In mathematics, topology generalizes the notion of triangulation in a natural way as follows: A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h: K → X. Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories. For topological manifolds, there is a stronger notion of triangulation: a piecewise-linear triangulation is a triangulation with the extra property – defined for dimensions 0, 1, 2... inductively – that the link of any simplex is a piecewise-linear sphere. The link of a simplex s in a simplicial complex K is a subcomplex of K consisting of the simplices t that are disjoint from s and such that both s and t are faces of some higher-dimensional simplex in K. For instance, in a two-dimensional piecewise-linear manifold formed by a set of vertices and triangles, the link of a vertex s consists of the cycle of vertices and edges surrounding s: if t is a vertex in this cycle, t and s are both endpoints of an edge of K, if t is an edge in this cycle, it and s are both faces of a triangle of K.
This cycle is homeomorphic to a circle, a 1-dimensional sphere. But in this article the word "triangulation" is just used to mean homeomorphic to a simplicial complex. For manifolds of dimension at most 4, any triangulation of a manifold is a piecewise linear triangulation: In any simplicial complex homeomorphic to a manifold, the link of any simplex can only be homeomorphic to a sphere, but in dimension n ≥ 5 the -fold suspension of the Poincaré sphere is a topological manifold with a triangulation, not piecewise-linear: it has a simplex whose link is the Poincaré sphere, a three-dimensional manifold, not homeomorphic to a sphere. This is the double suspension theorem, due to R. D. Edwards in the 1970s; the question of which manifolds have piecewise-linear triangulations has led to much research in topology. Differentiable manifolds and subanalytic sets admit a piecewise-linear triangulation, technically by passing via the PDIFF category. Topological manifolds of dimensions 2 and 3 are always triangulable by an unique triangulation.
As shown independently by James Munkres, Steve Smale and J. H. C. Whitehead, each of these manifolds admits a smooth structure, unique up to diffeomorphism. In dimension 4, the E8 manifold does not admit a triangulation, some compact 4-manifolds have an infinite number of triangulations, all piecewise-linear inequivalent. In dimension greater than 4, Rob Kirby and Larry Siebenmann constructed manifolds that do not have piecewise-linear triangulations. Further, Ciprian Manolescu proved that there exist compact manifolds of dimension 5 that are not homeomorphic to a simplicial complex, i.e. that do not admit a triangulation. An important special case of topological triangulation is that of two-dimensional surfaces, or closed 2-manifolds. There is a standard proof. Indeed, if the surface is given a Riemannian metric, each point x is contained inside a small convex geodesic triangle lying inside a normal ball with centre x; the interiors of finitely many of the triangles will cover the surface. Another simple procedure for triangulating differentiable manifolds was given by Hassler Whitney in 1957, based on his embedding theorem.
In fact, if X is a closed n-submanifold of Rm, subdivide a cubical lattice in Rm into simplices to give a triangulation of Rm. By taking the mesh of the lattice small enough and moving finitely many of the vertices, the triangulation will be in general position with respect to X: thus no simplices of dimension < s = m − n intersect X and each s-simplex intersecting X does so in one interior point. These points of intersection and their barycentres generate an n-dimensional simplicial subcomplex in Rm, lying wholly inside the tubular neighbourhood; the triangulation is given by the projection of this simplicial complex onto X. A Whitney triangulation or clean triangulation of a surface is an embedding of a graph onto the surface in such a way that the faces of the embedding are the cliques of the graph. Equivalently, every face is a triangle, every triangle is a face, the graph is not itself a clique; the clique complex of the graph is homeomorphic to the surface. The 1-skeletons of Whitney triangulations are the locally cyclic graphs other than K4.
