Open Problems

Mathematical Logic / Reverse Mathematics

Does the Carlson-Simpson lemma for 1-variable words have strictly lower proof-theoretic strength than the full Carlson-Simpson theorem, and can it be characterized by a precise arithmetical conservation class below Pi-1-1?

The Carlson-Simpson theorem is a powerful Ramsey-type result about variable words over finite alphabets, and it is known to require substantial set-existence assumptions. Recent work establishes that ...

Related: Carlson-Simpson theorem, Hales-Jewett theorem, Hindman's theorem in reverse mathematics

Probability / Random Matrix Theory

Does the pair correlation function of the Sine-beta process converge to that of a Poisson process as beta approaches zero, and if so, at what precise rate?

The Sine-beta process is a canonical point process on the real line that interpolates between highly structured eigenvalue repulsion at large beta and complete randomness at beta equal to zero. The pa...

Related: GUE hypothesis for Riemann zeta zeros, Selberg integral conjectures for beta ensembles, Forrester log-gas universality conjecture

Mathematical Logic

Does there exist a complete and decidable axiomatization of the modal logic governing the forcing relation over all models of ZFC simultaneously, rather than over any fixed ground model?

The internal modal logic of forcing assigns modal operators to set-theoretic statements by interpreting possibility as truth in some forcing extension and necessity as truth in all forcing extensions....

Related: Hamkins-Lowe Theorem on the modal logic of forcing being S4.2, Solovay's completeness theorem for provability logic GL, Woodin's Generic Multiverse Conjecture

Random Matrix Theory / Probability

Does the Marchenko-Pastur law hold for tensor powers of exchangeable vectors that are not unconditional, and if so under what moment or dependence conditions?

The Marchenko-Pastur law describes the limiting spectral distribution of sample covariance matrices formed from high-dimensional data. The paper under consideration establishes this law for tensor pow...

Related: Marchenko-Pastur theorem for independent entries, Wigner semicircle law universality, de Finetti theorem for exchangeable sequences

Probability Theory

For a general smooth stationary Gaussian process on the real line, can one derive the exact asymptotic joint distribution of the location of the minimum and the overshoot above a high threshold, conditioned on that minimum being unusually low?

Resolving this problem would provide a complete extreme value theory for smooth Gaussian processes analogous to the classical Fisher-Tippett-Gnedenko theorem for independent sequences, but capturing s...

Related: Pickands theorem on Gaussian tail asymptotics, Borell-TIS inequality for suprema of Gaussian processes, Talagrand's majorizing measures theorem

Mathematical Logic

Does there exist a complete classification of all Post complete extensions of the basic conditional logic system CK, analogous to the classical Post completeness theorem for propositional logic?

A complete Post-style classification would provide a definitive map of the expressive boundaries of conditional logic, telling us exactly which collections of conditional principles are internally con...

Related: Post's Functional Completeness Theorem, Lindenbaum-Tarski Theorem, Lewis's Completeness Results for Counterfactual Logic

Mathematical Logic

Does there exist a complete characterization of all Post-complete extensions of Lewis's conditional logic system VC in terms of algebraic or semantic invariants?

Post completeness in a logical system means that adding any new axiom not already derivable collapses the system into triviality, making it a maximally consistent logic. For classical propositional lo...

Related: Post's theorem on completeness in many-valued logic, Lewis's triviality results for conditional probability, Blok-Pigozzi algebraization theorem

Mathematical Logic

Does there exist a complete classification of all Post-complete extensions of the basic conditional logic CK, analogous to the classical Post lattice for propositional logic?

The problem is open because conditional logics admit a far richer and more complex semantic landscape than classical propositional logic. The semantics typically involve possible worlds with selection...

Related: Post's theorem on closed classes of Boolean functions, Blok-Pigozzi algebraizability theorem, Wolter and Zakharyaschev results on modal logic extension lattices

Probability

Does the Chernoff distribution belong to a broader class of strongly log-concave distributions arising from isotonic regression, and can strong log-concavity be characterized for the entire family of distributions generated by monotone function estimation problems?

