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Riemannian metrics, complex manifolds
the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954
Calabi_conjecture
Riemannian manifold with SU(n) holonomy
Ricci-flat Kähler metrics, and Shing-Tung Yau (1978), who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of
Calabi–Yau_manifold
Chinese-American mathematician (born 1949)
recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is
Shing-Tung_Yau
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
Eugenio_Calabi
Study of complex manifolds and several complex variables
proven with great success, including Shing-Tung Yau's proof of the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence
Complex_geometry
Mathematical conjecture
metrics involved. SYZ conjecture: Every 6-dimensional Calabi–Yau manifold X {\displaystyle X} has a mirror 6-dimensional Calabi–Yau manifold X ^ {\displaystyle
SYZ_conjecture
Partial differential equation with nonlinear terms
used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Algebro-geometric stability condition
conjecture about the existence of Kähler metrics on compact Kähler manifolds, now known as the Calabi conjecture. One formulation of the conjecture is
K-stability
French mathematician (1942–2009)
always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class
Thierry_Aubin
Chinese mathematician (born 1958)
Kähler-Einstein metrics. Shing-Tung Yau, in his renowned resolution of the Calabi conjecture, had settled the case of closed Kähler manifolds with nonpositive
Tian_Gang
Type of metric in Riemannian geometry
always a Kähler–Einstein metric, as Yau proved in the Calabi conjecture. That leads to the name Calabi–Yau manifolds. He was awarded with the Fields Medal
Kähler–Einstein_metric
Type of geometry in mathematics
Choquet-Bruhat. In Riemannian geometry, Shing-Tung Yau's resolution of the Calabi conjecture produced a number of Ricci-flat metrics on Kähler manifolds. A pseudo-Riemannian
Ricci-flat_manifold
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
Mathematics concept
Zürich, Kontsevich (1994) speculated that mirror symmetry for a pair of Calabi–Yau manifolds X and Y could be explained as an equivalence of a triangulated
Homological_mirror_symmetry
Chinese mathematician
in 1985, he showed that Yau's estimates in the resolution of the Calabi conjecture could be modified to the Kähler-Ricci flow context, to prove a convergence
Huai-Dong_Cao
Mathematics award
was found in 1993. In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal, stating "I'm not interested in money or fame;
Fields_Medal
On heights of points on algebraic varieties over number fields
bundle, for example, an abelian variety, a K3 surface or a Calabi-Yau variety. Vojta's conjecture predicts that if D {\displaystyle D} is an effective ample
Vojta's_conjecture
spaces. As a consequence of the work of Aubin and Yau on solution of Calabi Conjecture in the case of negative Ricci curvature, see Yau (1977, 1978), any
Fake_projective_plane
Type of Riemannian manifold
to I {\displaystyle I} . Conversely, Shing-Tung Yau's proof of the Calabi conjecture implies that a compact, Kähler, holomorphically symplectic manifold
Hyperkähler_manifold
Conjecture in symplectic geometry
category of the Calabi–Yau manifold, which is a triangulated category appearing in Kontsevich's homological mirror symmetry conjecture. The statement of
Thomas–Yau_conjecture
Topics referred to by the same term
relation between two Calabi–Yau manifolds in string theory Homological mirror symmetry, a mathematical conjecture about Calabi–Yau manifolds made by
Mirror symmetry (disambiguation)
Mirror_symmetry_(disambiguation)
Mathematical conjecture about elliptic curves
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational
Sato–Tate_conjecture
American mathematician (born 1989)
(MNOP) conjecture, which posited an equivalence between two different curve enumeration invariants of Calabi–Yau threefolds. He is currently
John_Pardon
English mathematician (born 1957)
since its proposal in the 1980s by Shing-Tung Yau after he proved the Calabi conjecture. It was later generalized by Gang Tian and Donaldson. The solution
Simon_Donaldson
Mathematical conjecture
certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold
Mirror_symmetry_conjecture
Type of smooth complex surface of kodaira dimension 0
over a field is projective.) By Shing-Tung Yau's solution to the Calabi conjecture, it follows that every complex analytic K3 surface has a Ricci-flat
K3_surface
Concept in algebraic geometry
c_{1}:Pic(X)\to H^{2}(X,\mathbb {Z} )} . By Yau's solution of the Calabi conjecture, a smooth complex variety admits Kähler metrics of positive Ricci
Fano_variety
American annual mathematics conference
Generalized Kahler Ricci flow and a generalized Calabi conjecture Jean-Pierre Bourguignon (IHES) Eugenio Calabi (Penn) Yakov Eliashberg (Stanford) Carolyn
