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Riemannian manifold with SU(n) holonomy
the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry
Calabi–Yau_manifold
Mathematical conjecture
In differential geometry, Yau's conjecture is a mathematical conjecture which states that any closed Riemannian 3-manifold has infinitely many smooth
Yau's_conjecture
Chinese-American mathematician (born 1949)
partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors
Shing-Tung_Yau
Conjecture in symplectic geometry
In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear
Thomas–Yau_conjecture
In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true
Yau's conjecture on the first eigenvalue
Yau's_conjecture_on_the_first_eigenvalue
Mathematical conjecture
The Strominger–Yau–Zaslow (SYZ) conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics
SYZ_conjecture
the Schoen–Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing-Tung Yau. It was inspired
Schoen–Yau_conjecture
Riemannian metrics, complex manifolds
the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on
Calabi_conjecture
2024) Maximal rank conjecture (Eric Larson, 2018) Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018) Yau's conjecture (Antoine Song,
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Vector bundles theorem
correspondence inspired conjectures leading to the nonabelian Hodge correspondence for Higgs bundles, as well as the Yau–Tian–Donaldson conjecture about the existence
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Theorem in geometric topology
In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about
Poincaré_conjecture
French mathematician
geometric analysis. He is a professor at Caltech. In 2018, he proved Yau's conjecture. Antoine Song was a student at the École Normale Supérieure de Paris
Antoine_Song
Brazilian mathematician
Freedman–He–Wang conjecture (Freedman–He–Wang, 1994) In December 2017, in cooperation with Kei Irie and André Neves, he solved Yau's conjecture (Yau, 1982) in
Fernando_Codá_Marques
Algebro-geometric stability condition
Calabi–Yau manifold. The Calabi conjecture was resolved in the case where c 1 ( X ) < 0 {\displaystyle c_{1}(X)<0} by Thierry Aubin and Shing-Tung Yau, and
K-stability
British mathematician
work with Paul Seidel. With Shing-Tung Yau he formulated a conjecture (now known as the Thomas–Yau conjecture) concerning the existence of a special Lagrangian
Richard Thomas (mathematician)
Richard_Thomas_(mathematician)
Russian mathematician (born 1989)
in harmonic analysis and differential geometry that proved conjectures by Shing-Tung Yau and Nikolai Nadirashvili. In 2018 he received the Salem Prize
Aleksandr Logunov (mathematician)
Aleksandr_Logunov_(mathematician)
Russian mathematician (born 1966)
analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous open problem
Grigori_Perelman
diffeomorphism preserving the symplectic forms. Thomas–Yau conjecture see Thomas–Yau conjecture virtual fundamental class A generalization of the fundamental
Glossary of symplectic geometry
Glossary_of_symplectic_geometry
Black hole conjecture
The hoop conjecture, proposed by Kip Thorne in 1972, states that an imploding object forms a black hole when, and only when, a circular hoop with a specific
Hoop_conjecture
statement is a consequence of Yau's differential geometric approach which is based on his resolution of the Calabi conjecture. Since c 2 ( X ) = e ( X )
Bogomolov–Miyaoka–Yau inequality
Bogomolov–Miyaoka–Yau_inequality
Seven mathematical problems with a US$1 million prize for each solution
unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem
Millennium_Prize_Problems
Mathematical conjectures attributed to Heinz Hopf
conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf. The Hopf conjecture is
Hopf_conjecture
analysis Minimal surface Freedman–He–Wang conjecture Willmore conjecture Yau's conjecture Global analysis Tobias Colding and Camillo De Lellis: "The min-max
Almgren–Pitts_min-max_theory
On heights of points on algebraic varieties over number fields
Vojta's conjecture is a conjecture introduced by Paul Vojta about heights of points on algebraic varieties over number fields. The conjecture was motivated
Vojta's_conjecture
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
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
Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yum-Tong Siu and Shing-Tung Yau. In
Frankel_conjecture
Ugandan Social Media influencer / blogger born 1995 in mbarara town
Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates
Chern's conjecture for hypersurfaces in spheres
Chern's_conjecture_for_hypersurfaces_in_spheres
Award of the American Mathematical Society
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
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
In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur (1995) about when p-adic representations of Galois
Fontaine–Mazur_conjecture
Negatively-curved minimal surface
do not exist. Nadirashvili, Nikolai (1996), "Hadamard's and Calabi–Yau's conjectures on negatively curved and minimal surfaces", Inventiones Mathematicae
Nadirashvili_surface
Existence of geodesic circles on surfaces
Yau posed the conjecture that every Riemannian 3-sphere contains at least 4 embedded minimal spheres. In 2023, Zhichao Wang and Xin Zhou proved Yau's
Theorem of the three geodesics
Theorem_of_the_three_geodesics
American mathematician (1943–2024)
of results and ideas for using it to prove the Poincaré conjecture and geometrization conjecture from the field of geometric topology. Hamilton's work on
Richard_S._Hamilton
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
symmetry program of Maxim Kontsevich, and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow and its algebraic analog — the Gross-Siebert
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
Periodic minimal surface
unit disc with the hyperbolic metric), thereby disproving the Schoen–Yau conjecture. Scherk's second surface looks globally like two orthogonal planes whose
Scherk_surface
Theorem in topology
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial
Smith_conjecture
Theorem in differential geometry
Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965
Willmore_conjecture
analogy of the conjecture with the four-vertex theorem for plane curves. Modern references to the conjecture are the problem list of Shing-Tung Yau, the books
Carathéodory_conjecture
Theorem in differential geometry
follows that all metrics involved must be flat. In 1982, Shing-Tung Yau conjectured that a particular Lorentzian version of Cheeger and Gromoll's theorem
Splitting_theorem
American mathematician
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
Complete manifolds of non-negative sectional curvature largely reduce to the compact case
generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years
Soul_theorem
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
Chinese-American mathematician
collaboration with Bong Lian and Shing-Tung Yau in which they establish some enumerative geometry conjectures motivated by mirror symmetry. Liu was born
Kefeng_Liu
2006 article in The New Yorker magazine
Poincaré Conjecture, Richard S. Hamilton's formulation of a strategy to prove the conjecture, and William Thurston's geometrization conjecture. Yau's long
Manifold_Destiny
Chinese mathematician (born 1958)
class, also known as "Fano manifolds." Tian and Yau extended Yau's analysis of the Calabi conjecture to non-compact settings, where they obtained partial
Tian_Gang
In 1968 John Milnor conjectured that the fundamental group of a complete manifold is finitely generated if its Ricci curvature stays nonnegative. In an
Milnor conjecture (Ricci curvature)
Milnor_conjecture_(Ricci_curvature)
Y=X} . Hermitian Yang–Mills connection Yang–Mills connection Thomas–Yau conjecture Marino, M., Minasian, R., Moore, G. and Strominger, A., Nonlinear instantons
Deformed Hermitian Yang–Mills equation
Deformed_Hermitian_Yang–Mills_equation
Partial differential equation
the proof of the Poincaré conjecture and geometrization conjecture. The terms on the right hand side of Perelman's Li–Yau inequality motivates the definition
Ricci_flow
Portuguese mathematician (born 1975)
with Kei Irie and Fernando Codá Marques, he solved Yau's conjecture (formulated by Shing-Tung Yau in 1982) in the generic case. He was awarded the Philip
André_Neves
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
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
Study of complex manifolds and several complex variables
results to be proven with great success, including Shing-Tung Yau's proof of the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge
Complex_geometry
Australian and American mathematician (born 1975)
resolved or made progress on a number of conjectures. In 2012, Green and Tao announced proofs of the conjectured "orchard-planting problem," which asks
Terence_Tao
Key result in general relativity
4171/JEMS/584. S2CID 119633794. Schoen, Richard; Yau, Shing-Tung (1979). "On the proof of the positive mass conjecture in general relativity". Communications in
Positive_energy_theorem
Mathematics concept
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called
Homological_mirror_symmetry
Type of metric in Riemannian geometry
Shing-Tung Yau proved independently. When the first Chern class is zero, there is always a Kähler–Einstein metric, as Yau proved in the Calabi conjecture. That
Kähler–Einstein_metric
Japanese mathematician
gives explicit values for the coefficients of the so-called Lang-Vojta conjecture relating the degree of a curve on a surface with its geometric genus.
Yoichi_Miyaoka
Field of mathematics
Manin–Mumford conjecture, proven by Michel Raynaud, and the Mordell–Lang conjecture, proven by Gerd Faltings. The following conjectures illustrate the
Arithmetic_dynamics
Chinese mathematician
This also provided a parabolic alternative to Yau's method of continuity in the proof of the Calabi conjecture, although much of the technical work in the
Huai-Dong_Cao
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
the Lichnerowicz conjecture is a generalization of a conjecture introduced by Lichnerowicz (1944). Lichnerowicz's original conjecture was that locally
Lichnerowicz_conjecture
English mathematician (born 1957)
geometry 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.
