Mathematics

Conjecture on the quantum cohomology ring of

K

-equivalent algebraic varieties5 questions

Monodromy of the crepant resolution identification versus derived autoequivalences for type A surface singularities
Open for about 19 years
Forbes–Jinzenji conjecture: $QH^*(\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(k)\oplus\mathcal{O}(-2-k)))\cong QH^*(\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)))$ for all integers $k$
Open for about 20 years
Quantum flips after Lee–Lin–Wang: globalizing the quantum-cohomology embedding under simple flips, and the accompanying singularity, uniqueness, linearity, and Cayley-combinatorics questions
Open for about 6 years
Ruan–Wang K-equivalence conjecture for Gromov–Witten theory: all-genus invariance for ordinary flops, decomposition of K-equivalent maps, and global convergence of the quantum potential
Open for about 25 years
Directed double-point degeneration: an algebraic-cobordism route to quantum-ring invariance under ordinary flops (Question 8.3.1 of Lee–Lin–Qu–Wang)
Open for about 12 years

A series of generalization patterns for an important congruence on semigroups5 questions

Erdős's problem on covering systems with distinct moduli8 questions

Erdős-Turán conjecture on reciprocal sums and divergent sequences6 questions

The conjecture on commutative equivalence of codes to prefix codes8 questions

Snevily conjecture on Abelian groups of odd order4 questions

Conjecture on holomorphic bisectional curvature6 questions

Decision problem of finite codes having finite completions9 questions

The Drinfeld–Vlăduţ bound on the number of points on curves over finite fields5 questions

Seifert's conjecture on the multiplicity of brake orbits5 questions

Two embedding problems of regular semigroups5 questions

Sullivan conjecture on smooth complex complete intersections6 questions

Generalized metric space problem4 questions

Two conjectures concerning generalized tilting modules8 questions

Generalized Smale conjecture4 questions

Hartshorne conjecture10 questions

Invariant tori near equilibrium points of Hamiltonian systems7 questions

Hunt hypothesis and Getoor conjecture8 questions

Hopf conjecture3 questions

Hopf problem4 questions

Certifying global optimality using sums of squares of rational polynomials5 questions

Minimal tower problem7 questions

Problem of commutation between tight closure and localization8 questions

Closed geodesic conjecture on compact manifolds7 questions

Estimation of classical Ramsey functions1 questions

The optimal constant for Ramsey goodness of cycles versus general graphs
Open for about 8 years

Finiteness problem of associated prime ideals of local cohomology modules7 questions

Topological similarity problem of matrices5 questions

Kaplansky's sixth conjecture5 questions

Cayley graphs and related problems7 questions

Cells of Coxeter groups7 questions

Existence of large sets of Kirkman triple systems10 questions

Transversal problem of Latin squares8 questions

The Ramras problem6 questions

Langlands Program9 questions

Gauss's conjecture on real quadratic fields of class number one8 questions

Discrete logarithm solving problems6 questions

Monodromy of the crepant resolution identification versus derived autoequivalences for type A surface singularities

Quantum flips after Lee–Lin–Wang: globalizing the quantum-cohomology embedding under simple flips, and the accompanying singularity, uniqueness, linearity, and Cayley-combinatorics questions

Ruan–Wang K-equivalence conjecture for Gromov–Witten theory: all-genus invariance for ordinary flops, decomposition of K-equivalent maps, and global convergence of the quantum potential

Directed double-point degeneration: an algebraic-cobordism route to quantum-ring invariance under ordinary flops (Question 8.3.1 of Lee–Lin–Qu–Wang)

Open problems on quasi-absorbing elements in semigroups: order generation by quasi-primitive elements, semigroup-level duality, and Rees-congruence reduction

The congruence lattice of NI(X) for the exceptional finite sets ∣X∣=3|X|=3∣X∣=3 and ∣X∣=5|X|=5∣X∣=5

Congruences, embeddings, and automorphisms of the generalized bicyclic semigroups BωFB_\omega^{\mathscr{F}}BωF​ for arbitrary ω\omegaω-closed families F\mathscr{F}F

An intrinsic characterization of Condition (K) for inverse semigroups and their tight C*-algebras

DSC semigroups: which groups are DSC, and is the DSC class closed under direct products?

