In the mathematical field of geometric topology, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research- and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults and inspiring their further study of the subject.
In vector calculus and differential geometry the generalized Stokes theorem, also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or and the divergence theorem is the case of a volume in Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus.
Topology is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Mornington Crescent is an improvisational comedy game featured in the BBC Radio 4 comedy panel show I'm Sorry I Haven't a Clue (ISIHAC), a series that satirises panel games.
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
Ian Nicholas Stewart is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of Warwick, England.
Sir Simon Kirwan Donaldson is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York, and a Professor in Pure Mathematics at Imperial College London.
Ronald Vincent Jaworski, nicknamed "Jaws", is an American former football quarterback who played 17 seasons in the National Football League (NFL). He played college football for the Youngstown State Penguins and was selected by the Los Angeles Rams in the second round of the 1973 NFL draft. After spending four seasons mostly as a backup for the Rams, Jaworski was traded to the Philadelphia Eagles in 1977, where he would lead the Eagles to four consecutive playoff appearances, including a division title and the franchise's first Super Bowl appearance in 1980. After ten seasons with the Eagles, Jaworski signed with the Miami Dolphins, where he would spend two seasons as a backup.
Dominic David Joyce FRS is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College since 1995. His undergraduate and doctoral studies were at Merton College, Oxford. He undertook a DPhil in geometry under the supervision of Simon Donaldson, completed in 1992. After this he held short-term research posts at Christ Church, Oxford, as well as Princeton University and the University of California, Berkeley in the United States.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… and David Gabai, Robert Meyerhoff, and Peter Milley showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by Jeffrey Weeks as well as Sergei V. Matveev and Anatoly T. Fomenko.
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover that is a Haken manifold.
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold. Important consequences of this theorem include the existence of an Exotic R4 and the failure of the smooth h-cobordism theorem in 4 dimensions. The results of Donaldson theory depend therefore on the manifold having a differential structure, and are largely false for topological 4-manifolds.
Ian Agol is an American mathematician who deals primarily with the topology of three-dimensional manifolds.
Finchley Central is a humorous game in which two players take turns naming stations in the London Underground. The first person to name Finchley Central is the winner, with humor coming from the fact that there is nothing stopping either player from naming the station at any time. Mathematics professor Jonathan Partington compares Finchley Central to the concept of polite refusal, describing the reciprocity and the game's solutions to be isomorphic as he somewhat facetiously notes:
An opening move of "Finchley Central" is too much of a cheat, and you might wish to start with, say, Liverpool Street, when, assuming that your opponent isn't rude enough to reply with Finchley Central, leaves you with a mate on your second move.