Geometry_Festival
Theory of subatomic structure
physics, the compact extra dimensions must be shaped like a Calabi–Yau manifold. A Calabi–Yau manifold is a special space which is typically taken to
String_theory
Mathematical inequality
differential geometric approach which is based on his resolution of the Calabi conjecture. Since c 2 ( X ) = e ( X ) {\displaystyle c_{2}(X)=e(X)} is the topological
Bogomolov–Miyaoka–Yau inequality
Bogomolov–Miyaoka–Yau_inequality
Manifold with Riemannian, complex and symplectic structure
are automatically projective varieties. Shing-Tung Yau proved the Calabi conjecture: every smooth projective variety with ample canonical bundle has a
Kähler_manifold
Mathematics concept
MR 0231827 Lecture slides (PDF) by Phillip Griffiths Mumford-Tate groups, families of Calabi-Yau varieties and analogue André-Oort problems I, preprint (PDF)
Mumford–Tate_group
Constructing a strictly convex compact surface with specified Gaussian curvature
equations, particularly for solving such difficult problems as the Calabi conjecture of 1954, and a problem of Hermann Minkowski in Euclidean spaces concerning
Minkowski_problem
Chinese-American mathematician
by eventual Fields Medalist Shing-Tung Yau after his proof of the Calabi conjecture. More precise versions were subsequently proposed by Chinese mathematician
Xiuxiong_Chen
Vector bundles theorem
and the Thomas–Yau conjecture about existence of special Lagrangians inside isotopy classes of Lagrangian submanifolds of a Calabi–Yau manifold. In 1965
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
3d hypersurface of degree 5
{\displaystyle \mathbb {P} ^{4}} . Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-singular quintic 3-fold is Physicist
Quintic_threefold
Equivalence of two physical theories
of the string tension. The SYZ conjecture generalizes this idea to the more complicated case of six-dimensional Calabi–Yau manifolds like the one illustrated
T-duality
label), bladder cancer. Eugenio Calabi, 100, Italian-born American mathematician (Calabi conjecture, Calabi–Yau manifold, Calabi flow). Bob Dahl, 54, American
Deaths_in_September_2023
obtained from the article CR manifold. Calabi conjecture Monge–Ampère equations Positive mass conjecture Yamabe conjecture Graham, C. Robin; Lee, John M. (1988)
Paneitz_operator
certain functional on the space of Kähler metrics, the Calabi functional, introduced by Calabi. The Calabi functional is a functional defined on the space of
Constant scalar curvature Kähler metric
Constant_scalar_curvature_Kähler_metric
Theory in physics
Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its
Donaldson–Thomas_theory
Extended physical object in string theory
homological mirror symmetry conjecture of Maxim Kontsevich states that the derived category of coherent sheaves on one Calabi–Yau manifold is equivalent
Brane
Chinese mathematician (born 1987)
version of it was conjectured in the 1980s by Fields Medalist Shing-Tung Yau, who had previously proved the Calabi conjecture. The conjecture was later given
Song_Sun
Framework of superstring theory
unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University
M-theory
facts label) (b. 1947) Eugenio Calabi, 100, Italian-born mathematician (Calabi conjecture, Calabi–Yau manifold, Calabi flow) (b. 1923) Gerry Shamray,
2023 deaths in the United States (July–September)
2023_deaths_in_the_United_States_(July–September)
Generalization of a manifold
known Calabi–Yau compactifications in string theory; this partially supports a conjecture by Reid (1987) whereby conifolds connect all possible Calabi–Yau
Conifold
Category in mathematics
have been proved or conjectured. For example, the homological mirror symmetry conjecture predicts that the derived category of a Calabi–Yau manifold is equivalent
Triangulated_category
Algebraic invariant of topological spaces
manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac
Elliptic_cohomology
Russian-French mathematician
volume growth. Invent. Math. 97 (1989), no. 2, 313–349. Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Chinese-American mathematician
geometry, topology and analysis of moduli spaces of Riemann surfaces and Calabi–Yau manifolds. He is a professor of mathematics at University of California
Kefeng_Liu
Symplectic topology tool
Homological Mirror Symmetry conjecture states there is a type of derived Morita equivalence between the Fukaya category of the Calabi–Yau X {\displaystyle X}
Floer_homology
\displaystyle \det(\partial _{i{\bar {j}}}\varphi )=} lower order terms Calabi conjecture Constant astigmatism 1+1 z y y + ( 1 z ) x x + 2 = 0 {\displaystyle
List of nonlinear partial differential equations
List_of_nonlinear_partial_differential_equations
British mathematician
theory in theoretical physics. Thomas obtained his PhD on gauge theory on Calabi–Yau manifolds in 1997 under the supervision of Simon Donaldson at the University
