Open Problems

Probability

Does propagation of chaos hold for coloured epidemic models with heterogeneous infectivity when the infection kernel has long-range dependence, and if so, at what rate does the empirical measure converge to the mean-field limit?

The coloured epidemic paper establishes functional law of large numbers and propagation of chaos for structured epidemic models where individuals carry distinct types or colours affecting their suscep...

Related: McKean-Vlasov propagation of chaos theorem, Dobrushin stability estimate for mean-field limits, Nualart-Peccati fourth moment theorem for Wiener chaos

Probability / Network Theory

Does the random walk mixing time on small-world networks in dimension d greater than or equal to 3 exhibit a sharp threshold as the rewiring probability crosses a critical value?

Small-world networks are constructed by taking a regular lattice in d dimensions and randomly rewiring or adding a small fraction of long-range edges. The random walk mixing time, which measures how l...

Related: Aldous-Fill conjecture on mixing times, Peres-Sousi theorem on hitting times and mixing, Benjamini-Berger small-world percolation threshold results

Mathematical Logic / Proof Theory

What is the exact proof-theoretic strength of Fraisse's conjecture restricted to well-quasi-orders of finite width, and does it fall strictly between ATR0 and Pi11-CA0?

Fraisse's conjecture, now a theorem proved by Laver, states that the class of countable linear orders is well-quasi-ordered under embeddability. The reverse mathematics of this theorem is known to req...

Related: Laver's theorem on countable linear orders, Kruskal's tree theorem, Simpson's reverse mathematics program for ATR0

Constructive Mathematics and Logic

Does there exist a constructive proof that the arithmetic mean strictly exceeds the geometric mean for all distinct positive reals, without invoking the law of excluded middle or dependent choice?

The classical AM-GM inequality states that the arithmetic mean of distinct positive real numbers strictly exceeds their geometric mean, and this is provable in standard mathematics. However, in constr...

Related: Bishop's constructive analysis, Brouwer's continuity theorem, Markov's principle

Probability

Does propagation of chaos hold quantitatively for memory-dependent particle systems on the same timescales as their memoryless counterparts, and if so with what rate?

Consider a large system of interacting particles where each particle's future motion depends not only on its current position and interactions with others, but also on its entire history of past posit...

Related: McKean-Vlasov propagation of chaos theorem, Sznitman coupling argument for mean-field limits, Dobrushin stability and contractivity for interacting diffusions

Mathematical Logic and Quantum Foundations

Can every pair of compatible states on a quantum logic be simultaneously extended to a single state on the full algebra, and if not, what is the minimal algebraic obstruction?

This problem is open because the failure of distributivity in orthomodular lattices destroys the measure-theoretic machinery that resolves the analogous classical question. Partial results exist for s...

Related: Gleason's theorem, Kochen-Specker theorem, Horn-Tarski extension theorem for Boolean algebras

Constructive Mathematics / Logic

Does there exist a constructive proof that the arithmetic mean equals the geometric mean if and only if all arguments are equal, without relying on excluded middle or dependent choice?

The arithmetic mean-geometric mean inequality is a cornerstone of classical analysis, and its equality condition, that AM equals GM precisely when all inputs are identical, is treated as trivial in cl...

Related: Bishop's constructive analysis framework, Markov's principle, Brouwer's theorem on continuity of constructive functions

Mathematical Logic and Algebra

For which ordered fields does the group of worldview transformations collapse to a unique group, and can a complete algebraic classification be given for all ordered fields?

The paper by Borisov (1978) and its extensions show that over the real numbers, Einstein's special principle of relativity forces the worldview transformation group to be either the Lorentz group or t...

Related: Borisov's 1978 classification theorem over the reals, Artin-Schreier theory of real closed fields, Witt's theorem on quadratic forms

Probability Theory

For exchangeable sign-invariant random walks in higher dimensions, do the persistence probabilities obey a universal polynomial decay law whose exponent depends only on the dimension and the exchangeability structure?

Resolving this question would establish a higher-dimensional analogue of the Sparre Andersen universality theorem, one of the landmark results in fluctuation theory, and would significantly advance th...

Related: Sparre Andersen theorem, Wendel formula for cone probabilities, Comtet-Majumdar persistence exponent conjecture

Logic and Algebra

For which rings R does there exist a torsion-free abelian group G of finite rank such that the endomorphism ring of G is isomorphic to R?

