Mathematics

Subgraph covering problem8 questions

Exact values of the cover numbers by perfect, generalized split, co-unipolar and unipolar graphs, and which functions are cover-number functions
Open for about 1 year
The Dau–Milenkovic–Puleo $K_t$ clique cover conjecture for $t\ge 5$
Open for about 9 years
Edge clique covers of graphs with independence number two: does ecc(G) ≤ n always hold?
Open for about 26 years
Is the local bipartite covering number bounded by a function of the fractional chromatic number? (Question 6.1 of Goetze–Schwebler)
Open for less than 1 year
The de Caen–Gregory–Pritikin conjecture on k-fold biclique partitions of the complete graph, and the extremal constant for clique partitions of complementary graphs
Open for about 40 years
Is the ordered biclique partition number of the complete graph $O(n^{1/2})$ ? Removing the sub-polynomial term in $bp_{1.5}(K_n)=n^{1/2+o(1)}$
Open for about 12 years
The biclique partition number of balanced split graphs: deciding $\mathrm{bp}(G)=\omega(G)-1$ versus $\mathrm{bp}(G)=\omega(G)$
Open for about 4 years
Explicit bipartite graphs of high resistance (Jukna): construct $G_n$ with $\rho(G_n) \ge n^\varepsilon$
Open for about 17 years

Counting problem of self-avoiding walks6 questions

Computational complexity of combinatorial optimization inverse problems2 questions

Hotspot conjecture7 questions

Computation of the polynomial of minimal differential dimension7 questions

Minimal resistance convex body problem4 questions

Relevant theory and computational problems of optimal triangulations8 questions

ABC conjecture7 questions

Integrability in AdS/CFT correspondence6 questions

Chern-number inequalities on Calabi–Yau moduli spaces5 questions

Cheeger-Goresky-MacPherson conjecture10 questions

Cherlin-Zilber conjecture9 questions

A problem of Chern-Moser7 questions

Bonnet rigidity problem of CR type and Siu's superrigidity problem of isolated complex normal singularities7 questions

The Relationship between Donaldson Invariants and Seiberg-Witten Invariants8 questions

The Dry Ten Martini Problem7 questions

The Virasoro Conjecture for Gromov-Witten Invariants8 questions

Hadwiger Conjecture7 questions

Kashaev-Murakami-Murakami Volume Conjecture7 questions

Koebe Problem7 questions

KZB Equations7 questions

Lichtenbaum Conjecture8 questions

Palis Conjecture9 questions

Shelah's Uniqueness Conjecture10 questions

Construction of Dirichlet forms on the Sierpiński carpet7 questions

Boundedness of Sierpiński hyperbolic components9 questions

Open Problems on Smashing Subcategories8 questions

Tate-Oort Problem8 questions

Tate Conjecture10 questions

Veech Conjecture8 questions

Virtual Haken Conjecture5 questions

Exact values of the cover numbers by perfect, generalized split, co-unipolar and unipolar graphs, and which functions are cover-number functions

The Dau–Milenkovic–Puleo KtK_tKt​ clique cover conjecture for t≥5t\ge 5t≥5

Edge clique covers of graphs with independence number two: does ecc(G) ≤ n always hold?

Is the local bipartite covering number bounded by a function of the fractional chromatic number? (Question 6.1 of Goetze–Schwebler)

The de Caen–Gregory–Pritikin conjecture on k-fold biclique partitions of the complete graph, and the extremal constant for clique partitions of complementary graphs

Is the ordered biclique partition number of the complete graph O(n1/2)O(n^{1/2})O(n1/2)? Removing the sub-polynomial term in bp1.5(Kn)=n1/2+o(1)bp_{1.5}(K_n)=n^{1/2+o(1)}bp1.5​(Kn​)=n1/2+o(1)

The biclique partition number of balanced split graphs: deciding bp(G)=ω(G)−1\mathrm{bp}(G)=\omega(G)-1bp(G)=ω(G)−1 versus bp(G)=ω(G)\mathrm{bp}(G)=\omega(G)bp(G)=ω(G)

