position."

O'Connor, John J.. Hellmuth Kneser at the Mathematics Genealogy Project

Joel Hass

Joel Hass is an American mathematician, a professor of mathematics and chair of the mathematics department at the University of California, Davis. Hass received his Ph. D. from the University of California, Berkeley in 1981 under the supervision of Robion Kirby. He joined the Davis faculty in 1988. In 2012 he became a fellow of the American Mathematical Society. Hass is known for proving the equal-volume special case of the double bubble conjecture, for proving that the unknotting problem is in NP, for giving an exponential bound on the number of Reidemeister moves needed to reduce the unknot to a circle. Research papersFreedman, Michael. Hass, Joel. Hass, Joel. Hass, Joel. "The number of Reidemeister moves needed for unknotting", Journal of the American Mathematical Society, 14: 399–428, arXiv:math/9807012, doi:10.1090/S0894-0347-01-00358-7, MR 1815217. BooksAdams, Colin. H. Freeman and Company, ISBN 0-7167-3160-6. Adams, Colin. H. Freeman and Company, ISBN 0-7167-4174-1. Home page at UC Davis Google scholar profile

Mathematics

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

ArXiv

ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.

Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.

This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.

Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.

Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev