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Does the six sphere have a complex structure
6-dimensional sphere S6 has a complex structure is known as the Hopf problem, after Heinz Hopf. S6, when considered as the set of imaginary octonions, with
Hopf_problem
American mathematician (born 1992)
using the model. Four days later, he announced a positive answer to Hopf's problem concerning the existence of a complex structure on a 6-dimensional sphere
Levent_Alpöge
German mathematician (1894–1971)
object Hopf algebra Quantum group Hopf fibration Hopf problem Alexandroff P., Hopf H. Topologie Bd.1 — B: 1935 Hopf, Heinz (1964), Selecta Heinz Hopf, Herausgegeben
Heinz_Hopf
Mathematical method for integrodifferential equations
Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method
Wiener–Hopf_method
Large language model and AI chatbot by Anthropic
complex structure on the six-dimensional sphere, S6, thereby solving the Hopf problem, which has been open since 1947. In November 2024, Anthropic partnered
Claude_(AI)
Homotopy invariant of maps between n-spheres
the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map η
Hopf_invariant
of the conjectures and open problems concerning The Riemann Hypothesis Some open problems in Hopf algebras Open Problems in Dynamical Systems & Ergodic
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Claude Helped Build a Complex Structure on S⁶, Taking Aim At The Unsolved Hopf Problem". OfficeChai. 24 August 2026. Retrieved 27 August 2026. "A tilted residue-class
List of mathematical discoveries made with artificial intelligence
List_of_mathematical_discoveries_made_with_artificial_intelligence
Simplest nontrivial knot link
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly
Hopf_link
Critical point where a periodic solution arises
In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the
Hopf_bifurcation
Mathematical conjectures attributed to Heinz Hopf
Hopf. The Hopf conjecture is an open problem in global Riemannian geometry. It goes back to questions by Heinz Hopf from 1931. A modern formulation is:
Hopf_conjectures
Manifold
question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.) Using an almost complex structure we can make sense of holomorphic
Complex_manifold
Smooth manifold
question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. Just as a complex structure on a vector space V allows a decomposition
Almost_complex_manifold
Generalized sphere of dimension n (mathematics)
question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. 7-sphere Topological quasigroup structure as the set of unit
N-sphere
differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on
Hopf_lemma
Construction in algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative)
Hopf_algebra
Partial differential equation
The Cole–Hopf transformation, developed independently by Julian D. Cole and Eberhard Hopf in 1950, is a change of variables that allows to transform a
Cole–Hopf_transformation
Gives equivalent statements about the geodesic completeness of Riemannian manifolds
The Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi
Hopf–Rinow_theorem
Mathematical problems related to differential equations
equations in the complex plane. Specifically, a Riemann–Hilbert problem is a boundary value problem for a holomorphic function on the complement of an oriented
Riemann–Hilbert_problem
Physical theory
(1944). "К проблеме турбулентности" [On the problem of turbulence]. Doklady Akademii Nauk SSSR. 44: 339–342. Hopf, E. (1948). "A mathematical example displaying
Landau–Hopf theory of turbulence
Landau–Hopf_theory_of_turbulence
Concept in mathematics
comultiplications that turn them into Hopf algebras. This is made precise in the article on the tensor algebra: the tensor algebra has a Hopf algebra structure on it
Universal_enveloping_algebra
Boundary value problem Dirichlet problem, Dirichlet boundary condition Neumann boundary condition Stefan problem Wiener–Hopf problem Separation of variables
List of partial differential equation topics
List_of_partial_differential_equation_topics
German serial killer
Schneider, who also began suffering from health problems. She divorced Hopf, left him, and died soon afterward. Hopf was unable, however, to collect the life
Karl_Hopf_(serial_killer)
Algebra in algebraic topology
{\displaystyle p} , the Steenrod algebra A p {\displaystyle A_{p}} is the graded Hopf algebra over the field F p {\displaystyle \mathbb {F} _{p}} of order p {\displaystyle
Steenrod_algebra
Basic unit of taxonomic classification, below genus
102.6600D. doi:10.1073/pnas.0502030102. PMC 1131873. PMID 15851674. Hopf, F. A.; Hopf, F. W. (1985). "The role of the Allee effect on species packing".
Species
Soviet theoretical physicist (1908–1968)
Landau quantization Landau theory Landau–Squire jet Landau–Levich problem Landau–Hopf theory of turbulence Ginzburg–Landau theory Darrieus–Landau instability
Lev_Landau
The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic
Hopf_maximum_principle
Collection of knots that do not intersect, but may be linked
nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The
Link_(knot_theory)
Equations of motion for viscous fluids
with no singularities comes from considering the flow along the lines of a Hopf fibration. Let r be a constant radius of the inner coil. One set of solutions
Navier–Stokes_equations
Mathematician
Manin–Drinfeld theorem Quantum group Chiral algebra Quasitriangular Hopf algebra Ruziewicz problem O'Connor, J. J.; Robertson, E. F. "Vladimir Gershonovich Drinfeld"
Vladimir_Drinfeld
How spheres of various dimensions can wrap around each other
interesting example is the group π3(S2) which is generated by the Hopf fibration. The problem of computing the groups πn+k(Sn) for positive k is very difficult
Homotopy_groups_of_spheres
English mathematician
general techniques for solving matrix Wiener–Hopf problems and this gave the solution to a basic problem of diffraction theory, namely, scattering by
David Abrahams (mathematician)
David_Abrahams_(mathematician)
Canadian-American mathematician (1925–2020)
existence. His work was later put into the setting of a boundary value problem by Eberhard Hopf. A breakthrough came with work of Vladimir Scheffer in the 1970s
Louis_Nirenberg
principle Landau quantization or Landau levels Landau–Hopf theory of turbulence Landau–Levich problem Landau–Lifshitz model Landau-Lifshitz pseudotensor
List of things named after Lev Landau
List_of_things_named_after_Lev_Landau
German-American mathematician
Sciences, and the recipient of the 1999 Ostrowski Prize and the 2013 Heinz Hopf Prize. Since 2009, he has been a faculty member at the Institute for Advanced
Helmut_Hofer
Orientable surface whose boundary is a knot or link
A Seifert surface for the Hopf link. This is an annulus, not a Möbius strip. It has two half-twists and is thus orientable.
Seifert_surface
Determining whether a knot is the unknot
Unsolved problem in mathematics Can unknots be recognized in polynomial time? More unsolved problems in mathematics In mathematics, the unknotting problem is
Unknotting_problem
Study of mathematical knots
applied. C() = C() + z C() gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated, C() = C() + z C() gives a link
Knot_theory
Prime knot named for John Horton Conway
2020.191.2.5. JSTOR 10.4007/annals.2020.191.2.5. Wolfson, John. "A math problem stumped experts for 50 years. This grad student from Maine solved it in
Conway_knot
renormalization as a Riemann–Hilbert problem, the fact that the Slavnov–Taylor identities of gauge theories generate a Hopf ideal, the quantization of fields
Combinatorics_and_physics
Infinite dimensional Lie group
"Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem" (PDF), Commun.
Butcher_group
Concept in differential topology
M^{4m+2}} (for m ≠ 0 , 1 , 3 {\displaystyle m\neq 0,1,3} ) and the mod 2 Hopf invariant of maps S 4 m + 2 + k → S 2 m + 1 + k {\displaystyle S^{4m+2+k}\to
Kervaire_invariant
Type of biological prediction
1073/pnas.91.1.98. PMC 42893. PMID 8278414. Marks DS, Colwell LJ, Sheridan R, Hopf TA, Pagnani A, Zecchina R, Sander C (2011). "Protein 3D structure computed
Protein_structure_prediction
American mathematician (born 1950)
shared with Charles Fefferman. In the same year, he was awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize by Kazan Federal University, and the
Richard_Schoen
Chaotic model of atmospheric convection
critical value, both equilibrium points lose stability through a subcritical Hopf bifurcation. When ρ = 28, σ = 10, and β = 8/3, the Lorenz system has chaotic
Lorenz_system
Group whose operation is a composition of braids
Affine symmetric group, a braid group on an annulus Artin–Tits group Braided Hopf algebra Braided monoidal category Braided vector space Knot theory Non-commutative
Braid_group
Dirichlet problem with U = 0 on the outer curve and −1 on the inner curve. By the maximum principle 0 < u(z) < 1 for z in Ω and so by the Hopf lemma the
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Method to avoid singularities in classical mechanics
