Mathematics

Transcendence of the Riemann

\zeta

function at odd positive integers10 questions

Determining the $\mathbb{Q}$ -dimension of the span of $1,\zeta(3),\zeta(5),\ldots,\zeta(s)$ : linear independence of odd zeta values
Open for about 48 years
Is Apéry's constant $\zeta(3)$ transcendental, and are $\zeta(3)$ and $\pi$ algebraically independent?
Open for about 47 years
Irrationality of odd zeta values: is ζ(5) irrational, and how many of ζ(3), ζ(5), …, ζ(s) are irrational?
Open for about 47 years
The Sondow–Zlobin conjecture on triangle integrals: prove that $I_n$ is never a rational multiple of $\zeta(n+2)$ for any integer $n>2$
Open for about 19 years
Irrationality of $\zeta(5)$
Open for about 47 years
Irrationality of the Riemann zeta values at odd integers: is ζ(s) irrational for any fixed odd s ≥ 5?
Open for about 48 years
Arithmetic nature of the odd Riemann zeta values: irrationality of $\zeta(2m+1)$ for $m\geq 2$
Open for about 47 years
Closed forms for odd zeta values: whether $\zeta(2n+1)$ is a rational multiple of $\pi^{2n+1}$
Open for about 47 years
Improving the lower bound for the number of irrational odd zeta values
Open for about 24 years
Irrationality of individual odd zeta values: is ζ(5) irrational?
Open for about 48 years

Riemann Conjecture6 questions

Lie algebras and their classification of nilpotent elements in dual spaces7 questions

Branching laws for representations of Lie groups5 questions

Classification of unitary representations of Lie groups8 questions

Decoding Problems of Reed-Solomon Code7 questions

Determining the cardinality of the continuum5 questions

Quantum Layer Conjecture7 questions

The quantum minimal model conjecture7 questions

Quantum Yang-Mills Millennium Problem4 questions

List Coloring Conjecture9 questions

Existence of a complex structure on the six-dimensional sphere6 questions

Does the travelling salesman problem admit an approximation ratio below 1.5?7 questions

The travelling salesman problem7 questions

The Rohlin problem8 questions

Several problems in Rota-Baxter algebra5 questions

Is the Fitting subgroup of a solvable group satisfying the minimal condition on normal subgroups nilpotent?3 questions

Existence of infinitely many solutions of nonlinear elliptic equations without symmetry7 questions

Semiprimeness of smash products of module algebras10 questions

Kac-Weisfeiler Conjecture in Modular Lie Algebra Representation Theory8 questions

Singularities induced by characteristic vectors in quasilinear hyperbolic systems6 questions

Computation and additivity of knot crossing numbers7 questions

Computation and additivity of unknotting numbers6 questions

Transcendence of Euler's constant10 questions

Well-posedness of global

L^{\infty}

weak solutions of the Euler equations8 questions

Schoen's conjecture on harmonic maps of the Poincaré disk4 questions

Penrose's Conjecture6 questions

Poincaré's problem on degrees of invariant curves of planar vector fields5 questions

Homogeneous Einstein manifolds8 questions

Resolution of singularities7 questions

Problem of singular cardinals4 questions

Embedding conjecture9 questions

Artificial boundary method for solving nonlinear partial differential equations in unbounded domains7 questions

Optimal methods for solving linear algebraic systems6 questions

Rigidity of spheres and algebraic domains6 questions

Raising Pieri-type formulas for zonal functions7 questions

Determining the Q\mathbb{Q}Q-dimension of the span of 1,ζ(3),ζ(5),…,ζ(s)1,\zeta(3),\zeta(5),\ldots,\zeta(s)1,ζ(3),ζ(5),…,ζ(s): linear independence of odd zeta values

Is Apéry's constant ζ(3)\zeta(3)ζ(3) transcendental, and are ζ(3)\zeta(3)ζ(3) and π\piπ algebraically independent?

