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In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group
Borel_conjecture
French mathematician (1871–1956)
theorem Borel right process Borel set Borel summation Borel distribution Borel's conjecture about strong measure zero sets (not to be confused with Borel conjecture
Émile_Borel
Unsolved problem in topology
{\displaystyle f} . The Novikov conjecture is equivalent to the rational injectivity of the assembly map in L-theory. The Borel conjecture on the rigidity of aspherical
Novikov_conjecture
uncountable set of Lebesgue measure 0 which is not of strong measure zero. Borel's conjecture states that every strong measure zero set is countable. It is now
Strong_measure_zero_set
Swiss mathematician (1923–2003)
452) Borel–Weil–Bott theorem Borel cohomology Borel conjecture Borel construction Borel subgroup Borel subalgebra Borel fixed-point theorem Borel's theorem
Armand_Borel
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
On distance sets of high-dimensional sets
to resolving several other unsolved conjectures. These include a conjecture of Paul Erdős on the existence of Borel subrings of the real numbers with fractional
Falconer's_conjecture
Chinese American mathematician
conjecture on homotopy invariance of higher signatures, the Baum–Connes conjecture on K-theory of group C*-algebras, and the stable Borel conjecture on
Guoliang_Yu
Unsolved problem in geometry
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular
Hodge_conjecture
Farrell-Jones Conjecture for hyperbolic groups", arXiv:math/0609685 Bartels, Arthur; Lück, Wolfgang; Reich, Holger (2009), The Borel Conjecture for hyperbolic
Farrell–Jones_conjecture
Bing–Borsuk conjecture: every n {\displaystyle n} -dimensional homogeneous absolute neighborhood retract is a topological manifold. Borel conjecture: aspherical
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Unsolved problem in mathematics
In mathematics, the Ramanujan-Petersson conjecture is a conjecture concerning the growth rate of coefficients of modular forms and more generally, automorphic
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
American mathematician
and dynamics to questions such as the Borel conjecture. The Farrell-Jones conjecture implies the Borel Conjecture for manifolds of dimension greater than
Lowell_E._Jones
Mathematical theorem
Borel–Cantelli lemma. The converse implication is the crux of the conjecture. There have been many partial results of the Duffin–Schaeffer conjecture
Duffin–Schaeffer_theorem
1929 mathematical conjecture
Margulis. Qualitative versions of the Oppenheim conjecture were later proved by Eskin–Margulis–Mozes. Borel and Prasad established some S-arithmetic analogues
Oppenheim_conjecture
Chinese-American mathematician
Thomas Farrell he worked on a program to prove the Novikov conjecture and the Borel conjecture with methods from geometric topology and gave proofs for
Wu-Chung_Hsiang
German mathematician
Farrell-Jones Conjecture for mapping class groups". arXiv:1606.02844 [math.GT]. Bartels, Arthur; Lück, Wolfgang (2012). "The Borel Conjecture for hyperbolic
Arthur_Bartels
H^{3}} are called Fuchsian groups and Kleinian groups, respectively. Borel conjecture Goldberg, Samuel I. (1998), Curvature and Homology, Dover Publications
Space_form
intervals (In) which covers X and such that In has length at most εn. Borel's conjecture, that every strong measure zero set is countable, is independent of
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Tool to classify manifolds within a homotopy type in dim > 4
with which they can be identified. See more details in Borel conjecture, Farrell-Jones Conjecture. Quinn, Frank (1971), A geomeric formulation of surgery
Surgery_exact_sequence
to show that Borel's conjecture, which says that all strong measure zero sets are countable, is consistent with ZFC. (Borel's conjecture is not consistent
List_of_forcing_notions
List of terms created from a person's name
Bordigism Armand Borel, French mathematician – Borel–Weil–Bott theorem, Borel conjecture, Borel fixed-point theorem, Borel's theorem Émile Borel, French mathematician
List_of_eponyms_(A–K)
American mathematician (born 1940)
of analytic determinacy (from the existence of a measurable cardinal), Borel determinacy (from ZFC alone), the proof (with John R. Steel) of projective
Donald_A._Martin
matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H
Stahl's_theorem
French mathematician (1906-1998)
Weil conjecture, around 1967, which later under pressure from Serge Lang (resp. of Jean-Pierre Serre) became known as the Taniyama–Shimura conjecture (resp
André_Weil
theory Borel-Serre Compactification Grothendieck-Serre Correspondence Serre class Quillen–Suslin theorem (sometimes known as "Serre's Conjecture" or "Serre's
