Mathematics

Spectral asymptotics of elliptic operators and the Weyl-Berry conjecture3 questions

Classification of generator-remainder behaviours for meromorphic continuation of the renewal transform in Koch-snowflake fractal sprays
Open for about 2 years
Converse spectral problem for fractal strings with regularly varying gauge functions
Open for about 29 years
Lattice-case eigenvalue counting asymptotics for Laplacians on generalized Sierpiński carpets
Open for about 16 years

Irreducible decomposition of differential algebraic varieties8 questions

Trivial strongly minimal sets over differentially closed fields8 questions

Weiss finite local primitive graph conjecture10 questions

Automorphism group problem of Witt algebras5 questions

Minimal representations of phase-type distributions6 questions

Weinstein conjecture7 questions

Stable equivalence conjecture7 questions

Seymour's second neighborhood conjecture4 questions

Hilbert's 16th problem6 questions

Hilbert's 9th problem: how Gauss's quadratic reciprocity law generalizes9 questions

Hilbert's 12th problem: construction of the maximal abelian extension of number fields9 questions

Do linearly degenerate quasi-linear hyperbolic systems prevent shock formation?7 questions

Statistical modeling problems for complex censored data under dependent structures3 questions

Soliton conjecture in the Schrödinger equation7 questions

Jacobian conjecture6 questions

Deficiency problems of meromorphic functions3 questions

Do meromorphic functions and their derivatives share common Borel directions?4 questions

Main conjecture of Iwasawa theory7 questions

Yang-Baxter equation8 questions

Can increasing the sample size maintain the asymptotic properties of the original estimate?6 questions

Green function of a class of second-order fully nonlinear partial differential equations3 questions

Some algebraic Gröbner-Shirshov bases8 questions

Quantum groups and canonical bases constructed from derived categories9 questions

Existence problem of wandering continua6 questions

Efficient parametrization and implicitization methods for rational algebraic surfaces4 questions

Classification of rational vertex operator algebras5 questions

Davenport constant of finite Abelian groups8 questions

Minimization problem of the maximum of finitely many quadratic functions7 questions

The finitistic dimension conjecture6 questions

Dirac operator in unitary representations6 questions

Cosmic censorship hypothesis8 questions

Nagata conjecture10 questions

Integer flow conjecture7 questions

Nakayama conjecture and generalized Nakayama conjecture9 questions

Fractal dimension of Julia sets10 questions

Classification of generator-remainder behaviours for meromorphic continuation of the renewal transform in Koch-snowflake fractal sprays

Converse spectral problem for fractal strings with regularly varying gauge functions

Lattice-case eigenvalue counting asymptotics for Laplacians on generalized Sierpiński carpets

The generalized Ritt problem: uniform bounds on differentiations for detecting components of differential ideals, equivalently definability in families of irreducibility of differential varieties

Definability of primality for radical differential ideals and the generalized Ritt problem (Harrison-Trainor–Klys–Moosa Conjecture 1.1)

The generalized Ritt problem: computing the essential prime components of a radical differential ideal

The Ritt problem: algorithmic containment of prime differential ideals and essential prime decomposition of radical differential ideals

Completeness of projective differential algebraic varieties: classification, the zero-dimensional test reduction, and the constant Kolchin polynomial question

Deciding strong irreducibility of ascending chains and computing prime decompositions of perfect ideals of differential-difference polynomials

Small irreducible components of arc spaces in positive characteristic: component status, transfer of arc-sharpness, curve selection at arc-blunt points, and birational elimination

The Jacobi bound conjecture for prime differential ideals presented by generators

Kodaira-Spencer rank distribution in Buium's delta-stratification of the moduli of principally polarized abelian varieties: simple abelian varieties, Jacobians, and Manin-kernel absolute dimensions

Geometrically trivial strongly minimal sets that fail ℵ0\aleph_0ℵ0​-categoricity: classification beyond arithmetic groups

Strong minimality and geometric triviality of generic differential equations (Conjectures 1.5 and 1.7 of DeVilbiss–Freitag)

Are geometrically trivial strongly minimal sets in differentially closed fields unimodular? (Nagloo's Conjecture 4.1)

Non-ℵ0\aleph_0ℵ0​-categorical geometrically trivial strongly minimal sets in DCF0DCF_0DCF0​ beyond arithmetic Fuchsian groups (Casale–Freitag–Nagloo, Question 2.15)

Statement (C2)(C_2)(C2​): binary algebraic independence of the solutions of a general planar algebraic vector field

Ax-Schanuel for the first Painlevé equation

Algebraic independence of Painlevé transcendents at non-generic parameter values

Existence of locally-3n:23^n{:}23n:2 graphs with nontrivial third iterated local kernel Gx[3]G_x^{[3]}Gx[3]​

The Weiss conjecture for locally primitive graphs with almost simple or product action local type

The Weiss conjecture: valency-only bounds on vertex stabilizers of locally primitive vertex-transitive graphs and its equivalent second-singular-value formulation

Is every semiprimitive transitive permutation group graph-restrictive? (the converse implication of the Potočnik–Spiga–Verret conjecture)

Uniform Kazhdan–Margulis neighbourhoods for cocompact lattices with dense projections in products of quasi just-non-compact tdlc groups (Caprace–Le Boudec, Question 5.1)

Tornier's finiteness conjecture for discrete groups acting on trees with semiprimitive local action and an involutive inversion

Graph-restrictiveness of semiprimitive groups with a regular normal nilpotent subgroup: the exceptional cases where 2 or 3 divides the degree (the Potočnik–Spiga–Verret conjecture)

Bounding vertex-stabilisers in finite vertex-transitive graphs: the Weiss, Praeger, and PSV conjectures

The Weiss conjecture for primitive local actions of almost simple, simple diagonal, and product action type

The Weiss conjecture: is the order of a vertex stabiliser in a finite locally primitive vertex-transitive graph bounded by the valency alone?

Determine the automorphism group of the rank-nnn Witt algebra Wn=Der C[x1,…,xn]W_n=\mathrm{Der}\,\mathbb{C}[x_1,\dots,x_n]Wn​=DerC[x1​,…,xn​] (equivalently the polynomial automorphism group Aut C[x1,…,xn]\mathrm{Aut}\,\mathbb{C}[x_1,\dots,x_n]AutC[x1​,…,xn​]) for n≥3n\ge 3n≥3

Rota--Baxter operators beyond the homogeneous block-diagonal class and Novikov--Poisson structures on the full transposed Poisson algebra W(a,−1)\mathcal{W}(a,-1)W(a,−1)

Central extensions, derivations, and automorphisms of multi-copy semi-direct sums of the Witt algebra with its intermediate series modules

Non-surjective local and 2-local automorphisms of simple generalized Witt algebras with non-finitely generated grading group

Automorphism groups of Lie algebras of polynomial vector fields in infinitely many variables (the Bezushchak–Kashuba conjectures)

Minimal-order phase-type representations beyond three distinct real poles: extending the invariant-polytope construction to more, repeated, and complex poles, with global guarantees

The minimal positive realization problem: characterizing the minimal dimension of positive state-space realizations

Canonical representation of general phase-type distributions of order four and higher

Minimal-Norm Phase-Type Representations and the Minimal Maier Parameter Beyond Order Two

Efficient identification of all equivalent representations of Coxian phase-type distributions: permutation feasibility and complexity

A computable upper bound on the triangular order of phase-type distributions with real poles

The Weinstein conjecture for closed contact manifolds of dimension at least five

The multiplicity and elliptic-orbit conjectures for arbitrary Reeb flows on the standard contact sphere

The Weinstein conjecture in exact cotangent bundles: closed characteristics on compact contact-type hypersurfaces

Characterizing the open contact manifolds that satisfy the Weinstein conjecture

Strong Weinstein conjecture in dimension three: homologically trivial weighted cycles of closed Reeb orbits (Abbas–Cieliebak–Hofer conjecture)

Invariant Reeb orbits of strict contactomorphisms: Mazzucchelli's generalized Weinstein conjecture

The Weinstein conjecture for closed contact manifolds of dimension at least five

Conjecture 5.18 (Zhang, 2025): every finite weakly simple-minded system over a self-injective algebra is a simple-minded system

Stable-equivalence classification, scalar problems, and the Auslander–Reiten conjecture for self-injective algebras of polynomial growth

Cardinality of simple-minded systems in stable module categories (Koenig–Liu Question 6.1): does every simple-minded system over an artin algebra have exactly as many objects as the non-projective simples?

Auslander–Reiten conjecture on stable equivalences: invariance of the number of non-projective simple modules

Completing the classification of Erdmann's algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type: the surviving scalar-parameter problems

The Auslander–Reiten conjecture for stable equivalences: preservation of non-projective simple modules and the gluing-of-idempotents characterization

Conditional closure of stable equivalences of Morita type under tensor products, and completeness of the syzygy-algebra invariant B(ΩCe;C)B(\Omega_{C^e};C)B(ΩCe​;C)

The unresolved density regimes for Seymour's second neighbourhood conjecture on G(n,p)G(n,p)G(n,p): the critical case p=1/2p=1/2p=1/2 and the p→1p\to 1p→1 regime

Characterizing the P⃗\vec{\mathcal{P}}P-forcing graphs and the Second Neighborhood Conjecture for oriented graphs missing them

Seymour's second neighborhood conjecture: pushing the universal constant beyond 0.715538 and the limit of the nested weighted-minimizer/CSP method

Seymour's and Sullivan's second-neighbourhood conjectures for oriented split graphs and multipartite tournaments

The sharp upper bound on the number of algebraic limit cycles of planar polynomial vector fields (the simple version of Hilbert's 16th problem for algebraic limit cycles)

Realizable real schemes of M-curves of degree 9 with deep nests (Hilbert's 16th problem, part I, at the frontier degree)

The second part of Hilbert's sixteenth problem: finiteness and value of the Hilbert number H(n)H(n)H(n) for planar polynomial vector fields

The monomial-count variant of Hilbert's 16th problem: bounds for the Hilbert monomial number HM(m)\mathcal{H}^M(m)HM(m) and the Newton-polygon bound on periodic annuli

Open problems completing the zero-dimensional infinitesimal and tangential 16th Hilbert problem: higher-order Melnikov functions, Bautin index, bifurcation diagrams, real zero counts, and chains

Refined Ragsdale conjecture for M-curves of even degree: extremal bounds on even and odd ovals

Explicit reciprocity laws for Hilbert symbols over higher-dimensional local fields at the prime 2

Realizing the full Hilbert symbol, including its wild part, as a boundary map of a K-theory localization sequence, and a localization-theoretic proof of Hilbert reciprocity for number fields

The second reciprocity law for Pablos Romo's 3-cocycle symbols on a curve over a discretely valued field

The residue-characteristic-2 case of Shafarevich-style explicit formulas for the Hilbert pairing on complete higher-dimensional fields

Extending K-theoretic flag-symbol reciprocity to arithmetic bases: realizing Hilbert reciprocity over Spec⁡Z\operatorname{Spec}\mathbb{Z}SpecZ

Explicit reciprocity laws for formal Drinfeld modules without the congruence hypothesis, and their extension to higher local fields

Obstruction-free regimes for qqqth-degree power-residue reciprocity laws from Brauer-group exact sequences

The two-dimensional Contou-Carrère symbol: adelic two-dimensional class field theory

Generalizing the supplement to Scholz's reciprocity law: an odd-prime analogue and higher-power reciprocity for the primes above 2 in Q(pq)\mathbb{Q}(\sqrt{pq})Q(pq​)

Algebraicity of higher elliptic units for number fields with exactly one complex place (the unproven half of Morain's optimal-settings conjecture)

Nondegeneracy of the half-system matrix NℓN_{\ell}Nℓ​ and the completion of Koo–Yoon's explicit construction of class fields over cyclotomic fields

Extended Hecke class-field inclusions and missing ring class field data in the measured-foliation approach to Hilbert's 12th problem

Stark's unit conjecture for the theta-null generators of class fields of primitive CM fields: integrality of the theta-null Stark units ϵ(R)\epsilon(R)ϵ(R)