Levi-Civita regularization, based on the real Hopf fibration Kustaanheimo–Stiefel regularization, based on the complex Hopf fibration Ligon–Schaaf regularization
Moser_regularization
Loop seen as a trivial knot
the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot
Unknot
Mathematical tool for studying knots
crossings (the Hopf link [1]). Patching the trefoil P() = A × P() + P() gives the unknot and, again, the Hopf link. Patching the Hopf link P() = A × P()
Skein_relation
were introduced by J. Frank Adams (1960) in his solution to the Hopf invariant problem. Similarly, one can define tertiary cohomology operations from the
Secondary cohomology operation
Secondary_cohomology_operation
Algebraic structure used in theoretical physics
structure of a Z2-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z2-graded comodules. This Hopf algebra gives the global
Supergroup_(physics)
American mathematician
geometry. He received his Ph.D. degree under the supervision of Hopf and George Pólya. Hopf later recommended Stoker to Richard Courant. In 1937 Stoker,
James_J._Stoker
Numerous conjectures by mathematician Irving Kaplansky
several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures. Let K be a field
Kaplansky's_conjectures
French mathematician, physicist and engineer (1854–1912)
Papers". Archived from the original on 1 August 2020.. Poincaré–Andronov–Hopf bifurcation Poincaré complex – an abstraction of the singular chain complex
Henri_Poincaré
Root vegetable related to carrot and parsley
with glyphosate-containing herbicides considered effective. Zohary and Hopf note that the archaeological evidence for the ancient cultivation of the
Parsnip
Knot invariant
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Alexander_polynomial
Theorem in differential topology
there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it
Hairy_ball_theorem
Mathematical invariant of a knot or link
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Jones_polynomial
Italian-born American mathematician (1923–2023)
Pierre-Louis Lions. By extending the strong maximum principle of Eberhard Hopf to his notion of viscosity solutions, Calabi was able to use his Laplacian
Eugenio_Calabi
British mathematician (1930–1989)
case. He used this spectral sequence to attack the celebrated Hopf invariant one problem, which he completely solved in a 1960 paper by making a deep analysis
Frank_Adams
Link that consists of finitely many unlinked unknots
free group on n generators, and is used in classifying Brunnian links. The Hopf link is a simple example of a link with two components that is not an unlink
Unlink
Individual living life form
have started". The Washington Post. Bernstein, Harris; Byerly, Henry C.; Hopf, Frederick A.; Michod, Richard A.; Vemulapalli, G. Krishna (1983). "The Darwinian
Organism
How many times curves wind around each other
Differentiable curve – Study of curves from a differential point of view Hopf invariant – Homotopy invariant of maps between n-spheres Kissing number –
Linking_number
Jewish-German born Dutch mathematician
visited the University of Paris. He completed his thesis work with Heinz Hopf at Berlin, defended a thesis on the ends of topological groups in 1930, and
Hans_Freudenthal
Ring that is also a vector space or a module
g., semisimple complex Lie group), then the coordinate ring of G is the Hopf algebra A corresponding to G. Many structures of G translate to those of
Associative_algebra
Mathematics award
2014. Retrieved 19 August 2014. McKinnon Riehm & Hoffman 2011, p. 212 H. Hopf. Proceedings of the International Congress of Mathematicians 1958. Archived
Fields_Medal
Poisson ring Poisson supermanifold Poisson–Charlier polynomials Poisson-Hopf algebra Poisson–Mellin–Newton cycle Poisson–Lie group Probability theory
List of things named after Siméon Denis Poisson
List_of_things_named_after_Siméon_Denis_Poisson
Type of mathematical object
comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme
Group_scheme
Swiss-born painter (1861–1918)
Annie Stebler-Hopf (4 September 1861 – 30 January 1918), also known as Anny Hopf or Anna Hopf, was a Swiss painter. She trained in Basel, Berlin, and
Annie_Stebler-Hopf
Millennium Prize Problems Painlevé conjecture Mathematical fallacy Superseded theories in science List of incomplete proofs List of unsolved problems in mathematics
List_of_conjectures
Branch of mathematics
the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's theorem, and the Poincaré conjecture.