The Chernoff distribution arises as the limiting distribution in isotonic regression and monotone function estimation, and the recent result establishing its strong log-concavity raises a natural and ...

Related: Groeneboom's theorem on the Chernoff distribution, Brascamp-Lieb inequality for log-concave measures, Prekopa-Leindler theorem

Mathematical Logic / Algebraic Logic

Does every variety of hoops admit a complete classification of its conuclei in terms of finitely many algebraic invariants?

Solving this problem would establish a uniform algebraic foundation for studying modal and epistemic extensions of fuzzy and many-valued logics, since conuclei correspond directly to certain logical m...

Related: Blok-Esakia theorem for Heyting algebras and modal algebras, Glivenko theorem for BL-algebras, McKenzie-Nation theorem on lattice varieties

Probability and Stochastic Processes

Does there exist an optimal swapping rate function for replica-exchange diffusions that minimizes asymptotic variance uniformly across all target distributions with a given mixing time?

Replica-exchange Monte Carlo methods work by running multiple diffusion processes at different temperatures and randomly swapping their states to help each process escape local traps in the probabilit...

Related: Peskun ordering theorem for Markov chains, Holley-Stroock perturbation lemma for spectral gaps, Diaconis-Holmes-Neal theorem on optimal tempering schedules

Logic and Combinatorics

Is there a decidable algebraic characterization of which matroids are realizable over some field, without fixing the field in advance?

Solving this problem would clarify the boundary between combinatorial and algebraic structure in matroid theory, which has been sought since the foundational work of Ingleton and others. A positive an...

Related: Hilbert's Tenth Problem, Ingleton's inequality for matroids, Existential theory of the reals

Probability and Stochastic Analysis

Can the optimal adapted approximation error for square integrable predictable processes be characterized by a universal convergence rate that is independent of the underlying filtration structure?

The NeuralChaos paper develops methods for approximating stochastic processes using neural network architectures that respect the information flow encoded in a filtration. A central open question emer...

Related: Doob-Meyer decomposition theorem, minimax optimal estimation in nonparametric statistics, Kolmogorov n-width for approximation classes

Mathematical Logic and Combinatorics

Is there a decidable axiomatization of the first-order theory of algebraic matroids representable over a fixed finite field?

The problem asks whether, when we restrict algebraic matroids to those representable over a specific finite field such as the field with two elements, the resulting class admits a computable and compl...

Related: Rota's conjecture on matroid representability, Tarski's theorem on the decidability of real closed fields, Matiyasevich's theorem on Hilbert's tenth problem

Probability

Does there exist an optimal swapping rate function for replica-exchange diffusions that simultaneously minimizes asymptotic variance across all observables in a provably dimension-free way?

Resolving this question would provide rigorous algorithmic guarantees for replica-exchange methods used throughout computational statistics, Bayesian inference, and statistical physics. It would also ...

Related: Peskun ordering theorem for Markov chains, Holley and Stroock spectral gap comparison for Glauber dynamics, Diaconis and Holmes optimal transport couplings for MCMC

Probability Theory

What are the precise transition density asymptotics for Levy processes with stochastic resetting when the underlying Levy process lacks a finite first moment?

The paper on Levy processes with stochastic resetting derives asymptotic formulas for transition densities, but these results appear to rely on moment conditions that exclude the most heavy-tailed Lev...

Related: Dynkin-Lamperti theorem for stable processes, Flajolet-Odlyzko Tauberian theorem for algebraic singularities, Blumenthal-Getoor index classification of Levy processes

Mathematical Logic

Does there exist a fully complete game semantics for intuitionistic constant domain logic that is also sound and complete with respect to its topological sheaf models?

The problem asks whether one can construct a single game-semantic framework that simultaneously captures the proof-theoretic content of constant domain intuitionistic logic and agrees with its topolog...

Related: Abramsky-Jagadeesan full completeness theorem for multiplicative linear logic, Kripke completeness for constant domain intuitionistic logic, Awodey-Kishida topological completeness for modal logic

Probability

For a higher order Markov chain on a finite state space, does the spectral gap of the lifted first order chain on the history space determine the optimal mixing time up to universal constants, independent of the order of the chain?