Whitehead, J. H. C.. "On C1-Complexes". Annals of Mathematics. Second Series. 41: 809–824. Doi:10.2307/1968861. JSTOR 1968861. Whitehead, J. H. C.. "Manifolds with Transverse Fields in Euclidean Space". The Annals of Mathematics. 73: 154–212. Doi:10.2307/1970
Selman Akbulut is a Turkish mathematician and a Professor at Michigan State University. His research is in topology. In 1975 he earned his Ph. D. from the University of California, Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds, symplectic topology, G2 manifolds. In the topology of real-algebraic sets, he and Henry C. King proved that every compact piecewise-linear manifold is a real-algebraic set, he was a visiting scholar several times at the Institute for Advanced Study. He has developed 4-dimensional handlebody techniques, settling conjectures and solving problems about 4-manifolds, such as a conjecture of Christopher Zeeman, the Harer–Kas–Kirby conjecture, a problem of Martin Scharlemann, problems of Sylvain Cappell and Julius Shaneson, he constructed an exotic compact 4-manifold from which he discovered "Akbulut corks". His most recent results concern the 4-dimensional smooth Poincaré conjecture, he has supervised 14 Ph. D students as of 2019.
He has more than 100 papers and three books published, several books edited. Selman Akbulut at the Mathematics Genealogy Project Akbulut's homepage Akbulut's papers at ArXiv Akbulut-King invariants Real algebraic geometry Akbulut cork
Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze was an Austrian mathematician, famous for the Tietze extension theorem on functions from topological spaces to the real numbers. He developed the Tietze transformations for group presentations, was the first to pose the group isomorphism problem. Tietze's graph is named after him. Tietze was the son of Emil Tietze and the grandson of Franz Ritter von Hauer, both of whom were Austrian geologists, he was born in Schleinz, Austria-Hungary, studied mathematics at the Technische Hochschule in Vienna beginning in 1898. After additional studies in Munich, he returned to Vienna, completing his doctorate in 1904 and his habilitation in 1908. From 1910 until 1918 Tietze taught mathematics in Brno, was promoted to ordinary professor in 1913, he served in the Austrian army during World War I, returned to Brno, but in 1919 he took a position at the University of Erlangen, in 1925 moved again to the University of Munich, where he remained for the rest of his career. One of his doctoral students was Georg Aumann.
Tietze retired in 1950, died in Munich, West Germany. Tietze was a fellow of the Bavarian Academy of Sciences and a fellow of the Austrian Academy of Sciences. Tietze, Heinrich, "Über Schachturnier-Tabellen", Mathematische Zeitschrift, 67: 188 Tietze, Heinrich, "Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen", DMV Annual Report Tietze, Heinrich, "Über Funktionen, die auf einer abgeschlossenen Menge stetig sind", Journal für die reine und angewandte Mathematik, 145 Über die mit Lineal und Zirkel und die mit dem rechten Zeichenwinkel lösbaren Konstruktionsaufgaben, Mathematische Zeitschrift vol.46, 1940 mit Leopold Vietoris Beziehungen zwischen den verschiedenen Zweigen der Topologie, Enzyklopädie der Mathematischen Wissenschaften 1929 Über die Anzahl der stabilen Ruhelagen eines Würfels, Elemente der Mathematik vol.3, 1948 Über die topologische Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte für Mathematik und Physik, vol. 19, 1908, p.1-118 Über Simony Knoten und Simony Ketten mit vorgeschriebenen singulären Primzahlen für die Figur und für ihr Spiegelbild, Mathematische Zeitschrift vol.49, 1943, p.351 Tietze, Famous problems of mathematics.
Solved and unsolved mathematical problems from antiquity to modern times. New York: Graylock Press, MR 0181558 Heinrich Franz Friedrich Tietze at the Mathematics Genealogy Project
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d; the conjecture was first put forth in a letter by Warren M. Hirsch to George B. Dantzig in 1957 and was motivated by the analysis of the simplex method in linear programming, as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method; the conjecture is now known to be false in general. The Hirsch conjecture was proven for d < 4 and for various special cases, while the best known upper bounds on the diameter are only sub-exponential in n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, from the University of Cantabria; the result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum and appeared in Annals of Mathematics.
The paper presented a 43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as it does not rule out the possibility of a larger but still linear or polynomial number of steps. Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d; the graph of a convex polytope P is any graph whose vertices are in bijection with the vertices of P in such a way that any two vertices of the graph are joined by an edge if and only if the two corresponding vertices of P are joined by an edge of the polytope. The diameter of P, denoted δ, is the diameter of any one of its graphs; these definitions are well-defined since any two graphs of the same polytope must be isomorphic as graphs. We may state the Hirsch conjecture as follows: Conjecture Let P be a d-dimensional convex polytope with n facets.