Hilbert's twelfth problem for totally real fields: complex-analytic generation of the maximal abelian extension

Algebraicity, reciprocity law, and Kronecker limit formula for higher elliptic units over number fields with exactly one complex place

The Charollois–Darmon conjecture: Stark unit arguments as periods of Hilbert Eisenstein series for ATR fields (Conjecture 4.1, 2008)

Geometrically trivial strongly minimal sets that fail $\aleph_0$ -categoricity: classification beyond arithmetic groups

Non- $\aleph_0$ -categorical geometrically trivial strongly minimal sets in $DCF_0$ beyond arithmetic Fuchsian groups (Casale–Freitag–Nagloo, Question 2.15)

Statement $(C_2)$ : binary algebraic independence of the solutions of a general planar algebraic vector field

Existence of locally- $3^n{:}2$ graphs with nontrivial third iterated local kernel $G_x^{[3]}$

Determine the automorphism group of the rank- $n$ Witt algebra $W_n=\mathrm{Der}\,\mathbb{C}[x_1,\dots,x_n]$ (equivalently the polynomial automorphism group $\mathrm{Aut}\,\mathbb{C}[x_1,\dots,x_n]$ ) for $n\ge 3$

Rota--Baxter operators beyond the homogeneous block-diagonal class and Novikov--Poisson structures on the full transposed Poisson algebra $\mathcal{W}(a,-1)$

Conditional closure of stable equivalences of Morita type under tensor products, and completeness of the syzygy-algebra invariant $B(\Omega_{C^e};C)$

The unresolved density regimes for Seymour's second neighbourhood conjecture on $G(n,p)$ : the critical case $p=1/2$ and the $p\to 1$ regime

Characterizing the $\vec{\mathcal{P}}$ -forcing graphs and the Second Neighborhood Conjecture for oriented graphs missing them

The second part of Hilbert's sixteenth problem: finiteness and value of the Hilbert number $H(n)$ for planar polynomial vector fields

The monomial-count variant of Hilbert's 16th problem: bounds for the Hilbert monomial number $\mathcal{H}^M(m)$ and the Newton-polygon bound on periodic annuli

Extending K-theoretic flag-symbol reciprocity to arithmetic bases: realizing Hilbert reciprocity over $\operatorname{Spec}\mathbb{Z}$

Obstruction-free regimes for $q$ th-degree power-residue reciprocity laws from Brauer-group exact sequences

Generalizing the supplement to Scholz's reciprocity law: an odd-prime analogue and higher-power reciprocity for the primes above 2 in $\mathbb{Q}(\sqrt{pq})$

Nondegeneracy of the half-system matrix $N_{\ell}$ and the completion of Koo–Yoon's explicit construction of class fields over cyclotomic fields

Stark's unit conjecture for the theta-null generators of class fields of primitive CM fields: integrality of the theta-null Stark units $\epsilon(R)$

Asymptotic stability of well-separated multi-solitons for the 1D $L^2$ -supercritical NLS in the full range $k>2$

Soliton resolution conjecture for the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$

Deciding the component dichotomy for the Jacobian-one varieties $V_n^k$ : automorphism versus counterexample components and the two-dimensional Jacobian conjecture

Convergence of the compositional inverse and the noncommutative Jacobian conjecture in Popescu's theory of free biholomorphic functions on $B(\mathcal{H})^n$

Common Borel directions of a function of finite positive order and its q-difference operator $D_qf$

The full Iwasawa Main Conjecture for the $\mathrm{GL}_2 \times \mathrm{GU}(1)$ $p$ -adic $L$ -function over totally real fields in the non-split case

Greenberg's $\mu = 0$ conjecture for elliptic curves over $\mathbb{Q}$ : vanishing of the Iwasawa $\mu$ -invariant of Selmer groups over the cyclotomic $\mathbb{Z}_p$ -extension

The Iwasawa main conjecture for CM fields: equality of the Fitting ideal of Selmer modules with Katz's analytic $p$ -adic $L$ -function

Existence of simple involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation of every cardinality $m^2 n$