Irrationality of odd zeta values: is ζ(5) irrational, and how many of ζ(3), ζ(5), …, ζ(s) are irrational?

The Sondow–Zlobin conjecture on triangle integrals: prove that InI_nIn​ is never a rational multiple of ζ(n+2)\zeta(n+2)ζ(n+2) for any integer n>2n>2n>2

Irrationality of ζ(5)\zeta(5)ζ(5)

Irrationality of the Riemann zeta values at odd integers: is ζ(s) irrational for any fixed odd s ≥ 5?

Arithmetic nature of the odd Riemann zeta values: irrationality of ζ(2m+1)\zeta(2m+1)ζ(2m+1) for m≥2m\geq 2m≥2

Closed forms for odd zeta values: whether ζ(2n+1)\zeta(2n+1)ζ(2n+1) is a rational multiple of π2n+1\pi^{2n+1}π2n+1

Improving the lower bound for the number of irrational odd zeta values

Irrationality of individual odd zeta values: is ζ(5) irrational?

Open analytical questions for Voros's discretized Keiper–Li sequence: transseries asymptotics, all-nnn positivity, and remainder bounds

The zeta spiral on the critical line: denseness of the values of ζ(1/2+it)\zeta(1/2+it)ζ(1/2+it) in C\mathbb{C}C and the unconditional curvature sign

The Alternative Hypothesis for zeta-zero pairs: determinacy of the limiting densities p_{k/2} under AH-Pairs alone

The missing direction of the Soundararajan and Radziwiłł equivalence between small gaps of zeros of the Riemann zeta-function and the horizontal distribution of zeros of ζ′(s), and the conjectured small-v asymptotics of m(v) and m′(v)

Denseness of the values of log⁡ζ(1/2+it)\log\zeta(1/2+it)logζ(1/2+it) on the critical line: the remaining case of the Bohr–Courant value-distribution problem

Unconditional discrete second moments of the Riemann zeta-function along polynomial shifts: closing the abscissa gap to the line σ=1/2\sigma=1/2σ=1/2

The normal-form map f↦wff\mapsto w_ff↦wf​ versus the Kazhdan–Lusztig map, and the even-depth analogue of the minimal reducing subalgebra theorem for nilpotent elements

Canonical forms for coadjoint orbits of maximal unipotent subgroups: Surkov's Conjecture 3.8 for root systems beyond G_2 and F_4

General classification of nilpotent orbits in real graded semisimple Lie algebras: completing the real carrier-algebra programme

Classification of coadjoint orbits of maximal dimension for Borel nilradicals in types BnB_nBn​, DnD_nDn​ and E8E_8E8​

Nilpotent orbit finiteness, separability, and Howe's conjecture for non-split reductive groups over local fields of positive characteristic

Spaltenstein's conjecture: every stabiliser sheet of a reductive Lie algebra contains a unique nilpotent orbit, in arbitrary characteristic

Coincidence of the two Springer correspondences for the dual of a classical Lie algebra in characteristic 2 (Xue's Remark 5.19 expectation)

Packet-level multiplicity-one for the branching of O(p,q)O(p,q)O(p,q) to O(p−1,q)O(p-1,q)O(p−1,q) beyond regular parameters, small-rank signatures, and a Gross–Prasad-type law for non-tempered packets

Classification of the triples (G,πλG,ϕ)(G,\pi_\lambda^G,\phi)(G,πλG​,ϕ) with admissible restriction of a discrete series representation to an embedded SU(2,1)SU(2,1)SU(2,1)

Finite multiplicity of the continuous spectrum in discrete series branching laws: a per-representation criterion and the finite-infinite boundary for symmetric pairs beyond the real-spherical class

Kobayashi's conjecture on admissible restrictions: does G′G'G′-admissibility imply K′K'K′-admissibility?