Simon_Donaldson
Type of geometry in mathematics
Yvonne Choquet-Bruhat. In Riemannian geometry, Shing-Tung Yau's resolution of the Calabi conjecture produced a number of Ricci-flat metrics on Kähler manifolds
Ricci-flat_manifold
Theory in physics
sheaves, however, these are integer valued invariants. There are deep conjectures due to Davesh Maulik, Andrei Okounkov, Nikita Nekrasov and Rahul Pandharipande
Donaldson–Thomas_theory
Mathematics concept
algebra of the Galois image. This conjecture is known only in particular cases. Through generalisations of this conjecture, the Mumford–Tate group has been
Mumford–Tate_group
American mathematician
mean curvature flow to prove the Riemannian Penrose conjecture, which is the fifteenth problem in Yau's list of open problems, and was resolved at the same
Tom_Ilmanen
Topological theorem
R3 is either constant or not contained within an open hemisphere. As conjectured by Louis Nirenberg and proved by Robert Osserman in 1959, in this form
Osserman–Xavier–Fujimoto theorem
Osserman–Xavier–Fujimoto_theorem
Hong Kong sweet bun
Sandhana, Lakshmi (16 May 2024). "What Is a Pineapple Bun". wisegeek. Conjecture Corporation. Retrieved 5 January 2014. Ngo, Hope (8 September 2022). "What
Pineapple_bun
Italian-born American mathematician (1923–2023)
Shing-Tung Yau began working on the Calabi conjecture, initially attempting to disprove it. After several years of work, he found a proof of the conjecture, and
Eugenio_Calabi
Chinese-American mathematician (born 1962)
proof of Chen's theorem which made substantial progress on Goldbach's conjecture. Zhang was admitted to the Sun Yat-sen University chemistry department
Shou-Wu_Zhang
American mathematician
work includes the discovery of the Li–Yau differential Harnack inequalities, and the proof of the Willmore conjecture in the case of non-embedded surfaces
Peter_Li_(mathematician)
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
Equivalence of two physical theories
in pure mathematics. Indeed, according to the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, T-duality is closely related to another
T-duality
American physicist
physical problem. The same year, Strominger, Shing-Tung Yau, and Eric Zaslow introduced the SYZ conjecture as a precise formulation of mirror symmetry. In 1997
Andrew_Strominger
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
Russian-French mathematician
positive scalar curvature, which had been a major conjecture previously resolved by Schoen and Yau in low dimensions. In 1981, Gromov identified topological
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Branch of algebraic geometry concerned with counting solutions
Gromov–Witten invariants and mirror symmetry gave significant progress in Clemens conjecture. Enumerative geometry is very closely tied to intersection theory. More
Enumerative_geometry
Measure of curvature in differential geometry
Novikov conjecture for the fundamental group, which deals with the K-theory of C*-algebras. This in turn is a special case of the Baum–Connes conjecture for
Scalar_curvature
3d hypersurface of degree 5
Liu, and S.-T. Yau (1997). The fact that the virtual number equals the actual number relies on confirmation of Clemens' conjecture, currently known
Quintic_threefold
Theory of subatomic structure
mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Group theory is the branch of mathematics
String_theory
French mathematician
Speaker with talk Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture at the International Congress of Mathematicians in Rio de Janeiro
Sébastien_Boucksom
Fano manifolds was made by Gang Tian in 1997, in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition which
K-stability_of_Fano_varieties
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
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 Maxim
Mirror symmetry (disambiguation)
Mirror_symmetry_(disambiguation)
Spectral graph theory concept
after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs. Let G {\displaystyle
Ramanujan_graph
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 in
Brane
French mathematician
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
Category in mathematics
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
American journalist (born 1947)
who solved the Poincaré conjecture and declined the 2006 Fields Medal. The article examined Fields Medalist Shing-Tung Yau's response to Perelman's proof
Sylvia_Nasar
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
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
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
proof of the restricted twin prime conjecture. List of mathematics awards Lizhen Ji; Yat Sun Poon; Lo Yang; Shing-Tung Yau, eds. (2010). Fifth International
Morningside_Medal
Vietnamese mathematician
anti-concentration. In 2007, with Johansson and Kahn, Vu solved the Shamir conjecture in random graph theory. Among others, they established the sharp threshold
Van_H._Vu
German mathematician (born 1958)
Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity. After finishing high school in 1977, Huisken took
Gerhard_Huisken
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
in specific mathematical fields. Grigori Perelman proved the Poincaré conjecture (one of the seven Millennium Prize Problems), and Yuri Matiyasevich gave
List of International Mathematical Olympiad participants
List_of_International_Mathematical_Olympiad_participants
Type of Riemannian manifold
respect to I {\displaystyle I} . Conversely, Shing-Tung Yau's proof of the Calabi conjecture implies that a compact, Kähler, holomorphically symplectic
Hyperkähler_manifold
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
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