clopen.solutions

Open problems

1131 registered problems.

OP-01001 Open
Popa's free flip problem
OA · automorphism groups · free products · interpolated free group factors · Peterson's operator algebra problems
OP-01002 Open
Biexactness of the bidual of the reduced C*-algebra of F₂
OA · biexactness · enveloping von Neumann algebras · Peterson's operator algebra problems · reduced group C*-algebras
OP-01003 Open
Discrete generating copies of F₂ in unitary groups of II₁ factors
OA · discrete subgroups · free group factors · Peterson's operator algebra problems · unitary groups
OP-01004 Open
Popa's tensor flip problem for II₁ factors
OA · automorphism groups · hyperfinite II₁ factor · Peterson's operator algebra problems · tensor products
OP-01005 Open
A mixing action whose crossed product has more than one Cartan subalgebra up to unitary conjugacy
OA · Cartan subalgebras · group measure space construction · mixing actions · Peterson's operator algebra problems
OP-01006 Open
Unique Cartan subalgebras for groups with nonzero first ℓ²-Betti number
OA · Cartan subalgebras · C-rigidity · ℓ²-Betti numbers · Peterson's operator algebra problems
OP-01007 Open
A II₁ factor with exactly two Cartan subalgebras
OA · Cartan subalgebras · II₁ factors · Peterson's operator algebra problems · unitary conjugacy
OP-01008 Open
Sharpness of the logarithmic factor in the higher-order Cheeger–Steklov inequality
SP · Cheeger inequality · Colbois–Girouard–Gordon–Sher Steklov problems · Steklov eigenvalues
OP-01009 Open
Existence of a metric maximising a higher normalised Steklov eigenvalue
SP · Colbois–Girouard–Gordon–Sher Steklov problems · eigenvalue optimisation · Riemannian surfaces · Steklov eigenvalues
OP-01010 Open
Embeddedness of first Steklov maximisers with many boundary components
DG · Colbois–Girouard–Gordon–Sher Steklov problems · eigenvalue optimisation · free boundary minimal surfaces · Steklov eigenvalues
OP-01011 Open
Steklov critical metrics versus critical metric–density pairs
SP · Colbois–Girouard–Gordon–Sher Steklov problems · critical metrics · harmonic maps · Steklov eigenvalues
OP-01012 Open
Uniform comparison of all Steklov eigenvalues with those of a discretisation
SP · Colbois–Girouard–Gordon–Sher Steklov problems · discretisation · graphs with boundary · Steklov eigenvalues
OP-01013 Open
Does the Steklov spectrum detect boundary singularities of orbifolds?
SP · Colbois–Girouard–Gordon–Sher Steklov problems · inverse spectral problems · orbifolds · Steklov eigenvalues
OP-01014 Open
Does the Steklov spectrum detect interior singularities of orbifolds?
SP · Colbois–Girouard–Gordon–Sher Steklov problems · inverse spectral problems · orbifolds · Steklov eigenvalues
OP-01015 Open
Can a mixed Steklov–Neumann problem be isospectral to a pure Steklov problem?
SP · Colbois–Girouard–Gordon–Sher Steklov problems · isospectral manifolds · orbifolds · Steklov eigenvalues
OP-01016 Open
Does the Steklov spectrum determine the Laplace spectrum of the boundary?
SP · Colbois–Girouard–Gordon–Sher Steklov problems · inverse spectral problems · isospectral manifolds · Steklov eigenvalues
OP-01017 Open
Steklov isospectral manifolds that are not Neumann isospectral
SP · Colbois–Girouard–Gordon–Sher Steklov problems · isospectral manifolds · Steklov eigenvalues
OP-01018 Open
Steklov isospectral manifolds that are not strongly Steklov isospectral
SP · Colbois–Girouard–Gordon–Sher Steklov problems · isospectral manifolds · Robin problem · Steklov eigenvalues
OP-01019 Open
Curvature properties determined by the Steklov spectrum in dimension three and higher
SP · Colbois–Girouard–Gordon–Sher Steklov problems · inverse spectral problems · spectral invariants · Steklov eigenvalues
OP-01020 Open
Steklov spectra on forms of different degrees for manifolds
SP · Colbois–Girouard–Gordon–Sher Steklov problems · differential forms · isospectral manifolds · Steklov eigenvalues
OP-01021 Open
A Pleijel theorem for Steklov eigenfunctions in dimension three and higher
SP · Colbois–Girouard–Gordon–Sher Steklov problems · nodal domains · Pleijel theorem · Steklov eigenfunctions
OP-01022 Partially resolved
Large normalised first Steklov eigenvalue with the boundary metric fixed
SP · Colbois–Girouard–Gordon–Sher Steklov problems · eigenvalue bounds · prescribed boundary · Steklov eigenvalues
OP-01023 Partially resolved
Does the first conformal Steklov eigenvalue exceed 2π off the disk?
SP · Colbois–Girouard–Gordon–Sher Steklov problems · conformal eigenvalues · Riemannian surfaces · Steklov eigenvalues
OP-01024 Partially resolved
The Kusner–McGrath conjecture on first Steklov eigenspaces of free boundary minimal hypersurfaces
DG · Colbois–Girouard–Gordon–Sher Steklov problems · Fraser–Li conjecture · free boundary minimal surfaces · Steklov eigenvalues
OP-01025 Open
Strict monotonicity of normalised conformal Steklov eigenvalues
SP · Colbois–Girouard–Gordon–Sher Steklov problems · conformal eigenvalues · Steklov eigenvalues
OP-01026 Open
Steklov asymptotics of curvilinear polygons with reflex angles
SP · Colbois–Girouard–Gordon–Sher Steklov problems · polygons · spectral asymptotics · Steklov eigenvalues
OP-01027 Open
Recovering the angles of a curvilinear polygon from its Steklov spectrum
SP · Colbois–Girouard–Gordon–Sher Steklov problems · inverse spectral problems · polygons · Steklov eigenvalues
OP-01028 Partially resolved
The Erdős–Moser problem: a large subset avoiding its own restricted sums in A
NT · additive combinatorics · Erdős–Moser problem · restricted sumsets · sum-free sets
OP-01029 Solution claimed
Open subsets of [0, 1] of measure above 1/3 contain a solution of xy = z
CA · additive combinatorics · measure theory · product-free sets · sum-free sets
OP-01030 Solution claimed
The largest sum-free subset of the lattice cube
CO · extremal combinatorics · lattice points · sum-free sets
OP-01031 Partially resolved
Four-term progressions in Fourier-uniform sets
CO · arithmetic progressions · counting lemmas · Fourier uniformity · higher-order Fourier analysis
OP-01032 Partially resolved
Large subsets of [N] with no solution of x + 3y = 2z + 2w in distinct integers
NT · linear equations in sets · Ruzsa's problem · solution-free sets
OP-01033 Partially resolved
The largest number of affine copies of {0, 1, 3} in a set of n integers
CO · affine copies · extremal combinatorics · linear equations in sets
OP-01034 Partially resolved
The smallest set modulo a prime with no uniquely represented sum
NT · cyclic groups of prime order · sumsets · unique representations
OP-01035 Partially resolved
Independent summands of a uniformly distributed sum
PR · 0-1 polynomials · sums of independent random variables · uniform distribution
OP-01036 Open
How many rotated pyjama sets cover the plane
CO · coverings · pyjama problem · quantitative bounds · topological dynamics
OP-01037 Partially resolved
A partition of the Boolean cube with no structured sumset
CO · additive combinatorics · finite field model · Hamming balls · partitions · sumsets
OP-01038 Solution claimed
Talagrand's convexity problem for Gaussian measure
PR · convex sets · Gaussian measure · infinite-dimensional probability · Minkowski sums
OP-01039 Open
Every residue modulo a prime as a product of two smaller primes
NT · multiplicative number theory · primes in residue classes · products of primes
OP-01040 Solution claimed
Positive density of Hofstadter's ab − 1 set
NT · density · Erdős problems · integer sequences · multiplicative recurrences
OP-01041 Partially resolved
The big-line-big-clique conjecture
CO · collinear points · discrete geometry · Ramsey theory · visibility graphs
OP-01042 Solution claimed
Five points of a codimension-two surface on a two-plane
MG · finite geometry of submanifolds · Hilbert cubes · plane sections · Thom's theorem
OP-01043 Partially resolved
Chowla's cosine problem
CA · Chowla's cosine problem · cosine polynomials · minimum of trigonometric sums
OP-01044 Partially resolved
Permutations fixing a set of size k
CO · asymptotic enumeration · invariant sets · multiplication table problem · random permutations
OP-01045 Solution claimed
Random polynomials with 0-1 coefficients are irreducible
NT · 0-1 polynomials · irreducibility · Odlyzko–Poonen conjecture · random polynomials
OP-01046 Solution claimed
An affine copy of the dyadic sequence in every set of positive measure
CA · affine copies · Erdős problems · Erdős similarity problem · Lebesgue measure
OP-01047 Solution claimed
Is every group sofic, or hyperlinear?
GR · approximate homomorphisms · Higman group · hyperlinear groups · sofic groups
OP-01048 Solution claimed
Kadison's generator problem for von Neumann algebras
OA · generator problem · Peterson's operator algebra problems · single generation · von Neumann algebras
OP-01049 Solution claimed
Is every von Neumann algebra with property (T) singly generated?
OA · generator problem · Peterson's operator algebra problems · property (T) · single generation
OP-01050 Solution claimed
The free group factor problem
OA · free group factors · II₁ factors · isomorphism problem · Peterson's operator algebra problems