Blattner-type multiplicity formula for discretely decomposable restrictions of Zuckerman modules Aq(λ)A_{\mathfrak q}(\lambda)Aq​(λ) to symmetric subgroups (Ørsted–Speh conjecture)

Positive energy versus semiboundedness for irreducible unitary representations of full hermitian Lie groups

The unitary duals of the complex exceptional simple Lie groups F4F_4F4​, E6E_6E6​, E7E_7E7​, and E8E_8E8​

General super imprimitivity theorem: Mackey induction and systems of imprimitivity for super homogeneous spaces with odd directions

Unitary dual of Drinfeld doubles of q-deformations of compact semisimple Lie groups beyond the almost-real regime

The Salamanca-Riba–Vogan unitarily-small infinitesimal-character conjecture for real reductive groups beyond U(p,q)U(p,q)U(p,q)

Structural theory of the FS-scattered part of the Dirac series for real exceptional Lie groups: u-small K-types, unipotent representations, and cancellation in the Dirac index

Making the FPP description of the unitary dual of real reductive groups algorithmically effective

Algebraic constructions of first-reduction-point and exceptional unitary highest weight modules

Hardware realization of quasi-linear FFT-based Reed–Solomon decoders (the modular-approach MA/FDMA/FMA family)

Low-complexity algebraic soft-decision decoding of Reed–Solomon codes: closing the performance–complexity gap of Koetter–Vardy decoding

Combinatorial List-Size Classification of Linearized Reed–Solomon Codes Across All Sum-Rank Parameter Regimes

Efficient decoding, list-decodability, and dual insdel distance of Reed–Solomon codes under insertion–deletion errors

Efficient list-decoding of Reed-Solomon codes at the generalized Singleton radius, and efficient certification of optimal list-decodability

Explicit Reed–Solomon codes achieving capacity-level list-decodability over small fields

Proximity gaps for Reed–Solomon codes above the Johnson radius: list- and line-decodability over prime fields up to the Elias radius

Consistency strength of the higher Solovay configuration at singular cardinals of countable cofinality: measurability of κ+\kappa^+κ+ in L(P(κ))L(\mathcal{P}(\kappa))L(P(κ)) and the ℵω\aleph_\omegaℵω​-perfect set property

Woodin's Ω Conjecture: completeness of Ω-logic under a proper class of Woodin cardinals

Optimal large-cardinal strength for blowing up the power of a singular cardinal of uncountable cofinality that is singular in the core model

Optimal hypotheses for Easton's theorem at measurable and strong cardinals

The Revision Hypothesis: must every limit cardinal be a revision cardinal?

Bound states of the Dirichlet Laplacian in the Fichera layer and generalized Fichera layers: exact counts

Geometric characterization of the discrete spectrum of quantum layers and their nonsmooth analogues: bound states, thickness uniformity, and the converse problem

Discrete spectrum of curved quantum layers: the Lu–Rowlett open problems on uniform thin-layer existence, the positive-total-Gauss-curvature domination inequality, and the thick-layer dichotomy

Empty discrete spectrum of generalized quantum tubes: the converse problem beyond thin layers and surfaces

Width-uniform existence of bound states in hard-wall quantum layers over finite-total-curvature surfaces

Curvature-induced bound states in quantum layers over non-parabolic hypersurfaces in dimensions four and higher

The quantum layer conjecture: bound states in hard-wall layers over nonplanar asymptotically flat surfaces with finite total curvature

Invariance and decomposition of quantum cohomology under flips and birational maps between minimal models

Reduction conjecture for arbitrary smooth GIT quotients and quantum D-module decomposition for arbitrary standard flips

Effective computation of quantum cohomology across conifold transitions: a blow-up formula along Weil divisors

Does blowing up a Fano manifold along a smooth Fano center strictly increase the spectral radius of quantum multiplication by the first Chern class (Conjecture 1.1 of Hu–Ke–Li–Song)?