Richard Thomas (mathematician)
Richard_Thomas_(mathematician)
American mathematician
calibrated geometry, which is instrumental in the formulation of the SYZ conjecture. He is Edgar Odell Lovett Professor and professor emeritus of mathematics
F._Reese_Harvey
Italian mathematician (born c.1981)
Weinkove, Ben; Yau, Shing-Tung (2008). "Taming symplectic forms and the Calabi-Yau equation". Proceedings of the London Mathematical Society. 97 (2): 401–424
Valentino_Tosatti
Process in particle physics
total energy of the D-branes, and all other tests have confirmed Sen's conjecture as well. Tachyons therefore became an active area of interest in the early
Tachyon_condensation
Negatively-curved minimal surface
disks do not exist. Nadirashvili, Nikolai (1996), "Hadamard's and Calabi–Yau's conjectures on negatively curved and minimal surfaces", Inventiones Mathematicae
Nadirashvili_surface
Objects of certain abelian categories associated to topological spaces
target space (i.e. four-dimensional Minkowski space with a six-dimensional Calabi-Yau (CY) manifold). The determination of the matter and interaction content
Perverse_sheaf
Concept in algebraic geometry
minimal model and abundance conjectures would imply that the general fiber of the Iitaka fibration can be arranged to be a Calabi–Yau variety, which in particular
Kodaira_dimension
American mathematician (born 1965)
differential-geometric, proposal of Strominger, Yau, and Zaslow, in which the Calabi–Yau manifold is fibred by special Lagrangian tori, and the mirror by dual
Mark_Gross_(mathematician)
Branch of algebraic geometry concerned with counting solutions
Philip; de la Ossa, Xenia; Green, Paul; Parks, Linda (1991). "A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory". Nuclear
Enumerative_geometry
American mathematician
Michael; Shepherd-Barron, Nicholas; Taylor, Richard (2010), "A family of Calabi–Yau varieties and potential automorphy", Annals of Mathematics, 171 (2):
Michael Harris (mathematician)
Michael_Harris_(mathematician)
as in the solution of the Poincaré conjecture, and Richard S. Hamilton's proof of the uniformization theorem Calabi flow, a flow for Kähler metrics Yamabe
Geometric_flow
Pseudometric of complex manifolds
In the opposite direction, Kobayashi conjectured that the Kobayashi pseudometric is identically zero for Calabi–Yau manifolds. This is true in the case
Kobayashi_metric
British mathematician
Michael; Shepherd-Barron, Nicholas; Taylor, Richard (2010), "A family of Calabi–Yau varieties and potential automorphy", Annals of Mathematics, 171 (2):
Nicholas_Shepherd-Barron
Duality between theories of gravity on anti-de Sitter space and conformal field theories
field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories. On one side are
AdS/CFT_correspondence
Complex manifold
Pierre de Fermat, is a Calabi–Yau manifold. The Hodge diamond of a non-singular quintic 3-fold is Herbert Clemens (1984) conjectured that the number of rational
Fermat_quintic_threefold
partial differential equations Universal differential equation Calabi flow in the study of Calabi-Yau manifolds Cauchy–Riemann equations Equations for a minimal
List of named differential equations
List_of_named_differential_equations
Russian American mathematician
United States in 1990. He provided the first proof of the mirror conjecture for Calabi–Yau manifolds that are complete intersections in toric ambient spaces
Alexander_Givental
French mathematician (born 1962)
proved that the generalization of the Hodge conjecture for compact Kähler varieties is false. The Hodge conjecture is one of the seven Clay Mathematics Institute
Claire_Voisin
Algebraic variety that is a moduli space for principally polarized abelian varieties
to the Barth–Nieto quintic which is birationally equivalent to a modular Calabi–Yau manifold with Kodaira dimension zero. Siegel modular varieties cannot
Siegel_modular_variety
Russian mathematician
Deligne conjecture about operations on the cohomological Hochschild complex, a direct construction of Calabi-Yau varieties based on SYZ conjecture and non-archimedean
Yan_Soibelman
American physicist
known for early work on celestial holography. He co-formulated the SYZ conjecture and the CGHS model. Strominger won the Breakthrough Prize in Fundamental
Andrew_Strominger
Danish mathematician
arXiv:math/0210119. Colding, Tobias H.; Minicozzi, William P. (2004). "The Calabi-Yau conjectures for embedded surfaces". arXiv:math/0404197. In the final paper cited
Tobias_Colding
Algebraic hypersurface
"rigid" Calabi–Yau threefolds, for which the family of Galois representations has dimension two, by the proof of Serre's modularity conjecture. The Consani–Scholton
Consani–Scholten_quintic
polyhedron models Jean-Louis Koszul (1921–2018) Isaak Yaglom (1921–1988) Eugenio Calabi (1923–2023) Benoit Mandelbrot (1924–2010) – fractal geometry Katsumi Nomizu
List_of_geometers