List of things named after Jean-Pierre Serre
List_of_things_named_after_Jean-Pierre_Serre
American mathematician
unions of scattered ordered sets. He proved the consistency of the Borel conjecture, i.e., the statement that every strong measure zero set is countable
Richard_Laver
French mathematician (born 1947)
différentiel étranger). "Resurgent functions" are divergent power series whose Borel transforms converge in a neighborhood of the origin and give rise, by means
Jean_Écalle
Mathematical conjectures in class field theory
expositions of their work. Borel and Vogan discuss the Langlands conjectures for more general groups. The Langlands conjectures for arbitrary reductive groups
Local_Langlands_conjectures
German mathematician
and his coauthors resolved many cases of the Farrell-Jones conjecture and the Borel conjecture. He has also contributed to the development of the theory
Wolfgang_Lück
Techniques in topology used to produce one finite-dimensional manifold from another
Examples are the classification of exotic spheres, and the proofs of the Borel conjecture for negatively curved manifolds and manifolds with hyperbolic fundamental
Surgery_theory
Concept in complex analysis
i {\displaystyle K=\bigcup _{i=1}^{\infty }K_{i}} , where each Ki is a Borel set, then γ ( K ) ≤ C ∑ i = 1 ∞ γ ( K i ) {\displaystyle \gamma (K)\leq
Analytic_capacity
Topics referred to by the same term
analytic subset of the reals that is not Borel An analytic set whose complement is also analytic is a Borel set, a special case of the Lusin separation
Suslin's_theorem
pp. 175–192 Howe, Roger E. (1979), "θ-series and invariant theory", in Borel, A.; Casselman, W. (eds.), Automorphic forms, representations and L-functions
Theta_correspondence
Mathematics glossary
See Betti number. Bing–Borsuk conjecture See Bing–Borsuk conjecture. Bockstein homomorphism Borel Borel conjecture. Borel–Moore homology Borsuk's theorem
Glossary of algebraic topology
Glossary_of_algebraic_topology
a standard Borel space X is a Borel equivalence relation E with countable classes, that can, in a certain sense, be approximated by Borel equivalence
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
Completes the Langlands program for general linear groups over algebraic function fields
order is pure of weight 0. Local Langlands conjectures Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic
Lafforgue's_theorem
strong measure zero. In the Laver model for the consistency of the Borel conjecture every Rothberger subset of the real line is countable. Rothberger,
Rothberger_space
American mathematician
Much of Farrell's work lies around the Borel conjecture. He and his co-authors have verified the conjecture for various cases, most notably flat manifolds
F._Thomas_Farrell
On generating functions from counting points on algebraic varieties over finite fields
In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them
Weil_conjectures
Mathematical models of strategic interactions
as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false
Game_theory
André Weil (1906 – 1998), a French mathematician. Bergman–Weil formula Borel–Weil theorem Chern–Weil homomorphism Chern–Weil theory De Rham–Weil theorem
List of things named after André Weil
List_of_things_named_after_André_Weil
Mathematical theorem
(2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 [math.AG]. Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber)
Theorem_of_absolute_purity
function, such as Deligne's conjecture on special values of L-functions, the Beilinson conjecture, and the Bloch–Kato conjecture (on special values of L-functions)
Motivic_L-function
Invariant of algebraic varieties and of more general schemes
subvarieties of X. The Chow groups of X have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for
Motivic_cohomology
Branch of algebraic geometry
proof of the torsion conjecture was extended to all number fields by Loïc Merel. In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that
Arithmetic_geometry
Linear representation in mathematics
ways of defining Steinberg representations are equivalent. Borel & Serre (1976) and Borel (1976) showed how to realize the Steinberg representation in
Steinberg_representation
Divergence in perturbative quantum field theory
put forward a conjecture that Lipatov's and Lautrup's contributions are associated with different types of singularities in the Borel plane, the former
Renormalon
Subgroup of the group of invertible n×n matrices
groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today. One
Linear_algebraic_group
fields by Helge Glöckner. Cartier 1979, §1.1 Bushnell & Henniart 2006, §1.1 Borel & Wallach 2000, Chapter X van Dantzig 1936, p. 411 Willis 1994 Caprace &
Totally_disconnected_group
Mathematical concept
regarded as a Borel space. A famous conjecture of G. Mackey proposed that a separable locally compact group is of type I if and only if the Borel space is
Spectrum_of_a_C*-algebra
to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Eduard
L²_cohomology
Integral polynomial
analogues of complex flag manifolds G/B where G is a complex Lie group and B a Borel subgroup. The original (K-L) case is then about the details of decomposing
Kazhdan–Lusztig_polynomial
Mathematical inequality
proved weaker versions with the constant 3 replaced by 8 and 4. Armand Borel and Friedrich Hirzebruch showed that the inequality is best possible by
Bogomolov–Miyaoka–Yau inequality
Bogomolov–Miyaoka–Yau_inequality
Concept in mathematics
1.8. Borel (1991), section 23.4. Borel (1991), section 23.2. Borel & Tits (1971), Corollaire 3.8. Platonov & Rapinchuk (1994), Theorem 3.1. Borel (1991)
Reductive_group
Analogs of homology groups for algebraic varieties
sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore) motivic homology groups, also known as higher Chow groups. For any
Chow_group
Averages of repeated trials converge to the expected value
also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. Markov showed that the law can apply
Law_of_large_numbers
Locally compact topological group with an invariant averaging operation
translation-invariant nontrivial ring measure, the Haar measure. (This is a Borel regular measure when G is second-countable; the left and right Haar measures
Amenable_group
Technique in mathematical group theory
parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to G). The representations of parabolic induction
Deligne–Lusztig_theory
Mathematician
topology, notably the conjectures of Novikov and Borel. Carlsson has proved, jointly with E. Pedersen and B. Goldfarb Novikov's conjecture for large classes
Gunnar_Carlsson
and Rolf Ebert Borel algebra, measure, set, space, summation, Borel's lemma, paradox – Émile Borel Borel–Cantelli lemma – Émile Borel and Francesco Paolo
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Map from algebraic K-theory to cohomology
also generalization of the Borel regulator. M. Rapoport, N. Schappacher and P. Schneider, ed. (1988). Beilinson's conjectures on special values of L-functions
Beilinson_regulator
Field of knowledge
across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994
Mathematics
of as an algebraic counterpart of equivariant cohomology. For example, Borel's theorem states that the cohomology ring of a classifying stack is a polynomial
Cohomology_of_a_stack
Japanese mathematician (1930–2019)
varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem. Gorō Shimura
Goro_Shimura
classes in its integral cohomology. It is the only case of the Hodge conjecture which has been proved for all Kähler manifolds. Let X be a compact Kähler
Lefschetz theorem on (1,1)-classes
Lefschetz_theorem_on_(1,1)-classes
projection of a Borel set is Borel. Suslin pointed out the error and was inspired by it to define analytic sets as continuous images of Borel sets. Dehn's
List_of_incomplete_proofs
Algebraic variety with a group structure
variety Borel subgroup Tame group Morley rank Cherlin–Zilber conjecture Adelic algebraic group Pseudo-reductive group Borel 1991, p.54. Borel 1991, p
Algebraic_group
Type of group in group theory
Borel, André Weil, Jacques Tits and others on algebraic groups. Shortly afterwards the finiteness of covolume was proven in full generality by Borel and
Arithmetic_group
Local Lie group Formal group law Hilbert's fifth problem Hilbert–Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification
List_of_Lie_groups_topics
Each semi-simple algebraic group is geometrically reductive
mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and
Haboush's_theorem
capacity Disk algebra Univalent function Ahlfors theory Bieberbach conjecture Borel–Carathéodory theorem Corona theorem Hadamard three-circle theorem Hardy
List of complex analysis topics
List_of_complex_analysis_topics
Mathematical concept
variety is described by the André–Oort conjecture. Conditional results have been obtained on this conjecture, assuming a generalized Riemann hypothesis
Shimura_variety
In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is
Spherical_variety
Mathematical group
Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup
Outer_automorphism_group
Way to divide polygon into smaller parts
mathematical analysis such as for the Bolzano–Weierstrass theorem and Heine–Borel theorem. A finite subdivision rule R {\displaystyle R} consists of the following
Finite_subdivision_rule
Concept in complex dynamics
known as Eremenko's conjecture. In 2021, a paper by Martí-Pete, Rempe, and Waterman constructed a counterexample to Eremenko's conjecture. Eremenko also asked
