Problem Directoryv1.0

Sum-Product Problem (Erdős #52)

Erdős #52 — how small can max(|A+A|, |A·A|) be? Current state of progress (June, 2026): the conjecture (exponent 2) is FALSE over ℝ but OPEN over ℤ; best lower…

math.CO · Combinatorics · math.NT

Computation of Stable Homotopy Groups of Spheres

Classifying the stable homotopy groups of spheres π^s_k. Current state of progress (June, 2026): Computations for p=2 reach dimension 90 (Isaksen et al. 2025).…

math.AT · Algebraic Topology · math.GT

Stochastic Delay System Stability (Shaikhet, 2010)

Finding exact necessary and sufficient algebraic conditions for mean-square stability of stochastic differential delay equations (SDDEs), especially with non-s…

math.OC · Optimization and Control · math.PR · math.AP

Singmaster's Conjecture

Singmaster's Conjecture asserts that there is a finite global bound on the multiplicity of any number (a > 1) in Pascal's triangle. Current state …

math.CO · Combinatorics · math.NT

Robustness and Generalization in Learning-Based Robotic Control

Optimal sample complexity for distributionally robust policies in learning-based robotic control remains elusive. Current state of progress (June, 2026): gener…

math.OC · Optimization and Control · math.ST · math.LG

Riemann Hypothesis (Riemann, 1859)

The Riemann Hypothesis posits all nontrivial zeros of $\zeta(s)$ lie on $\Re(s)=1/2$. Current state of progress (June, 2026): RH verified up to $3\cdot 10^{12}…

math.NT · Number Theory · math.CV · math.FA

Quantum Unique Ergodicity (Rudnick–Sarnak)

Does high-energy quantum intensity equidistribute on chaotic systems? Current state of progress (June, 2026): Proven for arithmetic surfaces, open for generic …

math.MP · Mathematical Physics · math.DS · math.NT

Robust Control Theory for Quantum Systems

For open quantum systems, how to design controls to maximize worst-case gate fidelity under uncertainty? Current state of progress (June, 2026): Universal anal…

math.OC · Optimization and Control · math.QA · math.PR

Percolation at the critical point

For Bernoulli bond percolation on $\mathbb{Z}^d$, the problem asks if the infinite cluster probability $\theta(p_c(d))$ is zero for $d \ge 3$. Current state of…

math.PR · Probability · math.CO · math.MP

Nearby Lagrangian Conjecture (Arnold)

Is every closed exact Lagrangian $L \subset T^*M$ Hamiltonian isotopic to the zero section $M_0$? Current state of progress (June, 2026): Proved homotopy and s…

math.SG · Symplectic Geometry · math.DG

Navier–Stokes Existence and Smoothness

Do smooth solutions to the 3D Navier–Stokes equations exist globally? Current state of progress (June, 2026): Global non-uniqueness is proven, but global smoot…

math.MP · Mathematical Physics · math.AP · math.FA

Magnetic Birkhoff Conjecture (Bialy et al.)

Is the circle the only domain where a billiard flow in a constant magnetic field is integrable? Current state of progress (June, 2026): Settled for algebraic i…

math.SG · Symplectic Geometry · math.DS · math.DG

Köthe's Conjecture

Does the absence of non-zero two-sided nil ideals imply the absence of one-sided nil ideals? Current state of progress (June, 2026): open, with counterexamples…

math.RA · Rings and Algebras · math.KT

The Kodaira Problem

The Kodaira problem asks if compact Kähler manifolds can be deformed to projective ones. Current state of progress (June, 2026): solved in dim 3, refuted in di…

math.AG · Algebraic Geometry · math.DG · math.CV

Katok's Entropy Conjecture

Katok's conjecture asserts that for a closed manifold with negative sectional curvature, equality of topological and Liouville entropy implies local symme…

math.SP · Spectral Theory · math.DS · math.DG

Kaplansky's Zero Divisor Conjecture

Kaplansky's conjecture asserts that the group ring K[G] of a torsion-free group G has no non-trivial zero divisors. Current state of progress (June, 2026)…

math.QA · Quantum Algebra · math.RA · math.GR

Kannan–Lovász–Simonovits (KLS) Conjecture

The Kannan–Lovász–Simonovits (KLS) conjecture (1995) posits a universal constant bound for the Cheeger isoperimetric constant of log-concave measures. Current …

math.PR · Probability · math.CO · math.FA

Kakeya Conjecture (n >= 4)

The Kakeya conjecture asserts that sets containing a unit line segment in every direction in R^n have dimension n. Current state of progress (June, 2026) shows…

math.CA · Classical Analysis and ODEs · math.DG · math.CO

Jacobian and Dixmier Conjectures

The Jacobian and Dixmier conjectures are stably equivalent and find deep quantum analogues. Current state of progress (June, 2026): Quantum cases settled; a po…

math.QA · Quantum Algebra · math.RA · math.AC

Jacobian Conjecture (Keller / Kraus #N)

If det(J(F)) is a constant non-zero, is F an automorphism? Current state of progress (June, 2026): proven for n=2 up to degree 104; open in general dimensions.…

math.RA · Rings and Algebras · math.AG · math.AC

Isomorphism Problem for Free Group Factors (Kadison 1967)

Is $L(\mathbb{F}_n) \not\cong L(\mathbb{F}_m)$ for $n \neq m$? Current state of progress (June, 2026): The problem for von Neumann algebras remains open, despi…

math.OA · Operator Algebras · math.GR

Huneke–Wiegand Conjecture

If $M \otimes_R \text{Hom}_R(M, R)$ is torsion-free, is $M$ free? The Huneke–Wiegand conjecture (1994) holds for hypersurfaces and Ulrich modules but remains o…

math.AC · Commutative Algebra · math.RA

Grothendieck–Katz p-curvature Conjecture

The Grothendieck–Katz conjecture links mod p reductions to algebraic solutions. Current state of progress (June, 2026) holds for rigid, solvable, and Gauss-Man…

math.AG · Algebraic Geometry · math.NT · math.CV

Gromov's Sofic Groups Conjecture

Gromov's 1999 conjecture asks if every countable discrete group is sofic. Current state of progress (June, 2026): the global Connes Embedding Conjecture i…

math.OA · Operator Algebras · math.GR · math.DS