Search references for SPHERICAL SPACE-FORM-CONJECTURE. Phrases containing SPHERICAL SPACE-FORM-CONJECTURE
See searches and references containing SPHERICAL SPACE-FORM-CONJECTURE!SPHERICAL SPACE-FORM-CONJECTURE
In geometric topology, the spherical space form conjecture (now a theorem) states that a finite group acting on the 3-sphere is conjugate to a group of
Spherical space form conjecture
Spherical_space_form_conjecture
Three dimensional analogue of uniformization conjecture
solving the Poincaré conjecture, though Perelman declined both awards. The Poincaré conjecture and the spherical space form conjecture are corollaries of
Geometrization_conjecture
equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture. The elliptization conjecture is a special case of Thurston's
Thurston elliptization conjecture
Thurston_elliptization_conjecture
Mathematical space
the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter
3-manifold
Russian mathematician (born 1966)
sphere Hyperbolic manifold "Manifold Destiny" Spherical space form conjecture Thurston elliptization conjecture Uniformization theorem The New Yorker authors
Grigori_Perelman
Lurie, 2008) Spherical space form conjecture (Grigori Perelman, 2006) Poincaré conjecture (Grigori Perelman, 2002) Geometrization conjecture (Grigori Perelman
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Aharoni-Korman conjecture also known as the fishbone conjecture Atiyah conjecture (not a conjecture to start with) Borsuk's conjecture Bunkbed conjecture Chinese
List_of_conjectures
Subclass of manifold
In mathematics, a spherical 3-manifold M is a 3-manifold of the form M = S 3 / Γ {\displaystyle M=S^{3}/\Gamma } where Γ {\displaystyle \Gamma } is a finite
Spherical_3-manifold
German mathematician (1894–1971)
constant sectional curvature is globally isometric to Euclidean, spherical, or hyperbolic space. He also studied the indices of zeros of vector fields on hypersurfaces
Heinz_Hopf
On distance sets of high-dimensional sets
-dimensional Euclidean space whose Hausdorff dimension is strictly greater than d / 2 {\displaystyle d/2} , then the conjecture states that the set of
Falconer's_conjecture
Unsolved problem in mathematics
Ramanujan-Petersson conjecture is a conjecture concerning the growth rate of coefficients of modular forms and more generally, automorphic forms. The name of
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Shape containing unit line segments in all directions
Kakeya needle sets of measure 0. The Kakeya conjecture states that Besicovitch sets in n-dimensional space must have Hausdorff dimension n; it remains
Kakeya_set
Set of points equidistant from a center
Hemisphere Octant of a sphere Spherical cap Spherical lune Spherical polygon Spherical sector Spherical segment Spherical wedge Spherical zone 3-sphere Affine
Sphere
Orthogonal symmetric polynomial family
to prove several conjectures made by Macdonald about them. First fix some notation: R is a finite root system in a real vector space V. R+ is a choice
Macdonald_polynomials
Type of measurement in quantum mechanics
n {\displaystyle n} , and another that has been conjectured to work for all dimensions. A spherical t-design is a set of vectors S = { | ϕ k ⟩ : | ϕ
SIC-POVM
1970 MIT notes The Adams conjecture I Johannes Ebert, The Adams Conjecture, after Edgar Brown Strunk, Florian. On motivic spherical bundles Is it true that
Sphere_bundle
Multiple proofs regarding Earth's approximately spherical shape
The roughly spherical shape of Earth can be empirically evidenced by many different types of observation, ranging from ground level, flight, or orbit
Empirical evidence for the spherical shape of Earth
Empirical_evidence_for_the_spherical_shape_of_Earth
Mathematical space
Examples: In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecture. If the form is the E8 lattice, this gives a
4-manifold
Topological space that locally resembles Euclidean space
functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry. The spherical Earth is navigated using
Manifold
Branch of topology
cusped manifolds. Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that
Low-dimensional_topology
Hypothetical FTL transportation by warping space
brings up the chronology protection conjecture and writes: "The conjecture has not been proven (it wouldn't be a conjecture if it had), but there are good
Alcubierre_drive
space Wirtinger inequality (2-forms) Gromov's systolic inequality for essential manifolds Essential manifold Filling radius Filling area conjecture Bolza
List of differential geometry topics
List_of_differential_geometry_topics
Relation between sides of a right triangle
finite spherical triangles on a sphere of infinite radius), the spherical relation between the sides of a right triangle reduces to the Euclidean form of
Pythagorean_theorem
On smallest surface enclosing two volumes
the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble
Double_bubble_theorem