The classical realization problem asks which rings can appear as endomorphism rings of abelian groups. While Baer initiated this question and Corner's theorem resolved it broadly for countable reduced...

Related: Corner's Realization Theorem, Butler's theorem on finite rank torsion-free groups, Shelah's Whitehead problem

Probability

For monotone aggregation dynamics on finite graphs with heterogeneous update rules, can one derive sharp phase transitions in consensus time as a function of network topology and the spectral gap of the opinion update operator?

A resolution would unify two largely separate bodies of work, namely algebraic and spectral graph theory on one side and nonlinear stochastic dynamics on the other. It would provide practical guarante...

Related: Aldous spectral gap conjecture for interchange processes, Holley-Liggett theorem for attractive interacting particle systems, Cheeger inequality for Markov chains

Mathematical Logic / Constructive Mathematics

Is Brouwer's fixed-point theorem for dimensions greater than two constructively equivalent to weak König's lemma over a base system weaker than RCA0?

The paper establishes constructive equivalence between Brouwer's fixed-point theorem and weak König's lemma, but it does so within RCA0 as a base theory. A genuinely open question is whether this equi...

Related: Fan theorem equivalences in constructive mathematics, Reverse mathematics hierarchy of WKL0, Sperner's lemma and its constructive content

Probability

Does the critical infection rate for epidemic survival in the avoidance-isolation model on Z^d exhibit a sharp phase transition that depends continuously on the strength of behavioral response?

Consider an epidemic model on the integer lattice Z^d where individuals not only spread infection but also actively avoid contact with visibly infected neighbors and isolate when symptomatic. The cent...

Related: Harris contact process phase transition theorem, Bezuidenhout-Grimmett theorem on critical contact processes, Liggett's theorem on survival of attractive interacting particle systems

Descriptive Set Theory / Logic

Is every finite-index extension of an essentially free countable Borel equivalence relation itself essentially free, and if not, which algebraic obstructions classify the failures?

Countable Borel equivalence relations are ubiquitous objects in descriptive set theory, encoding orbit structures of countable group actions on standard Borel spaces. An equivalence relation is essent...

Related: Dye's Theorem on hyperfinite equivalence relations, Gaboriau's theorem on cost and L2 Betti numbers, Feldman-Moore theorem on countable equivalence relations

Probability

Does the Maki-Thompson rumor model on the integer lattice in dimension two exhibit a sharp phase transition between extinction and survival as a function of the initial spreading rate?

The Maki-Thompson model describes rumor spreading among individuals arranged on an infinite graph, where spreaders contact neighbors and either convert them or retire from spreading. On trees and high...

Related: Harris contact process phase transition theorem, Bezuidenhout-Grimmett theorem on critical contact processes, Peierls argument for Ising model phase transition

Logic

For infinite Karchmer-Wigderson games, does communication complexity in the infinite setting characterize the proof complexity of the corresponding separation principle in intuitionistic logic?

The classical Karchmer-Wigderson theorem establishes a tight correspondence between the circuit depth needed to compute a Boolean function and the communication complexity of an associated two-player ...

Related: Karchmer-Wigderson theorem, Borel determinacy theorem, reverse mathematics of WKL0 and ATR0

Probability

Does the critical 2-LQG metric arise as a limit of natural discrete random planar map models in the same way that subcritical LQG metrics do?

Liouville Quantum Gravity at the critical parameter value of gamma equal to 2 sits at a phase boundary in the theory of random surfaces and random planar maps. For subcritical values of gamma strictly...

Related: KPZ relation for critical LQG, Brownian map universality theorem, Sheffield peanosphere construction

Probability / Combinatorics

For a random walk on a general graph, can the cover time decrease by an arbitrarily large multiplicative factor when a single edge is added between two previously disconnected components of the graph?

The cover time of a graph is the expected number of steps a random walker needs to visit every vertex at least once. When a new edge is added to a graph, intuition suggests the cover time should decre...

Related: Matthews method for cover time bounds, Aldous conjecture on cover times and the uniform spanning tree, Ding-Lee-Peres theorem relating cover time to the Gaussian free field

Mathematical Logic

Does there exist a complete axiomatization of the class of evidential argumentation frameworks that are coherent under transfinite chains of attack and support relations?