Explicit bipartite graphs of high resistance (Jukna): construct GnG_nGn​ with ρ(Gn)≥nε\rho(G_n) \ge n^\varepsilonρ(Gn​)≥nε

Ballisticity of the self-avoiding walk on nonamenable transitive graphs: the one-ended case

Making the lattice-point integral representation for self-avoiding walks on Zd\mathbb{Z}^dZd effective: rigorous main-term asymptotics and error control for cnc_ncn​

Universality class of the growing self-avoiding walk

Conformal invariance of the critical self-avoiding walk on the hexagonal lattice: convergence to chordal SLE(8/3) and the predicted critical exponents

Existence and value of the closing exponent for self-avoiding walks in low dimensions

Hammersley's program for the existence of critical exponents of two-dimensional self-avoiding walks: worm super-multiplicativity, two-layer slab asymptotics, and ratio-of-ratios decay

Unified Karp-Reduction Equivalence and Direct Algorithms for Inverse Mixed Integer Linear Optimization

Strongly polynomial solvability of the inverse maximum-weight perfect matching problem under the l1l_1l1​ norm

The asymptotic convex hot spots constant: does Cd→eC_d\to\sqrt{e}Cd​→e​? (Conjecture 1.10 of de Dios Pont, Hsu and Taylor)

No interior extrema for first mixed Dirichlet–Neumann eigenfunctions with a single-segment Dirichlet set (Hatcher's Conjecture 1.8), and the L-shaped eigenvalue inequalities of Remark 6.2

The strong hot spots conjecture in higher dimensions beyond lip and symmetric domains, and the spectrum of the associated vector-valued Laplacian

Unstable nonvertex critical points of second Neumann eigenfunctions on convex quadrilaterals: unconditional existence and consequences for the deformation approach to the hot spots conjecture

Necessity of connectivity for boundary Dirichlet regions in the mixed hot-spots problem

The hot spots conjecture for simply connected planar domains (Rauch's conjecture)

The hot spots conjecture for bounded convex planar domains

Algorithmic computation of multivariate differential dimension polynomials for nonlinear differential field extensions, and the classification of their invariants

Effective computation of the minimal differential dimension polynomial of a differential field extension and classification of Macaulay-constant boundedness under changes of generators

Sharper-than-Ackermann order bounds for differential elimination and effective estimates for the minimizing coefficients of the Kolchin polynomial

Optimality of the León Sánchez bounds for coefficients of the Kolchin differential dimension polynomial

Effective computation of multivariate difference–differential dimension polynomials via characteristic sets with respect to several term orderings

Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals with several commuting translations

Complete invariant theory for multivariable dimension polynomials of finitely generated filtered modules over Ore polynomial rings

Global minimizers of Newton's problem of minimal resistance in the class of convex bodies and the infinite-height limit

Fast rotation and constructive extremals in the Newton minimal resistance problem for rough bodies

Singular set of optimal bodies in Newton's least resistance problem: Plakhov's ridge-symmetry conjecture, conical singular points, and non-disk base domains

Singular-set structure and classification of minimizers in Newton's problem of minimal resistance

Computational complexity of Minimum Average Distance Triangulations with metric and Euclidean weights

Free-boundary regularity of optimal multiphase partitions for the first Robin eigenvalue

Asymptotic hexagonal optimality of optimal partitions for the first Dirichlet eigenvalue (the Caffarelli–Lin conjecture)

Free-boundary regularity, general partitions, and grouping for the optimal partition problem of the principal eigenvalue of linearly coupled elliptic systems

Acute triangulations in dimension four: existence for R4\mathbb{R}^4R4 and for the 16-cell, 24-cell, and 120-cell

Optimal spectral partitions for the sum and the maximum of the first Dirichlet eigenvalues on the square, the disk, and the equilateral triangle

A polynomial-time approximation scheme for minimum weight triangulation (PTAS for MWT)

Complexity of minimum-weight Steiner triangulation of convex polygons: NP-hard or polynomial-time?