Δ ≤ n − d. For example, a cube in three dimensions has six facets; the Hirsch conjecture indicates that the diameter of this cube cannot be greater than three. Accepting the conjecture would imply that any two vertices of the cube may be connected by a path from vertex to vertex using, at most, three steps. For all polytopes of dimension at least 8, this bound is optimal. In other words, for nearly all cases, the conjecture provides the minimum number of steps needed to join any two vertices of a polytope by a path along its edges. Since the simplex method operates by constructing a path from some vertex of the feasible region to an optimal point, the Hirsch conjecture would provide a lower bound needed for the simplex method to terminate in the worst case scenario; the Hirsch conjecture is a special case of the polynomial Hirsch conjecture, which claims that there exists some positive integer k such that, for all polytopes P, δ = O, where n is the number of facets of P. The Hirsch conjecture has been proven true for a number of cases.
For example, any polytope with dimension 3 or lower satisfies the conjecture. Any d-dimensional polytope with n facets. Other attempts to solve the conjecture manifested out of a desire to formulate a different problem whose solution would imply the Hirsch conjecture. One example of particular importance is the d-step conjecture, a relaxation of the Hirsch conjecture, shown to be equivalent to it. Theorem The following statements are equivalent: δ ≤ n − d for all d-dimensional polytopes P with n facets. Δ ≤ d for all d-dimensional polytopes P with 2d facets. In other words, in order to prove or disprove the Hirsch conjecture, one only needs to consider polytopes with twice as many facets as its dimension. Another significant relaxation is that the Hirsch conjecture holds for all polytopes if and only if it holds for all simple polytopes; the Hirsch conjecture is not true in all cases, as shown by Francisco Santos in 2011. Santos' explicit construction of a counterexample comes both from the fact that the conjecture may be relaxed to only consider simple polytopes, from equivalence between the Hirsch and d-step conjectures.
In particular, Santos produces his counterexample by examining a particular class of polytopes called spindles. Definition A d-spindle is a d-dimensional polytope P for which there exist a pair of distinct vertices such that every facet of P contains one of these two vertices; the length of the shortest path between these two vertices is c
Simply connected space
In topology, a topological space is called connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be connected: a path-connected topological space is connected if and only if its fundamental group is trivial. A topological space X is called connected if it is path-connected and any loop in X defined by f: S1 → X can be contracted to a point: there exists a continuous map F: D2 → X such that F restricted to S1 is f. Here, S1 and D2 closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is connected if and only if it is path-connected, whenever p: → X and q: → X are two paths with the same start and endpoint p can be continuously deformed into q while keeping both endpoints fixed. Explicitly, there exists a continuous homotopy F: × → X such that F = F = q.
A topological space X is connected if and only if X is path-connected and the fundamental group of X at each point is trivial, i.e. consists only of the identity element. X is connected if and only if for all points x, y ∈ X, the set of morphisms Hom Π in the fundamental groupoid of X has only one element. In complex analysis: an open subset X ⊆ C is connected if and only if both X and its complement in the Riemann sphere are connected; the set of complex numbers with imaginary part greater than zero and less than one, furnishes a nice example of an unbounded, open subset of the plane whose complement is not connected. It is simply connected, it might be worth pointing out that a relaxation of the requirement that X be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a open set has connected extended complement when each of its connected components are connected. Informally, an object in our space is connected if it consists of one piece and does not have any "holes" that pass all the way through it.
For example, neither a doughnut nor a coffee cup is connected, but a hollow rubber ball is connected. In two dimensions, a circle is not connected, but a disk and a line are. Spaces that are connected but not connected are called non-simply connected or multiply connected; the definition only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point though it has a "hole" in the hollow center; the stronger condition, that the object has no holes of any dimension, is called contractibility. The Euclidean plane R2 is connected, but R2 minus the origin is not. If n > 2 both Rn and Rn minus the origin are connected. Analogously: the n-dimensional sphere Sn is connected if and only if n > 2. Every convex subset of Rn is connected. A torus, the cylinder, the Möbius strip, the projective plane and the Klein bottle are not connected; every topological vector space is connected. For n ≥ 2, the special orthogonal group SO is not connected and the special unitary group SU is connected.