The problem asks whether one can write down a finite or recursively enumerable set of axioms that exactly captures when an evidential argumentation framework remains logically coherent as chains of su...

Related: Suslin Hypothesis, Zermelo-Fraenkel axioms and independence results, Dung's fundamental lemma on admissible argumentation semantics

Mathematical Logic / Set Theory

Is it consistent with ZF set theory without the Axiom of Choice that every ultrafilter on the natural numbers fails to be a P-point, while also failing to satisfy any of the standard chain condition properties used in forcing arguments?

The problem asks whether we can construct a model of set theory without the Axiom of Choice in which ultrafilters on the natural numbers exist but none of them are P-points, and simultaneously the sta...

Related: Rudin-Keisler ordering of ultrafilters, Martin's Axiom and the countable chain condition, Shelah's model with no P-points

Mathematical Logic / Set Theory

Is it consistent with ZF set theory without the Axiom of Choice that every ultrafilter on the natural numbers is a P-point?

In standard set theory with the Axiom of Choice, P-points are a special class of ultrafilters on the natural numbers that have strong regularity properties, allowing certain infinite partitions to be ...

Related: Rudin-Keisler ordering of ultrafilters, Shelah's theorem on the consistency of no P-points, Blass-Shelah theorem on ultrafilters and cardinal characteristics

Mathematical Logic / Set Theory

Does ZF set theory without the Axiom of Choice prove or refute the existence of P-points on the natural numbers?

A P-point is a type of ultrafilter on the natural numbers with special combinatorial regularity properties, meaning every partition of the naturals into finitely many pieces has one piece that belongs...

Related: Shelah's theorem on consistency of no P-points, Boolean Prime Ideal Theorem, Rudin-Keisler ordering on ultrafilters

Model Theory / Mathematical Logic

Does every omega-categorical structure admit a continuous realization of all its types over finite sets, and if not, which dividing lines in classification theory exactly characterize when this fails?

The problem asks for a complete structural characterization of which omega-categorical theories allow types over finite parameter sets to be realized in a topologically continuous and definably unifor...

Related: Ryll-Nardzewski theorem, Shelah's classification theory and stability spectrum, Kechris-Pestov-Todorcevic correspondence for automorphism groups

Computability Theory and Reverse Mathematics

Does every computable instance of the Laver Partition Theorem have a solution of low arithmetical complexity, or is there a computable instance requiring hyperarithmetic strength?

The Laver Partition Theorem guarantees the existence of highly structured homogeneous subtrees when an infinite tree-like object is partitioned into finitely many classes. The question is whether, whe...

Related: Milliken Tree Theorem, Hindman Theorem reverse mathematics, Galvin-Prikry Theorem computability

Descriptive Set Theory / Set Theory

Is the equivalence relation induced by mutual generic extensions of countable models of set theory classifiable by countable structures?

The paper on random generics establishes that the equivalence relation identifying two reals when they generate the same random generic extension is not essentially free. A central open problem emergi...

Related: Hjorth Turbulence Theorem, Borel Reducibility Hierarchy, Silver Dichotomy for co-analytic equivalence relations

Probability and Spectral Theory

Does the spectral gap of the colored interchange process on a general graph admit a universal product formula in terms of the underlying simple interchange spectral gap and the color-relabeling operator?

The colored interchange process on a graph combines two operations: swapping objects at adjacent vertices and randomly recoloring objects according to some Markov kernel. On specific graphs like compl...

Related: Aldous spectral gap conjecture (now theorem by Caputo-Liggett-Richthammer), Poincare inequality for product chains, Diaconis-Shahshahani upper bound lemma

Descriptive Set Theory / Ergodic Theory

For which countable linear orders does the automorphism group admit a free ergodic measure-preserving action that is totally weakly mixing but not strongly ergodic?

The papers on orbit equivalence of free group actions and the additive arithmetic of linear orders both touch on automorphism groups of highly structured countable objects, and a natural gap appears a...

Related: Connes embedding problem, Gaboriau cost conjecture, Glasner-Weiss theorem on total weak mixing

Ergodic Theory and Logic

Does every free group action that is orbit equivalent to a totally weakly mixing action also admit a totally weakly mixing representative in its orbit equivalence class?