Minimal least common multiple of a distinct covering system with given minimum modulus: the open values Lmin⁡(m)L_{\min}(m)Lmin​(m) for m≥8m\geq 8m≥8

The weight-function threshold in the Erdős covering problem: which multiplicative weights force a positive density of uncovered integers

Sharp values of the k-th smallest modulus bound B(k) for covering systems with distinct moduli, and Selfridge's problem on avoiding the modulus 2 in the squarefree case

Explicit effective bounds for covering systems of the polynomial ring Fq[x]\mathbb{F}_q[x]Fq​[x]: making the minimum-degree constants of Li–Wang–Wang–Yi quantitative

Nonexistence of distinct covering systems with all moduli in [n,12n][n,12n][n,12n] for n≥4n \geq 4n≥4 (the Dalton–Jones interval problem)

Minimum multiplicity of the repeated modulus 9 in covering systems with odd moduli: is t9≤2t_9\le 2t9​≤2, and is the residue arrangement {4,5,r}(mod9)\{4,5,r\}\pmod 9{4,5,r}(mod9) realizable?

A uniform constant-factor improvement on tp≤p−ct_p \le p-ctp​≤p−c for covering systems with one repeated odd prime modulus (Question 5.3 of Harrington–Sun–Wong)

Existence of infinite exact distinct saturated covering systems with minimum modulus at least 5

Graham's conjecture: sets of lattice points with divergent reciprocal-square sum must contain infinitely many axis-parallel squares

Erdős's conjecture on arithmetic progressions (Jensen's zero-solution-set formulation): does every set of natural numbers with divergent reciprocal sum contain arbitrarily long arithmetic progressions?

Erdős conjecture on arithmetic progressions: does every set of positive integers with divergent reciprocal sum contain arbitrarily long arithmetic progressions?

Finiteness and value of M4M_4M4​, the supremum of reciprocal sums of sets of integers with no 4-term arithmetic progression

Erdős's reciprocal-sum conjecture for arithmetic progressions: the open cases k≥4k \ge 4k≥4

Erdős's harmonic-set (divergent reciprocal sum) conjecture: must every set of natural numbers whose reciprocals sum to infinity contain arbitrarily long arithmetic progressions?

Cardinality control for prefix-free codes preserving weighted occurrence sums: Kraizberg's weak commutative-equivalence approximation

Bounded cardinality discrepancy between a finite code and a weight-equivalent prefix-free code

Complete bayonet codes and finite maximal codes: the strong triangle conjecture beyond Hajós numbers and the realization problem

Companion Hajós factorizations for finite maximal codes: extending De Felice's Ω(n)≤2 theorem, the existence of non-Hajós codes, and the Zhang–Shum pair correspondence

Strong triangle conjecture for finite maximal codes and the embeddability of Shor's counterexample

The Schützenberger factorization conjecture for finite maximal codes and the associated finite-completion decision problems

Commutative regularity of Kleene closures of bounded semi-linear languages: extending Carpi–D'Alessandro's Theorem 1 beyond codes and the two-distinct-letters condition

Deciding embeddability of finite codes into finite maximal codes, and the open instance of Shor's counterexample code

A purely combinatorial proof of Snevily's conjecture on transversals of Cayley addition tables of odd-order abelian groups

The Kézdy–Snevily conjecture: f(n,2)=nf(n,2)=nf(n,2)=n for even nnn and f(n,2)>nf(n,2)>nf(n,2)>n for odd nnn (covering radius of sets of permutations in the Hamming permutation space)

The all-orders classification of subsquares of group multiplication tables without transversals (Müyesser–Pokrovskiy Conjecture 7.1)

Power-saving error term for the count of additive triples of bijections of a finite abelian group (toroidal semiqueens / complete mappings)

Extending the sharp dimension-estimate rigidity for polynomial-growth holomorphic functions to nonnegative holomorphic sectional curvature

Positive orthogonal holomorphic bisectional curvature on compact complex manifolds: is the Hermitian Frankel property rigid beyond the conformally Kähler setting?