Dense arrangement of congruent spheres in an infinite, regular arrangement
conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was
Close-packing of equal spheres
Close-packing_of_equal_spheres
Chinese-American mathematician (born 1949)
Kontsevich. The viewpoint of the SYZ conjecture is on geometric phenomena in Calabi–Yau spaces, while Kontsevich's conjecture abstracts the problem to deal with
Shing-Tung_Yau
Method of analysis applied to problems wave propagation
of spherical wavelets and that the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms a new
Huygens–Fresnel_principle
Locally spherical point on a mathematical surface
characteristic, embedded smoothly into Euclidean space, has at least one umbilic. A famous conjecture of Constantin Carathéodory dating from 1924 states
Umbilical_point
Archaic conception of Earth's shape
historians, notably Joseph Needham, to conjecture that Chinese astronomers were, after all, aware of the Earth's sphericity. The egg reference, however, was
Flat_Earth
Family of infinite discrete groups
which are conjectured to be decidable, – determining torsion — which is conjectured to be trivial, – determining the center — which is conjectured to be trivial
Artin–Tits_group
million. Again, this applies to spherical geometry and hyperbolic geometry as well as to Euclidean geometry. Hadwiger's conjecture remains unproven for all dimensions
Dissection_into_orthoschemes
Partial differential equation
one of the following three singularity models: The shrinking round spherical space form S 3 / Γ {\displaystyle S^{3}/\Gamma } The shrinking round cylinder
Ricci_flow
Solid with six equal square faces
to each edge. Its dual is the spherical octahedron. The topological object three-dimensional torus is a topological space defined to be homeomorphic to
Cube
Branch of mathematics
related form of duality exists between a vector space and its dual space. Euclidean geometry is geometry in its classical sense. As it models the space of
Geometry
Italian-born American mathematician (1923–2023)
and the result became known as the Calabi conjecture. In 1957, Calabi published a paper in which the conjecture was stated as a proposition, but with an
Eugenio_Calabi
Differential equation for the description of waves or standing wave
problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used
Wave_equation
Arrangement of spheres within a space
container holding the spherical grains. When spheres are randomly added to a container and then compressed, they will generally form what is known as an
Sphere_packing
Conjecture in physics
weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities in the context of
Cosmic_censorship_hypothesis
Points with no three in a line
to the cap set problem can also be used to prove a partial form of the sunflower conjecture, namely that if a family of subsets of an n {\displaystyle
Cap_set
Euclidean space if the sum of the reciprocals of p, q, and r equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum
List_of_sums_of_reciprocals
American mathematician
the Double Bubble conjecture, which states that the minimum-surface-area enclosure of two given volumes is formed by three spherical patches meeting at
Frank_Morgan_(mathematician)
Keevash, Peter; Sudakov, Benny (2016). "Equiangular Lines and Spherical Codes in Euclidean Space". arXiv:1606.06620 [math.CO]. Sloane, N. J. A. (ed.). "Sequence
Equiangular_lines
Simplex formed from a right-angled path
and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the
Schläfli_orthoscheme
Branch of mathematics
algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy
Differential_geometry
Hungarian-Canadian mathematician
(2007), 626–641. A proof of the Kneser–Poulsen Conjecture (1955) for hemispheres in spherical d-space for all d > 1 (joint work with Robert Connelly,
Károly_Bezdek
Compact topological space
In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of
Dunce_hat_(topology)
Open problem in convex geometry
body can serve as the unit ball of a finite-dimensional Banach space, and the conjecture becomes the statement that if the unit ball is covered by planks
Affine_plank_problem
Field of higher mathematics
to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried
Geometric_analysis
Compact astronomical body
astrophysicist Karl Schwarzschild set out to apply the idea to stars. He assumed spherical symmetry with no spin and found a solution to Einstein's equations. A
Black_hole
Mathematical object
{\text{dim}}\ \pi ^{K}\leq 1} . In this case, the space G/H is called spherical space. It is conjectured that any spherical pair (G, K) over a local field satisfies
Gelfand_pair
Mathematics of smooth surfaces
immersed tori of constant mean curvature in Euclidean 3-space. Carathéodory conjecture: This conjecture states that a closed convex three times differentiable
Differential geometry of surfaces
Differential_geometry_of_surfaces
Polyhedron that tiles space by translation