The central problem asks whether total weak mixing is an orbit equivalence invariant for free group actions on standard probability spaces. Orbit equivalence is a coarse classification that identifies...

Related: Connes Embedding Problem, Zimmer Cocycle Superrigidity Theorem, Ornstein-Weiss Theorem on amenable group actions

Ergodic Theory and Logic

Does every free group action that is orbit equivalent to a totally weakly mixing action also admit a totally weakly mixing representative within its orbit equivalence class?

Resolving this problem would clarify the map between spectral ergodic theory and the orbit equivalence classification program for free group actions, potentially revealing new invariants that are fine...

Related: Gaboriau's theorem on L2 Betti numbers as orbit equivalence invariants, Ornstein-Weiss theorem on orbit equivalence for amenable groups, Popa's cocycle superrigidity theorem

Probability / Statistical Physics

Does the overlap distribution of the branching random walk at criticality converge to a universal limit that is independent of the offspring distribution, and if so, what is its explicit characterization?

Resolving this universality question would provide a mathematically rigorous foundation for physicists' predictions about mean-field spin glasses at their critical temperature, clarifying whether the ...

Related: Parisi ultrametricity conjecture for spin glasses, Gaussian multiplicative chaos phase transition theorem of Kahane, Bramson logarithmic correction for branching Brownian motion

Descriptive Set Theory

Does every bounded-to-one action of a finitely generated commutative monoid on a Polish space generate a hyperfinite equivalence relation?

Descriptive set theory studies the complexity of equivalence relations on Polish spaces, which are complete separable metric spaces. An equivalence relation is called hyperfinite if it can be written ...

Related: Weiss hyperfiniteness theorem for amenable group actions, Slaman-Steel theorem on hyperfinite relations, Dougherty-Jackson-Kechris classification of hyperfinite Borel equivalence relations

Set Theory / Mathematical Logic

For which successor cardinals of regular cardinals does a strong failure of club guessing imply the existence of a generic extension where the failure is witnessed by a uniformly definable family?

Club guessing principles assert that for certain infinite cardinals, one can find a sequence of clubs indexed along a cardinal such that the sequence anticipates any club at stationarily many places. ...

Related: Jensen's Square Principle, Shelah's Club Guessing Theorem, Todorcevic's Strong Negative Partition Relations at Successors

Probability / Statistical Mechanics

Does the Gibbs-non-Gibbs transition time for finite-alphabet spin models on trees depend continuously on the initial temperature, or can it exhibit discontinuous jumps?

Consider a finite-alphabet spin system evolving under a stochastic dynamics on a regular tree, starting from a Gibbs measure at some inverse temperature. As time progresses, the evolved measure may lo...

Related: Georgii-Haggstrom theorem on Gibbs measures, Mossel-Peres theory of information flow on trees, van Enter-Fernandez-Sokal theorem on renormalization and non-Gibbsianness

Probability and Analytic Number Theory

Does the multiplicative chaos measure arising from random multiplicative functions satisfy a nontrivial modulus of continuity with respect to the Riemann zeta function's critical line behavior, and can its multifractal spectrum be fully characterized in terms of analytic number theoretic data?

The multiplicative chaos measure built from random multiplicative functions is a probabilistic object that encodes deep arithmetic structure. While recent work has established its existence and basic ...

Related: Fyodorov-Hiary-Keating conjecture, Gaussian multiplicative chaos multifractal formalism due to Kahane, Selberg central limit theorem for the Riemann zeta function

Model Theory / O-minimal Geometry

For which o-minimal structures does the Hausdorff limit of a definable family of sets remain definable in that same structure, without passing to an expansion?

O-minimal structures provide a tame setting for real geometry, where every definable set has finitely many connected components and behaves nicely. A natural operation is to take a family of definable...

Related: Wilkie's theorem on o-minimality of the real exponential field, Cell Decomposition Theorem for o-minimal structures, Kurdyka-Lojasiewicz inequality

Probability Theory

Does a finite-moment criterion analogous to the four-moment theorem for Gaussian chaos characterize Poisson convergence on general discrete chaoses beyond the Poisson and Rademacher settings?

Resolving this question would establish a universal finite-moment principle for chaos convergence across the full landscape of discrete probability, mirroring the completeness that the fourth-moment t...