Differential_topology
Swedish mathematician
is also a juggler and magician. Ehrenborg, Richard (1996). "On posets and Hopf algebras". Advances in Mathematics. 119 (1): 1–25. doi:10.1006/aima.1996
Richard_Ehrenborg
Species of fig
their sycamore trees.[why?] According to botanists Daniel Zohary and Maria Hopf, cultivation of this species was "almost exclusively" by the ancient Egyptians
Ficus_sycomorus
Attempt to classify and tabulate all possible knots
of days. Knot theory Knot (mathematics) List of prime knots Unknotting problem Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998), "The first 1,701
Knot_tabulation
hyperbolic space, although a space form need not be simply connected. The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional
Space_form
Behavior in a nonlinear system
dynamical systems. Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the
Limit_cycle
Invariant of mathematical knots
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Khovanov_homology
Mathematical knot with crossing number 5
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Cinquefoil_knot
Smooth manifold with an inner product on each tangent space
{\displaystyle \operatorname {diam} (M,d_{g})=\sup\{d_{g}(p,q):p,q\in M\}.} The Hopf–Rinow theorem shows that if ( M , d g ) {\displaystyle (M,d_{g})} is complete
Riemannian_manifold
Vegetable
Archive.org. April 1942. p. 94. Retrieved 31 May 2026. Zohary, Daniel; Hopf, Maria; Weiss, Ehud (2012). Domestication of Plants in the Old World (Fourth ed
Beetroot
Invariant of framed knots
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Self-linking_number
Flowering plant and fruit
November 2024. "Cydonia oblonga Quince". Plants For A Future. Zohary, Daniel; Hopf, Maria; Weiss, Ehud (2012). Domestication of Plants in the Old World: The
Quince
Analog of the knot group
free product of the link groups of the components. The link group of the Hopf link, the simplest non-trivial link – two circles, linked once – is the free
Link_group
Simplest non-trivial closed knot with three crossings
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Trefoil_knot
Mathematical knot
example: The unknot, trefoil knot, and figure-eight knot are fibered knots. The Hopf link is a fibered link. The Alexander polynomial of a fibered knot is monic
Fibered_knot
Fundamental group of a knot complement
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Knot_group
Knot that can't be tied in a string of constant diameter
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Wild_knot
Polynomial invariant arising in knot theory
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
HOMFLY_polynomial
Type of mathematical link
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Hyperbolic_link
Hungarian and American mathematician and physicist (1903–1957)
University Press. p. 149. ISBN 978-0-19-537222-9. Murawski 2010, p. 209. Hopf, Eberhard (1939). "Statistik der geodätischen Linien in Mannigfaltigkeiten
John_von_Neumann
Branch of mathematics that studies dynamical systems
geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context
Ergodic_theory
Manner in which humans engage sexually
Oxford University Press.[ISBN missing] Bernstein, Harris; Byerly, Henry C.; Hopf, Frederic A.; Michod, Richard E. (20 September 1985). "Genetic Damage, Mutation
Human_sexual_activity
Type of mathematical knot
J} is the connect-sum of a non-trivial knot K ′ {\displaystyle K'} with a Hopf link. If the link K ′ ∪ J {\displaystyle K'\cup J} is the Whitehead link
Satellite_knot
Mathematical knot with crossing number 7
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
7_2_knot
Type of knot in knot theory
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
2-bridge_knot
Mathematical concept
of the variables. Another example shows how a center manifold models the Hopf bifurcation that occurs for parameter a ≈ 4 {\displaystyle a\approx 4} in
Center_manifold
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Crosscap_number
Polynomial invariant of framed links
Torus Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02 1) Hopf (22 1) Solomon's (42 1) Invariants Alternating Arf invariant Bridge no. 2-bridge
Bracket_polynomial
French mathematician (1927–2007)
intégrale généralisée et homotopie, was written under the direction of Heinz Hopf and Beno Eckmann. Kervaire was a professor at New York University's Courant
Michel_Kervaire
travel, tourism, insurance
HOPF PROBLEM
HOPF PROBLEM
Girl/Female
Christian & English(British/American/Australian)
Optimistic
Girl/Female
English American
One of the three Christian virtues (Faith, Hope and Charity).
Girl/Female
American, Anglo, Australian, British, Christian, Danish, English, German, Portuguese
Trust; Faith; Belief; Expectation Belief; Wish; Desire
Girl/Female
Hindu
Hope
Girl/Female
Indian
Hope
Surname or Lastname
Dutch and North German
Dutch and North German : variant of Hoff.North German : topographic name from a variant of Hoff.Dutch : nickname from hoofd ‘head’. Compare English Head 1.English : variant spelling of Huff.
Girl/Female
Muslim
Hope
Boy/Male
Muslim
Hope
Boy/Male
Indian
Hope
Girl/Female
Indian
Hope
Boy/Male
Muslim
Hope
Girl/Female
Indian
Hope
Girl/Female
Tamil
Pradiksha | பà¯à®°à®¤à¯€à®•à¯à®·à®¾
Hope
Pradiksha | பà¯à®°à®¤à¯€à®•à¯à®·à®¾
Boy/Male
Indian
Hope
Surname or Lastname
English
English : from the medieval personal name Hobb(e) (see Hobbs).Dutch : from Middle Dutch hoppe ‘hops’, hence a metonymic occupational name for a grower or seller of hops.Dutch : from a pet form of the personal name Hubrecht (see Hubert).South German : variant of Hoppe 3.North German form of Hopf.
Girl/Female
Muslim
Hope
Girl/Female
Muslim
Hope
Girl/Female
Muslim
Hope
Boy/Male
Muslim
Hope
Girl/Female
Muslim
Hope
HOPF PROBLEM
HOPF PROBLEM
HOPF PROBLEM
HOPF PROBLEM
HOPF PROBLEM
HOPF PROBLEM
HOPF PROBLEM
travel, tourism, insurance