polyhedron that tiles space so that all tiles are symmetric. The centers of the tiles in a tiling of space by parallelohedra form a Bravais lattice, and
Parallelohedron
Differential operator in mathematics
coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function
Laplace_operator
Property of a mathematical space
state of affairs was highly marked in the various cases of the Poincaré conjecture, in which four different proof methods are applied. The dimension of a
Dimension
Australian and American mathematician (born 1975)
problem related to the Twin Prime Conjecture by looking at prime number progressions—series of numbers equally spaced. (For example, 3, 7 and 11 constitute
Terence_Tao
Maximally symmetric Lorentzian manifold with a negative cosmological constant
holes. Mathematician Georgios Moschidis proved that given spherical symmetry, the conjecture holds true for the specific cases of the Einstein-null dust
Anti-de_Sitter_space
Type of vector space
algebra of a locally compact group and spherical Hecke algebra that arise when modular forms and other automorphic forms are viewed using adelic groups. These
Hecke_algebra
Metric for black holes in general relativity
simplest solution that has both. The metric of any spherically symmetric solution in Schwarzschild form is: d s 2 = − f ( r ) d t 2 + d r 2 f ( r ) + r 2
De Sitter–Schwarzschild metric
De_Sitter–Schwarzschild_metric
Integral lattice of determinant 1 or –1
in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is. The signature
Unimodular_lattice
Type of polygon
quadrangular algebras is to analyze two open questions. One is the Kneser-Tits conjecture that concerns the full group of linear transformations of a building (e
Moufang_polygon
Black holes are characterized only by mass, charge, and spin
mathematicians refer to it as the no-hair conjecture. Even in the case of gravity alone (i.e., zero electric fields), the conjecture has only been partially resolved
No-hair_theorem
On dissections between polyhedra
which all of three-dimensional space can be tiled periodically is zero. Unsolved problem in mathematics In spherical or hyperbolic geometry, must polyhedra
Hilbert's_third_problem
connected solvable Lie group by a lattice. Spherical geometry Submetry A short map f between metric spaces is called a submetry if there exists R > 0
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Representation theory
a hyperbolic space. The general case was reduced to two conjectures about the properties of the c-function and the so-called spherical Fourier transform
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Study of complex manifolds and several complex variables
The Hodge conjecture, one of the millennium prize problems, is a problem in complex geometry. Broadly, complex geometry is concerned with spaces and geometric
Complex_geometry
Differential geometry measure
independent vectors in parameter space then the mean curvature can be written in terms of the first and second quadratic form matrices as l G − 2 m F + n E
Mean_curvature
Polyhedron with 12 faces
only as a spherical polyhedron and is degenerate in Euclidean space. In crystallography, two important dodecahedra can occur as crystal forms in some symmetry
Dodecahedron
Topological space
Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture. A Seifert manifold
Seifert_fiber_space
Mathematical object
indices i. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres
3-sphere
field of graph theory, the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph
Sphericity_(graph_theory)
Cylindrical conformal map projection
to infinity in the vertical direction. A simple expression for the spherical form of the Mercator projection is: x = R ( λ − λ 0 ) , y = R ln ( tan (
Mercator_projection
Archimedean solid with 14 faces
truncated octahedron is a space-filling polyhedron; that is, it can tile space by translating its copies face-to-face in order to form a honeycomb. It is classified
Truncated_octahedron
Function describing an electron in an atom
fill in a volume of space around the nucleus so that the resulting collection ("electron cloud") tends toward a generally spherical zone of probability
Atomic_orbital
Bloch's theorem. Based on his Principle, Bloch was able to predict or conjecture several important results such as the Ahlfors's Five Islands theorem,
Bloch's_principle
Branch of algebraic geometry
Langlands conjectures for GLn was based on the geometry of certain Shimura varieties. In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology
Arithmetic_geometry
Mechanism in physical cosmology
In theoretical physics, the Fischler–Susskind holographic bound is a conjectured bound on the maximum amount of entropy that a region of spacetime could
Fischler–Susskind holographic bound
Fischler–Susskind_holographic_bound
French mathematician, physicist and engineer (1854–1912)
far-reaching consequences. Early in the 20th century he formulated the Poincaré conjecture, which became, over time, one of the famous unsolved problems in mathematics