Higher order Markov chains remember a fixed window of past states, and to analyze them one typically lifts them into a first order chain on a larger state space of histories. The question is whether t...

Related: Peres-Sousi theorem on mixing times and hitting times, Aldous spectral gap conjecture now theorem of Caputo-Liggett-Richthammer, Sinclair-Jerrum conductance bound

Mathematical Logic

Does every sparse random graph model satisfying a zero-one law admit a computable Scott sentence that almost surely describes the limit structure?

The paper on limit laws for sparse random graphs establishes that certain random graph models almost surely satisfy or violate specific logical sentences, producing a probabilistic limit theory. Separ...

Related: Fagin's theorem on zero-one laws for first-order logic, Scott isomorphism theorem for countable structures, Shelah's classification theory for countable models

Probability and Statistical Mechanics

Does the cutoff phenomenon for systematic scan Glauber dynamics in the mean-field Potts model persist at the critical temperature, and if so what is the precise window width?

Resolving this question would provide a template for understanding mixing at first-order phase transitions more broadly, a regime that is far less understood than continuous transitions. It would clar...

Related: Peres-Wormald cutoff criterion, Sokal-Thomas spectral gap conjecture for Potts models, Levin-Peres-Wilmer mixing time theory

Mathematical Logic

Does every sparse random graph model satisfying a zero-one law admit a computable Scott sentence that almost surely describes it?

The problem asks whether the almost-sure theory of a sparse random graph model, when that model obeys a zero-one law, can be witnessed by a single computable infinitary sentence in the sense of Scott ...

Related: Scott's isomorphism theorem for countable structures, Fagin's theorem on zero-one laws for existential second-order logic, Barwise compactness theorem

Mathematical Logic

Does every sparse random graph model satisfying a zero-one law admit a computable Scott sentence that captures its almost-sure theory?

The first paper studies limit laws for sparse random graphs, establishing which first-order properties hold with probability zero or one in certain random graph models. The second paper develops compu...

Related: Fagin's zero-one law for first-order logic on dense random graphs, Barany-Bollobas sparse zero-one law conjecture, Scott rank spectrum theorem for countable structures

Logic and Probabilistic Combinatorics

Does every sparse random graph model satisfying a zero-one law also satisfy a convergence law for all sentences of first-order logic with counting quantifiers?

The classical zero-one law for dense Erdos-Renyi random graphs tells us that every first-order sentence is almost surely true or almost surely false. For sparse random graphs, the situation is more de...

Related: Shelah-Spencer zero-one law for sparse random graphs, Fagin's zero-one law for first-order logic on dense random graphs, Lynch convergence law for random graphs

Probability and Statistical Mechanics

Does the cutoff phenomenon for systematic scan Glauber dynamics on the mean-field Potts model persist at the critical temperature, and if so, what is the precise window width?

Resolving this question would provide a template for understanding mixing at criticality more broadly, a regime largely absent from the rigorous cutoff literature. It would clarify whether algorithmic...

Related: Aldous-Diaconis cutoff conjecture, Peres cutoff criterion for reversible chains, Berger-Kenyon-Mossel-Peres spectral gap results for Glauber dynamics

Logic and Probabilistic Combinatorics

Does a zero-one law hold for first-order logic on the giant component of a sparse Erdos-Renyi random graph G(n, c/n) for all constants c greater than 1?

The classical zero-one law for first-order logic on dense random graphs says that every first-order sentence holds with probability tending to either 0 or 1. For sparse random graphs G(n, c/n), the si...

Related: Fagin zero-one law for dense random graphs, Lynch convergence law for sparse random graphs G(n, c/n) with c not equal to 1, Shelah-Spencer zero-one law for G(n, n^(-alpha))

Logic and Combinatorics

Does every component-pruned sparse random graph satisfy a zero-one law with respect to first-order logic for all pruning thresholds?

The central open problem is whether limit laws, specifically zero-one laws or convergence laws, hold universally for sparse random graphs that have been pruned by removing connected components below v...