Globalizing the quantum-flips F-embedding: big quantum cohomology under general global simple (r,r′) flips

Establish the GGG-equivariant noncommutative minimal model program: quasi-convergent paths of equivariant stability conditions from the equivariant quantum differential equation

Blow-up compatibility of reflection vectors in quantum cohomology: convergence of the quantum cohomology of Bl(X) and the reflection-vector property for pullbacks of exceptional objects

Extending the provable area-law regime for lattice Yang-Mills: the linear-in-N ('t Hooft) regime with uniform string tension, and fixed small N at large coupling

Deconfinement for trivial-center lattice Yang–Mills in the strong-coupling regime and mass-gap criteria beyond the Bakry–Émery approach

Deconfinement of trivial-center lattice Yang-Mills theories: uniform perimeter law for all strong-coupling beta, extensions to SO(2N+1) and general representations, the weak-coupling regime, and the large-N reconciliation

Yang–Mills existence and mass gap: a fully quantized, background-independent four-dimensional theory with a gauge-invariant positive mass gap

The List Equitable Total Coloring Conjecture

Enumerative chromatic-choosability of joins with complete graphs (the enumerative Ohba conjecture of Allred and Mudrock)

The Proportional Choice Number of Complete Bipartite and Complete Multipartite Graphs: Exact Value, Classification, and Asymptotics

Is every line graph {1⋆(χ′(G)−3), 3}\{1\star(\chi'(G)-3),\,3\}{1⋆(χ′(G)−3),3}-choosable?

Generalized Ohba conjecture for r-uniform hypergraphs: chromatic-choosability at high chromatic number

The List Coloring Conjecture: is the list chromatic index of every multigraph equal to its chromatic index?

List and correspondence colouring reconfiguration conjectures: optimal diameters n(G)+μ(G)n(G)+\mu(G)n(G)+μ(G) and n(G)+τ(G)n(G)+\tau(G)n(G)+τ(G)

Paintability of balanced complete multipartite graphs with parts of size three: is χP(K3⋆n)=⌈(4n−1)/3⌉\chi_P(K_{3\star n})=\lceil(4n-1)/3\rceilχP​(K3⋆n​)=⌈(4n−1)/3⌉ for every nnn, and what is χP(K3⋆7)\chi_P(K_{3\star 7})χP​(K3⋆7​)?

Open problems on the λ-choosability of planar graph families from Zhu's refinement of choosability

Integrability of the Clifford-induced almost complex structure on level hypersurfaces of OT-FKM isoparametric families

The Hopf problem in generalized geometry: existence of an integrable generalized complex structure on the six-sphere S6\mathbb{S}^6S6

Hopf's problem: does the six-sphere admit a complex structure?

Vanishing and Chern-class constraints for compact complex threefolds with b2=0b_2=0b2​=0 and holomorphic algebraic reduction in full generality

Complex-structure rigidity without the Kähler hypothesis for the two exceptional G2G_2G2​ flag manifolds QQQ and ZZZ

Determining the $\mathbb{Q}$ -dimension of the span of $1,\zeta(3),\zeta(5),\ldots,\zeta(s)$ : linear independence of odd zeta values

Is Apéry's constant $\zeta(3)$ transcendental, and are $\zeta(3)$ and $\pi$ algebraically independent?

The Sondow–Zlobin conjecture on triangle integrals: prove that $I_n$ is never a rational multiple of $\zeta(n+2)$ for any integer $n>2$

Irrationality of $\zeta(5)$

Arithmetic nature of the odd Riemann zeta values: irrationality of $\zeta(2m+1)$ for $m\geq 2$

Closed forms for odd zeta values: whether $\zeta(2n+1)$ is a rational multiple of $\pi^{2n+1}$

Open analytical questions for Voros's discretized Keiper–Li sequence: transseries asymptotics, all- $n$ positivity, and remainder bounds

The zeta spiral on the critical line: denseness of the values of $\zeta(1/2+it)$ in $\mathbb{C}$ and the unconditional curvature sign

Denseness of the values of $\log\zeta(1/2+it)$ on the critical line: the remaining case of the Bohr–Courant value-distribution problem

Unconditional discrete second moments of the Riemann zeta-function along polynomial shifts: closing the abscissa gap to the line $\sigma=1/2$