Henri_Poincaré
Branch of differential geometry
surface could be measured, was not how the surface sat in space, but how this quadratic form varied from point to point. Consider the simple case of a
Riemannian_geometry
Isogonal polyhedron with regular faces
are given. The spherical tilings include the set of hosohedra and dihedra which are degenerate polyhedra. These symmetry groups are formed from the reflectional
Uniform_polyhedron
Ukrainian mathematician (born 1984)
8- and 24-dimensional spaces. As well as for her work on sphere packing, Viazovska is also known for her research on spherical designs with Bondarenko
Maryna_Viazovska
Geometric inequality applicable to any closed curve
in Hadamard manifolds, which has become known as the Cartan–Hadamard conjecture. In dimension 2 this had already been established in 1926 by André Weil
Isoperimetric_inequality
Study of random spatial patterns
pairs are viewed as points in a larger product space formed as the product of the original space and the space of parametrization. Suppose we are concerned
Stochastic_geometry
Hypothetical topological feature of spacetime
was put forth by Juan Maldacena and Leonard Susskind in their ER = EPR conjecture. The quantum foam hypothesis is sometimes used to suggest that tiny wormholes
Wormhole
Matrix of partial derivatives of a vector-valued function
(see inverse function theorem for an explanation of this and Jacobian conjecture for a related problem of global invertibility). The Jacobian determinant
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Solid with eight equal triangular faces
through space. The spherical octahedron represents a regular octahedron projected to a sphere, a part of spherical polyhedron. There are eight spherical triangles
Regular_octahedron
Gives sufficient condition for Dehn filling to result in a negatively curved 3-manifold
431, MR 1799796. Bleiler, Steven A.; Hodgson, Craig D. (1996), "Spherical space forms and Dehn filling", Topology, 35 (3): 809–833, doi:10.1016/0040-9383(95)00040-2
2π_theorem
Branch of mathematics
geometries: positive curvature/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions
Topology
Measure of curvature in differential geometry
shows that the connected sum of an arbitrary number of copies of spherical space forms and generalized cylinders Sm × Sn has a Riemannian metric of positive
Scalar_curvature
Type of topological space
topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications
Topological_manifold
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
Ramanujan-Petersson conjecture. Maass cusp forms can be regarded as automorphic forms on GL(2). It is natural to define Maass cusp forms on GL(n) as spherical automorphic
Maass_wave_form
Classical statement of gravity as force
the square of the distance between their centers of mass. Separated, spherically symmetrical objects attract and are attracted as if all their mass were
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
Surface that locally minimizes its area
the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. the Smith conjecture, the Poincaré conjecture, the Thurston Geometrization
Minimal_surface
Theory of gravitation as curved spacetime
anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture). Given the difficulty of
General_relativity
Hypothetical phenomenon
the cosmic censorship conjecture by mathematically proving a generic condition under which a locally naked singularity could form. Naked singularities
Naked_singularity
Tensor in differential geometry
proof of the Poincaré conjecture. More recently, lower bounds for Ricci curvature have been extended to certain metric-measure spaces, connecting the subject
Ricci_curvature
Ordered mesophases in soft materials mimicking complex alloy structures
often described as a response to packing frustration. When spherical domains cannot fill space efficiently as a body-centered cubic (BCC) or face-centered
Frank-Kasper phases (soft matter)
Frank-Kasper_phases_(soft_matter)
Type of monoidal category
\mathbb {C} } of vector spaces, where 1 {\displaystyle {\bf {1}}} is the tensor unit of C {\displaystyle {\mathcal {C}}} . (Spherical axiom) Given an object
Modular_tensor_category
Type of non-Euclidean geometry
geometries. The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining
Hyperbolic_geometry
Equations of motion for viscous fluids
purely mathematical sense. Despite their wide range of practical uses, the conjecture that they have smooth (meaning infinitely differentiable) or bounded solutions
Navier–Stokes_equations
General relativity model near spacetime singularities
containing space derivatives equal to zero, one can define the so-called truncated theory of the system (truncated equations). Then, the BKL conjecture can be
BKL_singularity
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE
SPHERICAL SPACE-FORM-CONJECTURE