Status of Letendre's HHH-function strengthening of the abcabcabc conjecture: equivalence with abcabcabc and truth

Quantitative bounds for the exceptional set in the abc conjecture

Unconditional quantitative bounds for the four-term abcdabcdabcd equation over Z\mathbb{Z}Z without size hypotheses, and the subexponential shape of the rad⁡(bc)\operatorname{rad}(bc)rad(bc) bound towards abcabcabc

Sharp constants and second-order terms in the Robert–Stewart–Tenenbaum refinement of the abc conjecture (Conjectures A, B, C)

Baker's explicit abc-conjecture and the unconditional resolution of its conditional Diophantine applications

Unconditional uniform boundedness of preperiodic points for unicritical polynomials zd+cz^d+czd+c over number fields

Watkins's conjecture: divisibility of the modular degree of an elliptic curve by the power of two attached to its Mordell–Weil rank

Justification and regime of validity of the factorized-scattering criterion for (non)integrability of AdS/CFT string backgrounds

Gap and exact finite-volume spectrum of the long-string continuum in pure-NSNS AdS3×S3×T4_3\times S^3\times T^43​×S3×T4 strings at WZW level k≥2k\ge 2k≥2

Completing the hexagon-with-handles formalism: double-trace mixing, multi-mirror integrands, and the strong-coupling genus expansion for nonplanar N=4 supersymmetric Yang-Mills theory

Exactness of the Quantum Spectral Curve for planar AdS3×S3×T4AdS_3\times S^3\times T^4AdS3​×S3×T4 strings: massless modes at finite coupling and first-principles derivation

Completing the hexagon bootstrap for AdS3/CFT2: finite-size corrections with massless modes and the pure-NSNS long-string limit

Rigorous foundations of the planar AdS5_55​/CFT4_44​ quantum spectral curve: pole-freeness for arbitrary operators, a first-principles derivation beyond the TBA, and exact state-universal equivalence with the Y-system/TBA

Hodge degeneracy of simply connected Calabi–Yau manifolds in dimension greater than two

Volume finiteness and Weil–Petersson incompleteness: the Douglas–Lu questions on projective special Kähler manifolds and Calabi–Yau moduli spaces

Direct uniform estimates of Bott–Chern transgression forms at infinity of Calabi–Yau moduli spaces

The AMWP curvature identity for complexified index cones of arbitrary cubic forms

Rationality of the Weil–Petersson Chern numbers of Weil–Petersson subvarieties of Calabi–Yau moduli (Lu–Sun conjecture)

Hodge–Riemann relations for combinatorial intersection cohomology: arbitrary perverse sheaves on fans

The Cheeger–Goresky–MacPherson conjecture: L2L^2L2-de Rham cohomology of the Fubini–Study metric and intersection cohomology of singular projective varieties

Compatibility of the L2L^2L2 pure Hodge structure on intersection cohomology with Saito's pure Hodge structure (Shentu–Zhao, Conjecture 8.5)

Complete Kähler metric on a singular projective variety whose full-degree L2-cohomology is the intersection cohomology (Cheeger–Goresky–MacPherson question, complete-metric variant)

The Cheeger–Goresky–MacPherson conjecture: L²-cohomology of the smooth part of a complex projective variety in the Fubini–Study metric versus intersection cohomology

Cheeger–Goresky–MacPherson question: complete Kähler metrics on projective varieties whose L2L^2L2-cohomology computes intersection cohomology

The Cheeger–Goresky–MacPherson conjecture for complex projective varieties with non-isolated singularities

Compatibility of the L2L^2L2-Hodge structure with Saito's pure Hodge structure on the intersection cohomology of a singular Kähler space (Shentu–Zhao, Conjecture 8.5)

The Cheeger–Goresky–MacPherson conjecture for the Fubini–Study metric: does the L2L^2L2-cohomology of the regular part of a singular projective variety compute its intersection cohomology?