Russian and American mathematician
his early papers Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkähler. He proved a decomposition theorem, used for the classification
Fedor_Bogomolov
Hermitian manifold Newlander–Nirenberg theorem Generalized complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex manifold Quaternion-Kähler
List of differential geometry topics
List_of_differential_geometry_topics
Principle in theoretical physics
inspired by the Bekenstein bound of black hole thermodynamics, which conjectures that the maximum entropy in any region scales with the radius squared
Holographic_principle
He received the Fields Medal in 1982 for his contribution to the Calabi conjecture in algebraic geometry. He also received a MacArthur Fellowship in
Meanings of minor-planet names: 64001–65000
Meanings_of_minor-planet_names:_64001–65000
Theory in theoretical physics
A-model on a Calabi–Yau. In this case, the M2-branes wrap associative 3-cycles. Strictly speaking, the topological M-theory conjecture has only been
Topological_string_theory
Algebraic curve in mathematics
p. 160 Harris, M.; Shepherd-Barron, N.; Taylor, R. (2010). "A family of Calabi–Yau varieties and potential automorphy". Annals of Mathematics. 171 (2):
Elliptic_curve
Russian mathematician
MR 1265307. Batyrev, Victor V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties". Journal of Algebraic Geometry: 493–535
Victor_Batyrev
Strong-weak duality in supersymmetric theories of theoretical physics
carried magnetic rather than electric charges. In subsequent work this conjecture was refined by Ed Witten and David Olive, they showed that in a supersymmetric
Montonen–Olive_duality
Generalized manifold
compactified space must be a 6-dimensional Calabi–Yau manifold. There are a large number of possible Calabi–Yau manifolds (tens of thousands), hence the
Orbifold
British mathematician
mathematical physics, and representation theory. In his article on generalized Calabi–Yau manifolds, he introduced the notion of generalized complex manifolds
Nigel_Hitchin
theory Newton polygon Weil conjectures Kähler manifold Calabi–Yau manifold Stein manifold Hodge theory Hodge cycle Hodge conjecture Algebraic geometry and
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Award of the American Mathematical Society
simply connected. Ann. of Math. (2) 160 (2004), no. 2, 573–615. The Calabi-Yau conjectures for embedded surfaces. Ann. of Math. (2) 167 (2008), no. 1, 211–243
Oswald Veblen Prize in Geometry
Oswald_Veblen_Prize_in_Geometry
Branch of mathematics
examples of spaces studied in complex geometry include Riemann surfaces, and Calabi–Yau manifolds, and these spaces find uses in string theory. In particular
Geometry
Theory of strings with supersymmetry
are compactified, then the extra six dimensions must be in the form of a Calabi–Yau manifold. Within the more complete framework of M-theory, they would
Superstring_theory
Measure of curvature in differential geometry
has zero scalar curvature; the best-known spaces in this class are the Calabi–Yau manifolds. In the pseudo-Riemannian context, this also includes the
Scalar_curvature
Branch of mathematics
related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory, as different
Topology
Awarded every year by the American Mathematical Society
1016/0022-1236(73)90091-8. Branges, Louis (1985). "A proof of the Bieberbach conjecture" (PDF). Acta Mathematica. 154 (1–2): 137–152. doi:10.1007/BF02392821.
Leroy_P._Steele_Prize
Mathematics timeline
2018. Morgan, John W.; Tian, Gang (2007). Ricci Flow and the Poincaré Conjecture. American Mathematical Society. p. ix. ISBN 9780821843284. Manolescu,
Timeline_of_manifolds
to Gromov–Witten invariants. Marino and BKMP conjectured that Gromov–Witten invariants of a toric Calabi–Yau 3-fold X {\displaystyle {\mathfrak {X}}}
Topological_recursion
Compact astronomical body
no-hair conjecture proposes that dynamic gravitational collapse always results in an object characterized with only these three properties. The conjecture is
Black_hole
considers surfaces traced out by strings as they travel along paths in a Calabi–Yau 3-fold. Following the path integral formulation of quantum mechanics
Pseudoholomorphic_curve
Korean educator (born 1978)
Geraghty, David; Harris, Michael; Taylor, Richard (2011). "A family of Calabi–Yau varieties and potential automorphy. II". Publ. Res. Inst. Math. Sci
Sug_Woo_Shin
American mathematician
(link) Colding, Tobias H.; Minicozzi, William P., II (2008). "The Calabi-Yau Conjectures for Embedded Surfaces". Ann. of Math. 167 (1): 211–243. arXiv:math/0404197
William_Minicozzi
Theory of gravitation as curved spacetime
recently come to prominence in the context of what is called the Maldacena conjecture). Given the difficulty of finding exact solutions, Einstein's field equations
General_relativity
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