Yau's uniformization conjecture for complete simply connected Kähler manifolds with negatively pinched sectional curvature

Does nonnegative holomorphic bisectional curvature imply nonnegative isotropic curvature on Kähler manifolds of complex dimension at least three?

Holomorphic convexity and polynomial-growth holomorphic functions on complete Kähler manifolds with nonpositive bisectional curvature and Euclidean volume growth (Question 1.10 of Wu–Yang)

Converse of the pseudoindex inequality for m-positive bisectional curvature on compact Kähler manifolds

Finite completions of codes within structured families: θ-invariant codes and strong alt-induced codes

Decidability of the inclusion problem for finite codes: deciding whether a finite code can be embedded in a finite maximal code

Good arrangements of the a∗wa∗a^*wa^*a∗wa∗-components of finite maximal codes and Krasner factorizations of cyclic groups

The embedding problem for the class of strong alt-induced codes: maximal extensions preserving regularity and finiteness

Decidability of the inclusion problem for finite codes: embedding a finite code into a finite maximal code

Finite completions of theta-invariant codes: which finite theta-invariant codes embed into a finite complete theta-invariant code?

Realizability of complete bayonet codes as slices CM(ω)C_M(\omega)CM​(ω) of finite maximal codes

Does Shor's bayonet code have a finite completion? The triangle conjecture for finite maximal codes

Completing codes with bounded deciphering delay inside submonoids of the free monoid

Attainable spectra of Tsfasman–Vlăduţ invariants of towers of global fields: infinite supports, order-rrr optimality over non-square finite fields, and minimal spectra

Exact limit of the Bassa–Beelen–Nguyen Drinfeld modular tower over an elliptic-curve coordinate ring

Improving the lower bounds for the Ihara constants A(2)A(2)A(2) and A(3)A(3)A(3) (toward determining A(q)A(q)A(q) for nonsquare finite fields)

Explicit optimal Drinfeld-modular towers beyond the polynomial ring: the remaining degree-two principal ideals

Closed-form Serre-type arithmetic refinements of Weil–Oesterlé bounds of order n≥3n\ge 3n≥3

Multiplicity of orthogonal geodesic chords in ball-diffeomorphic Riemannian manifolds with boundary focal points

Seifert's conjecture on the multiplicity of brake orbits in potential wells

At least n brake orbits for classical-type Hamiltonians: the generalized Seifert conjecture for disk-shaped potential wells

Hypothesis-free multiplicity of orthogonal geodesic chords in strongly concave Riemannian disks, and generic multiplicity among smooth metrics

Dispensability of the special-geodesic exclusion assumption in the multiplicity theorem for orthogonal geodesic chords in strongly concave Riemannian disks

Boone–Higman embeddings for semigroups: the word problem and finitely presented congruence-free monoids, with Maltcev's regular and bisimple embedding questions

Fundamental bisimple embeddings for solid regular and locally inverse semigroups (the Uniform Bands problem)

Problem 6.1 of Dékány–Szendrei–Szittyai: classifying the varieties of completely simple semigroups for which every E-solid locally inverse extension embeds into a λ-semidirect product

Explicit semiband embedding for semigroups whose regular D-classes are aperiodic semigroups

The Boone–Higman programme for semigroups: embedding finitely presented semigroups into finitely presented congruence-free, bisimple, and regular monoids

Diffeomorphism classification of smooth weighted complete intersections beyond the odd pairwise-coprime regime

The Sullivan Conjecture for complex projective complete intersections of dimension at least five: diffeomorphism classification beyond the homeomorphism theorem of Fang and Wang

Inertia groups of non-spin 4-dimensional complete intersections (Crowley–Nagy Conjecture 1.15)

Classification of smooth complete intersections by total degree, Pontrjagin classes, and Euler characteristic in complex dimension at least 8

The Sullivan conjecture for complete intersections: diffeomorphism classification in complex dimension at least 5

The Sullivan Conjecture for complete intersections of complex dimension at least 8

Existence and uniqueness of φ-fixed circles and φ-fixed discs for self-mappings on 𝔘-metric spaces (Özgür–Taş open problem)

Are the shortest-Hamiltonian-path, smallest-enclosing-ball, and entrywise ℓp\ell_pℓp​ matrix-norm functionals ggg-metrics of order n≥3n\ge 3n≥3 (for ℓp\ell_pℓp​, n≥2n\ge 2n≥2)?