Related: Nualart-Peccati fourth-moment theorem, Stein-Chen method for Poisson approximation, de Jong central limit theorem for degenerate U-statistics

Probability Theory

Does a universal fourth-moment theorem hold for Poisson approximation on general discrete chaoses beyond the Poisson and Rademacher settings?

The classical fourth-moment phenomenon, originally discovered by Nualart and Peccati for Gaussian limits, says that on Gaussian Wiener chaos, convergence in distribution to a Gaussian is equivalent to...

Related: Nualart-Peccati fourth-moment theorem, Peccati-Taqqu product formula for Poisson chaos, Hypercontractivity on discrete probability spaces

Set Theory and Combinatorics

For which uncountable cardinals kappa does Martin's axiom imply the partition relation kappa squared arrows (kappa squared, 3) squared?

Martin's axiom (MA) is a combinatorial principle that holds in many forcing extensions of set theory and has powerful consequences for cardinal arithmetic and partition relations. The paper studies wh...

Related: Erdos-Rado theorem, Baumgartner's theorem on partition relations under PFA, Todorcevic's theory of walks on ordinals

Probability

Does the most likely geodesic in last passage percolation with general weight distributions converge to a universal deterministic limiting shape under appropriate rescaling, and if so, what geometric properties characterize that shape?

Last passage percolation assigns random weights to points on a grid, and the geodesic is the path from corner to corner that maximizes the total collected weight. While the maximum value itself is wel...

Related: KPZ universality conjecture, Johansson's theorem on geodesic transversal fluctuations, Busemann function convergence in first passage percolation

Model Theory / Descriptive Set Theory

Can every invariant measure on a definable group in a NIP theory be decomposed canonically into idempotent measures in a way that is functorial with respect to definable homomorphisms?

The problem asks whether the decomposition of invariant measures on definable groups in NIP theories into idempotent components can be made canonical and natural. More precisely, when a definable grou...

Related: Idempotent Measure Conjecture in NIP theories, Ellis Semigroup Theorem, Furstenberg Structure Theorem for distal systems

Set Theory / Mathematical Logic

Can the strong tree property hold simultaneously at every successor of a singular cardinal across an arbitrary class of such cardinals without any large cardinal upper bound stronger than a supercompact?

The strong tree property and its sibling the super tree property are combinatorial principles that generalize the tree property, and researchers have been working to force these principles to hold at ...

Related: Mitchell's theorem on the tree property at omega two, Magidor-Shelah theorem on successors of singular cardinals, Unger's results on the tree property along many cardinals

Probability / Statistical Mechanics

For Ising models on random sparse graphs whose adjacency spectra satisfy a given bounded spectral condition, does the free energy converge almost surely to the annealed free energy in the thermodynamic limit, and if so, at what rate?

Resolving this problem would bridge the spectral approach to Ising models with the rich tradition of studying spin glasses and disordered systems on random graphs, potentially unifying tools from rand...

Related: Guerra-Toninelli interpolation theorem, Benjamini-Schramm local convergence conjecture for free energy, Wigner semicircle law for sparse random matrices

Mathematical Logic

Is the logic J strongly complete with respect to its intended neighborhood or topological semantics when the premise set is uncountable?

The paper on strong completeness of the logic J establishes that derivability from an arbitrary set of premises matches semantic consequence, but the boundary conditions of this result leave open a pr...

Related: Godel incompleteness theorems, Compactness theorem for first-order logic, Kripke completeness for intuitionistic logic

Probability / Random Matrix Theory

Does the heat flow conjecture for random matrices extend to infinite-dimensional operator algebras, and if so, what is the correct formulation of the limiting zero distribution?

Solving this problem would unlock a unified framework connecting infinite-dimensional stochastic processes with free probability and random operator theory, potentially providing new tools for quantum...

Related: Circular Law Theorem, Free Entropy conjecture of Voiculescu, Wigner semicircle law

Probability / Random Matrix Theory

Does the heat flow conjecture for random matrices extend to non-Hermitian ensembles, and if so does the limiting zero distribution remain on the real line?

The heat flow conjecture predicts that applying the heat operator to the characteristic polynomial of a large random Hermitian matrix produces a polynomial whose zeros, after appropriate rescaling, co...