Related: Fagin zero-one law for sparse random graphs, Lynch convergence law for random graphs, Shelah and Spencer zero-one laws for rational exponent probabilities

Probability and Mathematical Physics

Does the four-dimensional Anderson Hamiltonian exhibit a sharp spectral phase transition between localized and delocalized eigenstates at a critical disorder strength, and can this transition be characterized via renormalization group fixed points of the corresponding singular SPDE?

The four-dimensional Anderson model sits precisely at the upper critical dimension for the localization-delocalization transition, where the disorder is a random generalized function (white noise) and...

Related: Anderson Localization Conjecture (Frohlich-Spencer), Yau Nodal Length Conjecture, Aizenman-Molchanov Fractional Moment Method for localization

Mathematical Statistical Mechanics

Does the critical aspect ratio threshold for nematic phase formation in hard rod systems on Z^2 admit a sharp universal lower bound that depends only on dimension, and does this bound persist in the infinite-volume limit for systems with weak long-range interactions?

The rigorous aspect ratio bound for nematic formation in hard rod and hard rectangle systems on Z^2 provides a sufficient condition for the nematic phase, but the exact critical aspect ratio remains u...

Related: Heilmann-Lieb theorem on monomer-dimer systems, Pirogov-Sinai theory for lattice spin systems, KAM theorem for infinite-dimensional Hamiltonian systems

Functional Analysis

Does the Ball Covering Property of the operator space L(X,Y) characterize when both X and Y have finite Szlenk index?

The Ball Covering Property (BCP) of a Banach space asks whether the unit sphere can be covered by countably many balls none of which is centered at the origin. For the space L(X,Y) of all bounded line...

Related: Crouzeix Conjecture, Lindenstrauss-Tzafriri theorem on complemented subspaces, Szlenk index and asymptotic smoothness characterization of Banach spaces

Probability and Statistical Physics

Does the orthogonally-invariant SK model exhibit a sharp cutoff phenomenon in its Langevin dynamics near the critical temperature, and if so, what is the window size?

The question asks whether the Langevin or Glauber-type dynamics for the orthogonally-invariant Sherrington-Kirkpatrick spin glass model, where the interaction matrix is drawn from an orthogonally inva...

Related: Aldous-Diaconis cutoff conjecture for Markov chains, Parisi formula for the SK free energy, Ben Arous-Dembo-Guionnet dynamics for the spherical SK model

Statistical Mechanics and Combinatorics

Can one establish a sharp threshold aspect ratio for nematic ordering in hard rectangle systems on Z^2 that matches the conjectured mean-field prediction as rod length tends to infinity?

The problem asks for a rigorous, quantitatively sharp lower bound on the critical aspect ratio k* (width-to-length ratio) of hard rectangles on the integer lattice Z^2 above which a nematic phase prov...

Related: Heilmann-Lieb theorem on monomer-dimer systems, Peierls argument for Ising ferromagnets, Dobrushin uniqueness theorem for Gibbs measures

Geometric Measure Theory / Differential Geometry

Do area-minimizing tensor varieties on the universal cover of a hyperbolic manifold satisfy a uniform volume comparison principle analogous to the Bishop-Gromov inequality?

The question asks whether area-minimizing cones and tensor varieties, when lifted to the universal cover of a closed hyperbolic manifold, satisfy a uniform comparison theorem relating the volume of me...

Related: Bishop-Gromov Volume Comparison Theorem, Allard Regularity Theorem for Varifolds, Almgren Dimension Reduction for Area-Minimizing Currents

Functional Analysis

Does the ball covering property of the operator space L(X,Y) admit a multilinear interpolation theorem with respect to the measure of noncompactness?

Let X and Y be Banach spaces and consider the space L(X,Y) of all bounded linear operators from X to Y. The ball covering property (BCP) asks whether the unit sphere of L(X,Y) can be covered by counta...

Related: Calderon interpolation theorem, Zafran's theorem on measure of noncompactness, Lindenstrauss-Tzafriri theorem on complemented subspaces