Existence and homeomorphic classification of (m,n)(m,n)(m,n)-equidistant metric continua (Popvassilev's Problem 5.2 and Conjecture 5.1)

The open problems of Gaba-Agyingi-Leko on Chatterjea-type fixed point theory in partial b-metric spaces: sharp coefficient regime, T-stability, the P property, partial ultra-metric metrization, and the theta-phi-Chatterjea program

Submodule-closure of higher Ext-images En(RR)E_n(R_R)En​(RR​) over noetherian rings: Auslander's question for n≥3n \ge 3n≥3

Finiteness of injective dimension for (quasi-)Gorenstein generalized tilting modules over Artin algebras (Huang's Conjecture 4.7)

Tensorly faithful Wakamatsu tilting modules: classifying rings that force tensorly faithfulness and conditions weaker than the generalized Auslander-Reiten conjecture implying the Wakamatsu tilting conjecture

Wakamatsu Tilting Conjecture and Huang's Conjecture 4.7: finiteness and symmetry of the injective dimensions of generalized tilting modules over Artin algebras

The Wakamatsu tilting conjecture: is every Wakamatsu tilting module of finite projective dimension a classical tilting module?

The congruence lattice of NI(X) for the exceptional finite sets $|X|=3$ and $|X|=5$

Congruences, embeddings, and automorphisms of the generalized bicyclic semigroups $B_\omega^{\mathscr{F}}$ for arbitrary $\omega$ -closed families $\mathscr{F}$

Minimal least common multiple of a distinct covering system with given minimum modulus: the open values $L_{\min}(m)$ for $m\geq 8$

Explicit effective bounds for covering systems of the polynomial ring $\mathbb{F}_q[x]$ : making the minimum-degree constants of Li–Wang–Wang–Yi quantitative

Nonexistence of distinct covering systems with all moduli in $[n,12n]$ for $n \geq 4$ (the Dalton–Jones interval problem)

Minimum multiplicity of the repeated modulus 9 in covering systems with odd moduli: is $t_9\le 2$ , and is the residue arrangement $\{4,5,r\}\pmod 9$ realizable?

A uniform constant-factor improvement on $t_p \le p-c$ for covering systems with one repeated odd prime modulus (Question 5.3 of Harrington–Sun–Wong)

Finiteness and value of $M_4$ , the supremum of reciprocal sums of sets of integers with no 4-term arithmetic progression

Erdős's reciprocal-sum conjecture for arithmetic progressions: the open cases $k \ge 4$

The Kézdy–Snevily conjecture: $f(n,2)=n$ for even $n$ and $f(n,2)>n$ for odd $n$ (covering radius of sets of permutations in the Hamming permutation space)

Good arrangements of the $a^wa^$ -components of finite maximal codes and Krasner factorizations of cyclic groups

Realizability of complete bayonet codes as slices $C_M(\omega)$ of finite maximal codes

Attainable spectra of Tsfasman–Vlăduţ invariants of towers of global fields: infinite supports, order- $r$ optimality over non-square finite fields, and minimal spectra

Improving the lower bounds for the Ihara constants $A(2)$ and $A(3)$ (toward determining $A(q)$ for nonsquare finite fields)

Closed-form Serre-type arithmetic refinements of Weil–Oesterlé bounds of order $n\ge 3$

Are the shortest-Hamiltonian-path, smallest-enclosing-ball, and entrywise $\ell_p$ matrix-norm functionals $g$ -metrics of order $n\ge 3$ (for $\ell_p$ , $n\ge 2$ )?

