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McShane's Identity
In geometric topology, McShane's identity for a once punctured torus \mathbb with a complete, finite-volume hyperbolic structure is given by :\sum_\gamma \frac=\frac where * the sum is over all simple closed geodesics γ on the torus; and * ''ℓ''(''γ'') denotes the hyperbolic length of ''γ''. This identity was generalized by Maryam Mirzakhani Maryam Mirzakhani ( fa, مریم میرزاخانی, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller theory, hyperbolic geometry, ... on her PhD thesis References {{Reflist *''Necessary and Sufficient Conditions for McShane's Identity and Variations'' Ser Peow Tan, Yan Loi Wong, and Ying Zhang eprint arXiv:math/041118*McShane, G. Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632. Geometric topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Once Punctured
In topology, puncturing a manifold is removing a finite set of points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ... from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With the removal of a second point, it becomes twice-punctured, and so on. Examples of punctured manifolds include the disk (mathematics), open disk (which is a sphere with a single puncture), the cylinder (geometry), cylinder (which is a sphere with two punctures), and the Möbius strip (which is a projective plane with a single puncture). References Topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