The normal-form map $f\mapsto w_f$ versus the Kazhdan–Lusztig map, and the even-depth analogue of the minimal reducing subalgebra theorem for nilpotent elements

Classification of coadjoint orbits of maximal dimension for Borel nilradicals in types $B_n$ , $D_n$ and $E_8$

Packet-level multiplicity-one for the branching of $O(p,q)$ to $O(p-1,q)$ beyond regular parameters, small-rank signatures, and a Gross–Prasad-type law for non-tempered packets

Classification of the triples $(G,\pi_\lambda^G,\phi)$ with admissible restriction of a discrete series representation to an embedded $SU(2,1)$

Kobayashi's conjecture on admissible restrictions: does $G'$ -admissibility imply $K'$ -admissibility?

Blattner-type multiplicity formula for discretely decomposable restrictions of Zuckerman modules $A_{\mathfrak q}(\lambda)$ to symmetric subgroups (Ørsted–Speh conjecture)

The unitary duals of the complex exceptional simple Lie groups $F_4$ , $E_6$ , $E_7$ , and $E_8$

The Salamanca-Riba–Vogan unitarily-small infinitesimal-character conjecture for real reductive groups beyond $U(p,q)$

Consistency strength of the higher Solovay configuration at singular cardinals of countable cofinality: measurability of $\kappa^+$ in $L(\mathcal{P}(\kappa))$ and the $\aleph_\omega$ -perfect set property

Establish the $G$ -equivariant noncommutative minimal model program: quasi-convergent paths of equivariant stability conditions from the equivariant quantum differential equation

Is every line graph $\{1\star(\chi'(G)-3),\,3\}$ -choosable?

List and correspondence colouring reconfiguration conjectures: optimal diameters $n(G)+\mu(G)$ and $n(G)+\tau(G)$

Paintability of balanced complete multipartite graphs with parts of size three: is $\chi_P(K_{3\star n})=\lceil(4n-1)/3\rceil$ for every $n$ , and what is $\chi_P(K_{3\star 7})$ ?

The Hopf problem in generalized geometry: existence of an integrable generalized complex structure on the six-sphere $\mathbb{S}^6$

Vanishing and Chern-class constraints for compact complex threefolds with $b_2=0$ and holomorphic algebraic reduction in full generality

Complex-structure rigidity without the Kähler hypothesis for the two exceptional $G_2$ flag manifolds $Q$ and $Z$

Constant-Factor Approximability of the Traveling Salesman Problem with Line Neighborhoods in $\mathbb{R}^3$

Rohlin's multiple mixing problem for $\mathbb{Z}$ -actions: does mixing imply mixing of all orders?

The Rohlin problem: does mixing imply mixing of all orders for measure-preserving $\mathbb{Z}$ -actions?

Rohlin's problem: does strong mixing imply mixing of all orders for measure-preserving $\mathbb{Z}$ -actions?

Rota-Baxter indices of upper-triangular and semisimple matrix algebras and the derived-algebra nilpotency bound for $M_n(F)$

Infinitely many solutions of the Dirichlet $p$ -Laplacian with nonzero boundary data throughout the subcritical range

Persistence of infinitely many solutions for the perturbed elliptic system $-\Delta u=v^{p-1}$ , $-\Delta v=f(u)+h$ in the full subcritical range (Salvatore's open problems)

The super Kac-Weisfeiler divisibility property for the strange classical Lie superalgebras $q(n)$ and $\tilde p(n)$

Tikaradze's Kac–Weisfeiler-type conjecture for filtered algebras in positive characteristic: $i(\chi)\ge\tfrac{1}{2}s(\chi)$

Minimal-dimensional representations of finite $W$ -superalgebras and attainability of the super Kac–Weisfeiler bounds (Zeng–Shu Conjecture 0.3)

No shock formation for totally linearly degenerate quasilinear hyperbolic systems: the $C^0\Rightarrow C^1$ conjecture beyond block-structured classes