Existence and homeomorphic classification of $(m,n)$ -equidistant metric continua (Popvassilev's Problem 5.2 and Conjecture 5.1)

Submodule-closure of higher Ext-images $E_n(R_R)$ over noetherian rings: Auslander's question for $n \ge 3$

When does the $\omega$ -torsionfree class $T_\omega(R)$ equal the generalized Gorenstein class $G_\omega(R)$ for a Wakamatsu tilting bimodule?

Cohomology of $B\mathrm{Diff}$ for connected sums of lens spaces beyond the generic non-diffeomorphic case

Hartshorne's conjecture on the Chow groups of hypersurfaces: is $A^i(X)=\mathbb{Q}[h^i]$ for $i<n/2$ ?

Rational smoothability of algebraic cycle classes: is $H^{2c}(X,\mathbb{Q})_{\mathrm{alg}}$ generated by classes of smooth subvarieties?

Is the smooth rational quartic curve in $\mathbb{P}^3$ a set-theoretic complete intersection?

Rational-curve determination of the nef-cone slope of rank-two vector bundles on Fano manifolds (the $\tau=\tau_0$ question of Muñoz–Occhetta–Solá Conde)

Existence of a smooth quadratic non-complete-intersection 11-fold in $\mathbb{P}^{17}$ of Fano index 8: the remaining case of Watanabe's classification of quadratic varieties with $n=2c-1$

Symmetry-rank thresholds for the Hopf conjecture under positive intermediate Ricci curvature: existence of a bound $d(k)$ for $k\ge 3$

Torus-symmetry thresholds for the Hopf conjecture under positive $k$ -th intermediate Ricci curvature with $k\ge 3$

The $\mathbb{Z}_2$ and $\mathbb{Z}_p$ Periodicity Conjectures: extending Adams's Hopf invariant one theorem from singly generated to periodic cohomology

Descent of nonnegativity certificates to rational coefficients: finite generation of $Pos(K)$ as a preordering of $\mathbb{Q}[x]$ for specific compact intervals with algebraic boundary, and archimedeanity of rational quadratic modules

Archimedean descent for finitely generated quadratic modules from $\mathbb{R}[X_1,\dots,X_n]$ to $\mathbb{Q}[X_1,\dots,X_n]$

Pseudointersection and tower numbers of topological Ramsey spaces: the Dobrinen–Navarro Flores questions on $\mathfrak{p}_\mathcal{R}\le\mathfrak{p}$ , $\mathfrak{p}_\mathcal{R}=\mathfrak{t}_\mathcal{R}$ , and values outside $\{\omega_1,\mathfrak{p}\}$

Generalized tower numbers at uncountable cardinals: the exact consistency strength of $\mathfrak{t}_\kappa>\kappa^+$ and the $\pi$ -depth equality $\pi\mathfrak{t}(U)=\pi\mathfrak{p}(U)$

Is the auxiliary cardinal $\mathfrak{d}^*$ equal to $\mathfrak{d}$ ? Optimal tower-refinement of filter-bases after $\mathfrak{p}=\mathfrak{t}$

Does $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$ hold for every uncountable regular cardinal $\kappa$ ? (the higher analogue of Malliaris–Shelah's $\mathfrak{p}=\mathfrak{t}$ )

Equality of the finitistic and full tight-closure annihilators in $F$ -pure local rings

Principal localization of tight closure and the existence of its failure in characteristic $p>2$

The closed geodesics conjecture for compact rank-one symmetric spaces: does every Riemannian metric on $S^n$ ( $n\ge3$ ), $\mathbb{CP}^m$ , $\mathbb{HP}^m$ , or the Cayley plane carry infinitely many geometrically distinct closed geodesics?

Finitely presented almost cyclic groups (Kourovka Notebook Problem 8.8b) and the second closed geodesic on closed $K(\pi,1)$ -manifolds

Duan–Long–Wang conjecture on the dichotomy of the number of prime closed geodesics on compact simply-connected Finsler manifolds with rational cohomology $T_{d,n+1}(x)$