The one-point compactification of R is not connected. The long line L is connected, but its compactification, the extended long line L* is not. A surface is connected if and only if it is connected and its genus is 0. A universal cover of any space X is a connected space which maps to X via a covering map. If X and Y are homotopy equivalent and X is connected so is Y; the image of a connected set under a continuous function need not be connected. Take for example the complex plane under the exponential map: the image is C -, not connected; the notion of simple connectedness is important in complex analysis because of the following facts: The Cauchy's integral theorem states that if U is a connected open subset of the complex plane C, f: U → C is a holomorphic function f has an antiderivative F on U, the value of every line integral in U with integrand f depends only on the end points u and v of the path, can be computed as F - F. The integral thus does not depend on the particular path connecting u and v.
The Riemann mapping theorem states that any non-empty open connected subset of C is conformally eq
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
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within society at large; the press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton, its first book was a new 1912 edition of John Witherspoon's Lectures on Moral Philosophy. Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two existing local publishers, that of the Princeton Alumni Weekly and the Princeton Press; the new press printed both local newspapers, university documents, The Daily Princetonian, added book publishing to its activities. Beginning as a small, for-profit printer, Princeton University Press was reincorporated as a nonprofit in 1910.
Since 1911, the press has been headquartered in a purpose-built gothic-style building designed by Ernest Flagg. The design of press’s building, named the Scribner Building in 1965, was inspired by the Plantin-Moretus Museum, a printing museum in Antwerp, Belgium. Princeton University Press established a European office, in Woodstock, north of Oxford, in 1999, opened an additional office, in Beijing, in early 2017. Six books from Princeton University Press have won Pulitzer Prizes: Russia Leaves the War by George F. Kennan Banks and Politics in America from the Revolution to the Civil War by Bray Hammond Between War and Peace by Herbert Feis Washington: Village and Capital by Constance McLaughlin Green The Greenback Era by Irwin Unger Machiavelli in Hell by Sebastian de Grazia Books from Princeton University Press have been awarded the Bancroft Prize, the Nautilus Book Award, the National Book Award. Multi-volume historical documents projects undertaken by the Press include: The Collected Papers of Albert Einstein The Writings of Henry D. Thoreau The Papers of Woodrow Wilson The Papers of Thomas Jefferson Kierkegaard's WritingsThe Papers of Woodrow Wilson has been called "one of the great editorial achievements in all history."
Princeton University Press's Bollingen Series had its beginnings in the Bollingen Foundation, a 1943 project of Paul Mellon's Old Dominion Foundation. From 1945, the foundation had independent status and providing fellowships and grants in several areas of study, including archaeology and psychology; the Bollingen Series was given to the university in 1969. Annals of Mathematics Studies Princeton Series in Astrophysics Princeton Series in Complexity Princeton Series in Evolutionary Biology Princeton Series in International Economics Princeton Modern Greek Studies The Whites of Their Eyes: The Tea Party's Revolution and the Battle over American History, by Jill Lepore The Meaning of Relativity by Albert Einstein Atomic Energy for Military Purposes by Henry DeWolf Smyth How to Solve It by George Polya The Open Society and Its Enemies by Karl Popper The Hero With a Thousand Faces by Joseph Campbell The Wilhelm/Baynes translation of the I Ching, Bollingen Series XIX. First copyright 1950, 27th printing 1997.
Anatomy of Criticism by Northrop Frye Philosophy and the Mirror of Nature by Richard Rorty QED: The Strange Theory of Light and Matter by Richard Feynman The Great Contraction 1929–1933 by Milton Friedman and Anna Jacobson Schwartz with a new Introduction by Peter L. Bernstein Military Power: Explaining Victory and Defeat in Modern Battle by Stephen Biddle Banks, Eric. "Book of Lists: Princeton University Press at 100". Artforum International. Staff of Princeton University Press. A Century in Books: Princeton University Press, 1905–2005. ISBN 9780691122922. CS1 maint: Uses authors parameter Official website Princeton University Press: Albert Einstein Web Page Princeton University Press: Bollingen Series Princeton University Press: Annals of Mathematics Studies Princeton University Press Centenary Princeton University Press: New in Print