Hodge theorem for iterated fibred cusp metrics on non-Witt stratified spaces of arbitrary depth

The positive-characteristic case of the log-CZ conjecture: linearization of absolutely simple Lie rings definable in finite-dimensional theories

The uniqueness case for odd-type groups of finite Morley rank: strong embedding and quasithin classification in the tameness-free L∗L^*L∗-theory

The Cherlin–Zilber algebraicity conjecture in odd type: simple groups of finite Morley rank with normal 2-rank at most 2

Algebraicity of proper Moufang sets of finite Morley rank (Borovik-Nesin Question B.17)

Existence of bad groups of finite Morley rank in the generalized sense of Corredor, Borovik–Poizat, and Jaligot

Removing the L-group hypothesis: bounding Alt(n)-actions and the degree of generic sharp transitivity for permutation groups of finite Morley rank

Standardness of sharply 2-transitive groups of finite Morley rank (Borovik–Nesin conjecture)

The characteristic-3 case of the classification of Lie rings of finite Morley rank: Rosengarten's Fact 1 as an obstruction, and simple Lie rings of Morley rank 4 in characteristic 3

The Borovik–Burdges conjecture on strongly real elements in simple groups of finite Morley rank

Convergence of Chern–Moser-type normal forms for real-analytic non-integrable almost CR structures

The Dau–Milenkovic–Puleo $K_t$ clique cover conjecture for $t\ge 5$

Is the ordered biclique partition number of the complete graph $O(n^{1/2})$ ? Removing the sub-polynomial term in $bp_{1.5}(K_n)=n^{1/2+o(1)}$

The biclique partition number of balanced split graphs: deciding $\mathrm{bp}(G)=\omega(G)-1$ versus $\mathrm{bp}(G)=\omega(G)$

Explicit bipartite graphs of high resistance (Jukna): construct $G_n$ with $\rho(G_n) \ge n^\varepsilon$

Making the lattice-point integral representation for self-avoiding walks on $\mathbb{Z}^d$ effective: rigorous main-term asymptotics and error control for $c_n$

Strongly polynomial solvability of the inverse maximum-weight perfect matching problem under the $l_1$ norm

The asymptotic convex hot spots constant: does $C_d\to\sqrt{e}$ ? (Conjecture 1.10 of de Dios Pont, Hsu and Taylor)

Acute triangulations in dimension four: existence for $\mathbb{R}^4$ and for the 16-cell, 24-cell, and 120-cell

Status of Letendre's $H$ -function strengthening of the $abc$ conjecture: equivalence with $abc$ and truth

Unconditional quantitative bounds for the four-term $abcd$ equation over $\mathbb{Z}$ without size hypotheses, and the subexponential shape of the $\operatorname{rad}(bc)$ bound towards $abc$

Unconditional uniform boundedness of preperiodic points for unicritical polynomials $z^d+c$ over number fields

Gap and exact finite-volume spectrum of the long-string continuum in pure-NSNS AdS $_3\times S^3\times T^4$ strings at WZW level $k\ge 2$

Exactness of the Quantum Spectral Curve for planar $AdS_3\times S^3\times T^4$ strings: massless modes at finite coupling and first-principles derivation

Rigorous foundations of the planar AdS $_5$ /CFT $_4$ quantum spectral curve: pole-freeness for arbitrary operators, a first-principles derivation beyond the TBA, and exact state-universal equivalence with the Y-system/TBA

The Cheeger–Goresky–MacPherson conjecture: $L^2$ -de Rham cohomology of the Fubini–Study metric and intersection cohomology of singular projective varieties

Compatibility of the $L^2$ pure Hodge structure on intersection cohomology with Saito's pure Hodge structure (Shentu–Zhao, Conjecture 8.5)

Cheeger–Goresky–MacPherson question: complete Kähler metrics on projective varieties whose $L^2$ -cohomology computes intersection cohomology

Compatibility of the $L^2$ -Hodge structure with Saito's pure Hodge structure on the intersection cohomology of a singular Kähler space (Shentu–Zhao, Conjecture 8.5)

The Cheeger–Goresky–MacPherson conjecture for the Fubini–Study metric: does the $L^2$ -cohomology of the regular part of a singular projective variety compute its intersection cohomology?