Related: Polya-Schur theorem on zero-preserving operators, Lee-Yang circle theorem, Circular Law for non-Hermitian random matrices

Set-Theoretic Graph Theory and Logic

Does Frucht's theorem remain provable in ZF set theory without the axiom of choice when restricted to infinite graphs, or does it require some weak choice principle?

Resolving this would sharpen our understanding of which classical combinatorial theorems are truly choice-free and which secretly encode fragments of choice, contributing to the broader project of rev...

Related: Frucht's theorem (1939), Axiom of Dependent Choice and its combinatorial consequences, Hajnal-Meredith results on graphs in choiceless set theory

Probability / Random Matrix Theory

Does the heat flow conjecture for random matrices extend to infinite-dimensional operator algebras where the characteristic polynomial is replaced by a spectral zeta function?

The heat flow conjecture, as studied in the finite-dimensional random matrix setting, asks whether applying the heat semigroup to the characteristic polynomial of a large random matrix preserves or tr...

Related: Riemann Hypothesis analogy for random matrix zeros, Wigner semicircle law, Free probability Voiculescu transform

Logic and Set Theory

Does Frucht's theorem remain provable in ZF set theory without the axiom of choice when restricted to countably infinite groups?

Frucht's theorem guarantees that every group arises as the automorphism group of some graph, but the classical proof relies on combinatorial constructions that implicitly invoke the axiom of choice in...

Related: Frucht's theorem (1939), Cayley's theorem on group representations, independence results for the axiom of choice in combinatorics

Probability / Random Matrix Theory

What is the limiting spectral distribution of a uniformly random correlation matrix drawn from the elliptope as the dimension grows to infinity?

A correlation matrix is a positive semidefinite matrix with ones on its diagonal, and the set of all such matrices of a given size is called the elliptope. When one draws a correlation matrix uniforml...

Related: Marchenko-Pastur law for Wishart matrices, Wigner semicircle law, Benaych-Georges and Nadler spiked correlation matrix results

Mathematical Logic / Model Theory

For which locally compact groups does the convolution algebra have a decidable first-order theory?

Solving this problem would bridge computable structure theory, harmonic analysis, and model theory in a fundamental way. A positive answer for broad classes of groups would give logicians new tools to...

Related: Tennenbaum's theorem on nonstandard models, Decidability of the theory of abelian groups by Szmielew, Ax-Kochen theorem on decidability of p-adic fields

Set Theory and Mathematical Logic

Is the Katetov order on uncountable coordinate ideals fully determined by the combinatorial structure of their dense sets?

The Katetov order is a way of comparing ideals on infinite sets by asking whether one ideal can be mapped into another via a finite-to-one function. For ideals built from uncountable product spaces, c...

Related: Solecki's theorem on analytic ideals and the Katetov order, Tukey reducibility for directed sets and its relationship to Katetov comparability, Fremlin's work on ideals and measurability in uncountable products

Mathematical Logic

Does every minimalist distinction-based foundation for arithmetic admit a canonical proof-theoretic normalization theorem that aligns with the symmetry-definability hierarchy of ultrapowers?

Solving this problem would unlock a unified framework connecting proof theory, model theory, and foundations of arithmetic in a novel way. It would clarify whether there is a natural correspondence be...

Related: Keisler's ultrapower theorem, Takeuti's fundamental conjecture on proof normalization, reverse mathematics over RCA0

Probability

Does the small value probability of the derivative martingale in supercritical branching Brownian motion exhibit a universal power-law exponent that is independent of the offspring distribution?

In supercritical branching Brownian motion, the derivative martingale converges to a nontrivial limit that plays a central role in describing the extremal process and the Gibbs measures near the front...

Related: Bramson's logarithmic correction conjecture for BBM maxima, Seneta-Heyde normalization for supercritical branching processes, Mandelbrot canonical cascades and Kahane-Peyriere theory

Mathematical Logic / Reverse Mathematics

Does the Carlson-Simpson lemma for 1-variable words admit a Pi-0-3 conservation result over RCA-0, or is Pi-0-4 the precise bound?

The Carlson-Simpson lemma describes a Ramsey-type structural result about colorings of variable words over finite alphabets. The paper establishes that a specific instance of this lemma for 1-variable...

Related: Hales-Jewett theorem, Hindman's theorem, Paris-Harrington theorem

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