Escaping_set
French mathematician (born 1926)
contributions were: his "Conjecture II" (still open) on Galois cohomology; his use of group actions on trees (with Hyman Bass); the Borel–Serre compactification;
Jean-Pierre_Serre
On the cohomology ring of the moduli stack of principal bundles
cohomology ring of Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} . Borel's theorem, which says that the cohomology ring of a classifying stack is
Atiyah–Bott_formula
Statement in probability theory
strengthening of the classical strong law of large numbers in the direction of the Borel–Cantelli lemma. The idea of such a result is probably due to Robbins, but
Hsu–Robbins–Erdős_theorem
Geometry formula
Quillen's papers. An alternative name for the formula is Borel cohomology, after Armand Borel The localization theorem states that the equivariant cohomology
Localization formula for equivariant cohomology
Localization_formula_for_equivariant_cohomology
British-Lebanese mathematician (1929–2019)
K-theory to improve the lower bounds found using ordinary cohomology by Borel and Serre for the James number, describing when a map from a complex Stiefel
Michael_Atiyah
Mathematical concept
of a modular form. They were introduced by Langlands (1967, 1970, 1971). Borel (1979) and Arthur & Gelbart (1991) gave surveys of automorphic L-functions
Automorphic_L-function
Subfield of set theory
with the Borel determinacy theorem of Martin, implies that all Blackwell games with Borel payoff functions are determined. Martin conjectured that ordinary
Determinacy
Russian mathematician
His superrigidity theorem from 1975 clarified an area of classical conjectures about the characterisation of arithmetic groups amongst lattices in Lie
Grigory_Margulis
Function type in graph theory
attacking inequalities related to homomorphisms. For example, Sidorenko's conjecture is a major open problem in extremal graph theory, which asserts that for
Graphon
Rational-number approximation of a real number
further improved without excluding some irrational numbers (see below). Émile Borel (1903) showed that, in fact, given any irrational number α, and given three
Diophantine_approximation
Semitopological group in abstract algebra
groups by Ono (1957) followed the algebraic group theory founded by Armand Borel and Harish-Chandra. Let K {\displaystyle K} be a global field, and let A
Adelic_algebraic_group
(1979), "A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations
Θ10
Possible axiom for set theory
closed set, then the game is determined. By the Borel determinacy theorem, games whose winning set is a Borel set are determined. It follows from the existence
Axiom_of_determinacy
Topological continuum undefinable as the union of any two proper subcontinua
{C}}_{i}} is the bucket handle. The bucket handle admits no Borel transversal, that is there is no Borel set containing exactly one point from each composant
Indecomposable_continuum
Branch of geometry that studies combinatorial properties and constructive methods
totality of all lattices are relatively well understood. Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M. S. Raghunathan, Margulis, Zimmer obtained
Discrete_geometry
theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical
List_of_theorems
Number with all digits equally frequent
base. The concept of a normal number was introduced by Émile Borel (1909). Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal
Normal_number
American mathematician (1916–1991)
Picard–Vessiot theory Kolchin-Schmidt Conjecture Jacobi Bound Conjecture Ritt Problem Kolchin Catenary Conjecture Kolchin, E. R. (1985), Differential algebraic
Ellis_Kolchin
French mathematician (born 1979)
Sophie (2006). "Complexes pondérés sur les compactifications de Baily-Borel: Le cas des variétés de Siegel". Journal of the American Mathematical Society
Sophie_Morel
Constructing a strictly convex compact surface with specified Gaussian curvature
problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere Sn-1 to be the surface area measure of a convex
Minkowski_problem
showed how to use the base change lifting for GL2 to prove the Artin conjecture for tetrahedral and some octahedral 2-dimensional representations of the
Base_change_lifting
American mathematician
representations are of type I) if and only if the Borel structure of its dual is a standard Borel space. He has written numerous survey articles connecting
George_Mackey
Russian mathematician (born 1959)
jointly with A. S. Merkurjev and A. R. Wadsworth, generalized, to arbitrary Borel varieties, results proved by David Tao concerning index reduction formulas
Ivan_Panin_(mathematician)
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