The uniqueness case for odd-type groups of finite Morley rank: strong embedding and quasithin classification in the tameness-free $L^*$ -theory

CR analogue of Hamburger's theorem: at least two CR umbilical points on compact strongly convex real-analytic hypersurfaces in $\mathbb{C}^2$

Convergent normal forms for Levi-degenerate real hypersurfaces beyond the everywhere 2-nondegenerate case in $\mathbb{C}^3$

A ball-like bounded strictly pseudoconvex domain in $\mathbb C^2$ without CR umbilical points (the refined Chern–Moser question)

Global classification of obstruction-flat strictly pseudoconvex boundaries in $\mathbb{C}^2$ (the strong Ramadanov conjecture in complex dimension two)

Construction, existence, and wall-crossing of generalized multi-monopole Seiberg–Witten invariants $SW_N(c_1,\dots,c_N;J)$ at the $E_1$ point of 5d $SU(2)$ super Yang–Mills

Witten's conjecture: Donaldson invariants of four-manifolds with $b_1=0$ and odd $b^+>1$ are determined by Seiberg–Witten invariants

Conjecture of Lin–Ruberman–Saveliev: equality of the Furuta–Ohta and Casson–Seiberg–Witten invariants for homology $S^1\times S^3$ 4-manifolds

The Dry Ten Martini Problem for the critical almost Mathieu operator (remaining case: frequencies with $\beta(\alpha)=0$ )

The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators at arbitrary irrational frequencies

Genus-1 Virasoro conjecture for Gromov–Witten invariants without semisimplicity: vanishing of the obstruction $\Psi$ on the small phase space

Virasoro conjecture for stable pairs on simply-connected 3-folds: the full descendent algebra beyond $(p,p)$ -cohomology and the vanishing of $(p,0)$ -descendants

The $\lambda_g$ conjecture with target varieties: unconditional universal Hodge-integral constraints for arbitrary smooth projective targets

The $\kappa^+$ -Hadwiger conjecture at successors of singular cardinals

Forcing a $K_t$ minor from polylogarithmically many disjoint $Ct$ -connected subgraphs at linear connectivity (Question 6.1 of Norin–Postle–Song)

Six-colorability of graphs excluding the near-complete minor $K_7^-$ : the surviving level-7 instance of Seymour's H-Hadwiger conjecture

The Rolek–Song edge-density conjecture: at $(t-2)n-\binom{t-1}{2}+1$ edges every graph has a $K_t$ minor or is $(t-1)$ -colorable

Clustered colouring of graphs excluding a fixed minor: the Norin–Scott–Seymour–Wood conjecture $\chi_\star(\mathcal{M}_H)\le 2\,\overline{td}(H)-2$

One-parameter-family asymptotic expansions of colored Jones polynomials and Turaev–Viro invariants for general hyperbolic knots (Wong–Au Conjectures 3–5, and the figure-eight regime $s\sim 1/2$ )

The Andersen–Kashaev volume conjecture for the Teichmüller TQFT: existence and volume asymptotics of the Jones function $J_{M,K}$ for arbitrary hyperbolic knots in closed oriented 3-manifolds

Modular-transformation matrices for higher-point solutions of the elliptic $(q,t)$ -Knizhnik–Zamolodchikov equation of the Ding–Iohara–Miki algebra

Completing the dynamical elliptic Bethe algebra correspondence: elliptic Wronski map geometry beyond the asymptotic regime, Bethe ansatz completeness for arbitrary $\mu$ , and the source-deferred extension beyond $\mathfrak{sl}_2$

Is the natural map from the modified Bloch group to indecomposable $K_3$ an isomorphism for every number field (Conjecture 3.33 of Burns–de Jeu–Gangl–Rahm–Yasaki)?

Completing Ausoni's computation: an unconditional description of $V(1)_*K(KU_p)$