Open problems

1131 registered problems.

OP-00951 Open

Many collinear triples force a cubic curve with many points

CO · collinear triples · cubic curves · incidence geometry · orchard problem

OP-00952 Open

Many collinear four-point sets force five points on a line

CO · collinear points · Erdős problems · incidence geometry · rich lines

OP-00953 Open

The Kupitz–Perles balanced line conjecture

CO · balanced lines · discrete geometry · halving lines · lines through point sets

OP-00954 Open

Weak nets of linear size for axis-parallel rectangles

CO · axis-parallel rectangles · discrepancy theory · discrete geometry · weak epsilon-nets

OP-00955 Open

Random generators of the alternating group give fast mixing

GR · alternating groups · Cayley graphs · expanders · mixing time · random walks on groups

OP-00956 Open

Carbery's rectangle problem

CA · axis-parallel rectangles · harmonic analysis · measure theory · sublevel sets

OP-00957 Open

A dilate with no short interval holding many points

NT · dilates · discrepancy · uniform distribution modulo one

OP-00958 Open

Large sets of pairwise comparable triples

CO · comparability · Delsarte bound · extremal combinatorics · independent sets · tripod packing

OP-00959 Open

The dichotomy problem for Sidon sets and sets of analyticity

CA · commutative harmonic analysis · Fourier algebra · functions operating on Fourier coefficients · sets of analyticity · Sidon sets

OP-00960 Open

Indices of irreducible subfactors of the hyperfinite II₁ factor

OA · hyperfinite II₁ factor · Jones index · Peterson's operator algebra problems · subfactors

OP-00961 Open

Does every nonamenable II₁ factor contain the free group factor L(F₂)?

OA · amenability · free group factors · II₁ factors · Peterson's operator algebra problems

OP-00962 Open

The Stone–Weierstrass problem for C*-algebras

OA · C*-algebras · Peterson's operator algebra problems · pure states · Stone–Weierstrass theorem

OP-00963 Open

Are the radial and generator masas of L(F₂) conjugate by an automorphism?

OA · free group factors · maximal abelian subalgebras · Peterson's operator algebra problems · radial masa

OP-00964 Open

Do reduced group C*-algebras remember torsion-free groups?

OA · C*-superrigidity · Peterson's operator algebra problems · reduced group C*-algebras · torsion-free groups

OP-00965 Open

Tensor factors of McDuff II₁ factors

OA · central sequences · McDuff factors · Peterson's operator algebra problems · tensor products

OP-00966 Open

Uniqueness of McDuff decompositions up to amplification

OA · amplifications · McDuff decomposition · McDuff factors · Peterson's operator algebra problems

OP-00967 Open

Tensor products of maximal injective subalgebras

OA · II₁ factors · maximal injective subalgebras · Peterson's operator algebra problems · tensor products

OP-00968 Open

Kadison's problem on finitely generated *-algebras of operators with finite spectra

OA · *-algebras · finite spectrum · operator algebras · Peterson's operator algebra problems

OP-00969 Open

Cocycle superrigidity of Bernoulli actions of groups with vanishing first ℓ²-Betti number

OA · Bernoulli actions · cocycle superrigidity · ℓ²-Betti numbers · Peterson's operator algebra problems

OP-00970 Open

Trivial fundamental group for Bernoulli crossed products

OA · Bernoulli actions · fundamental group of II₁ factors · ℓ²-Betti numbers · Peterson's operator algebra problems

OP-00971 Open

Popa's factorial commutant embedding problem

OA · hyperfinite II₁ factor · Peterson's operator algebra problems · relative commutants · ultrapowers

OP-00972 Open

Isomorphism of matrix ultraproducts under the continuum hypothesis

OA · continuous model theory · continuum hypothesis · matrix algebras · Peterson's operator algebra problems · ultraproducts

OP-00973 Open

Cost of a group and its first ℓ²-Betti number

GR · cost · ℓ²-Betti numbers · measured group theory · Peterson's operator algebra problems

OP-00974 Open

Nonamenable subfactors of the free group factor L(F₂)

OA · free group factors · interpolated free group factors · Peterson's operator algebra problems · subfactors

OP-00975 Open

A discrete copy of the free group in the unitary group of the hyperfinite II₁ factor

OA · discrete subgroups · free groups · hyperfinite II₁ factor · Peterson's operator algebra problems · unitary groups

OP-00976 Open

Unitary representations of higher-rank lattices in the hyperfinite II₁ factor

OA · hyperfinite II₁ factor · lattices in Lie groups · Peterson's operator algebra problems · rigidity · unitary representations

OP-00977 Open

Unique Cartan subalgebra for Bernoulli crossed products

OA · Bernoulli actions · Cartan subalgebras · group measure space construction · Peterson's operator algebra problems

OP-00978 Open

Unique Cartan subalgebra for SLₙ(ℤ) acting on the n-torus

OA · Cartan subalgebras · group measure space construction · Peterson's operator algebra problems · toral automorphisms

OP-00979 Open

Unique Cartan subalgebras for groups with a nonzero ℓ²-Betti number

OA · Cartan subalgebras · C-rigidity · ℓ²-Betti numbers · Peterson's operator algebra problems

OP-00980 Open

Are locally reflexive C*-algebras exact?

OA · C*-algebras · exactness · local reflexivity · Peterson's operator algebra problems

OP-00981 Open

Power-saving bounds for four-term progressions in F_5^n

CO · arithmetic progressions · finite field model · polynomial method · Szemerédi's theorem

OP-00982 Open

A Sidorenko-type count for the Möbius ladder in abelian groups

CO · abelian groups · Cayley graphs · graph homomorphisms · Möbius ladder · Sidorenko's conjecture

OP-00983 Open

Three-term progressions in F_3^n with common difference in {0,1}^n

CO · arithmetic progressions · density Hales–Jewett theorem · finite field model · restricted differences

OP-00984 Open

Polynomial bounds for a large set S with S^8 inside A^4 in approximate groups

GR · approximate groups · Croot–Sisask lemma · nonabelian additive combinatorics · product sets

OP-00985 Open

An additive construction that would give matrix multiplication exponent two

CO · abelian groups · group-theoretic method · matrix multiplication · sumsets

OP-00986 Open

Small sets with a k-term progression of every difference up to N

CO · arithmetic Kakeya conjecture · arithmetic progressions · Kakeya problem

OP-00987 Open

The polynomial Bogolyubov conjecture over F_2^n

CO · Bogolyubov's lemma · finite field model · polynomial Freiman–Ruzsa conjecture · sumsets

OP-00988 Open

The largest coset in A + A for a dense subset of F_2^n

CO · Bogolyubov's lemma · cosets · finite field model · sumsets

OP-00989 Open

The no-three-in-line problem: 2N points in the N × N grid

CO · collinear points · discrete geometry · lattice points · no-three-in-line problem

OP-00990 Open

Large subsets of the grid with no five points on a plane

CO · conics · discrete geometry · general position · lattice points

OP-00991 Open

Zeros of cosine polynomials with 0-1 coefficients

CA · cosine polynomials · Littlewood's problems · zeros of trigonometric polynomials

OP-00992 Open

An asymptotic formula for the number of n-queens configurations

CO · asymptotic enumeration · entropy methods · n-queens problem · permutations

OP-00993 Open

Bounded tensor length for group factors of property (T) groups

OA · group von Neumann algebras · Peterson's operator algebra problems · property (T) · tensor product decompositions

OP-00994 Open

Outer automorphisms of group factors of hyperbolic property (T) groups

OA · hyperbolic groups · outer automorphism groups · Peterson's operator algebra problems · property (T) · W*-rigidity

OP-00995 Open

Fundamental groups of ultrapowers of II₁ factors

OA · continuous model theory · fundamental group of II₁ factors · Peterson's operator algebra problems · ultrapowers

OP-00996 Open

A II₁ factor whose ultrapowers differ from those of its opposite

OA · elementary equivalence · opposite algebras · Peterson's operator algebra problems · ultrapowers

OP-00997 Open

Peterson's conjecture on property (T) subfactors of ultrapowers of II₁ factors with the Haagerup property

OA · Haagerup property · Peterson's operator algebra problems · property (T) · ultrapowers

OP-00998 Open

Embeddings into matrix ultraproducts with trivial relative commutant

OA · full factors · matrix algebras · Peterson's operator algebra problems · relative commutants · ultraproducts

OP-00999 Open

Existential embedding of L(F₂) into a matrix ultraproduct

OA · continuous model theory · existential embeddings · free group factors · Peterson's operator algebra problems · ultraproducts

OP-01000 Open

Is L(SL₃(ℤ)) pseudomatricial?

OA · matrix algebras · Peterson's operator algebra problems · property (T) · pseudomatricial factors · ultraproducts