Open Problems

Probability / Stochastic PDE

For stochastic conservation laws with nonlocal interactions, does the rate of convergence in the nonlocal-to-local limit depend on the regularity of the initial data in a sharp and quantifiable way?

The nonlocal-to-local limit for stochastic conservation laws establishes that as the interaction kernel concentrates to a Dirac delta, solutions of nonlocal stochastic conservation laws converge to so...

Related: Kružkov entropy solution theorem, DiPerna-Lions renormalization theory, regularization by noise for transport equations

Mathematical Logic and Set Theory

Does New Foundations set theory NF admit a natural well-ordering of its universe that is provably consistent without assuming large cardinal axioms?

New Foundations, the set theory invented by Quine in 1937, has a notoriously strange universe of sets governed by a stratification condition on formulas. While Randall Holmes recently announced a proo...

Related: Specker's theorem on the failure of Choice in NF, Holmes's consistency proof of NF relative to ZFC, Scott's theorem on cardinal arithmetic in NF

Mathematical Logic and Foundations

Does New Foundations set theory NF admit a natural hierarchy of proof-theoretic ordinals analogous to those of the Zermelo-Fraenkel subsystems, and if so how do they compare to the ordinals arising from iterated beta-model reflection?

New Foundations (NF), Quine's stratification-based alternative to ZF set theory, has resisted ordinal analysis for decades. While subsystems of second-order arithmetic and ZF have been assigned precis...

Related: Specker's theorem on NF and the axiom of choice, Rathjen's ordinal analysis of KPM and iterated admissibility, Holmes's proof of Con(NF)

Probability / Stochastic PDEs

Does the non-local to local limit for stochastic conservation laws hold at the level of invariant measures when the noise is multiplicative and the kernel is not of convolution type?

The paper on non-local to local limits studies how solutions of stochastic conservation laws with non-local interaction kernels converge to solutions of local conservation laws as the kernel shrinks t...

Related: Kružkov entropy solutions for conservation laws, DiPerna-Lions renormalization theory, Ergodicity of stochastic Burgers equation

Mathematical Logic and Set Theory

Does New Foundations set theory NF admit a natural well-ordering of its universe that is provable within NF itself, and if so what is the proof-theoretic ordinal of NF?

New Foundations, Quine's alternative set theory based on stratified comprehension, was proven consistent by Randall Holmes in a 2010 manuscript that is still undergoing verification. Even granting con...

Related: Holmes consistency proof of NF, Specker theorem on the negation of Choice in NF, Bachmann-Howard ordinal as proof-theoretic ordinal of KP

Probability

Does the product of random matrices satisfy a sharp cutoff phenomenon analogous to the coupon collector problem, and if so, what is the precise window of convergence to the limiting distribution?

The Berry-Esseen bounds for products of random matrices describe how quickly the distribution of a long matrix product converges to a Gaussian-type limit, but they do not resolve whether there is a sh...

Related: Furstenberg-Kesten theorem on Lyapunov exponents, Aldous-Diaconis cutoff conjecture for Markov chains, Le Page central limit theorem for random matrix products

Set Theory / Combinatorial Logic

Is there a model of ZFC in which every Jonsson cardinal is also a Magidor cardinal, or can these two large cardinal properties be consistently separated at every relevant cardinal simultaneously?

Resolving this question would clarify the structural relationship between algebraic and combinatorial characterizations of large cardinals, giving set theorists a sharper map of the large cardinal hie...

Related: Rowbottom's theorem on Jonsson cardinals and Silver's theorem, the Erdos-Hajnal partition calculus, Shelah's PCF theory

Mathematical Logic / Computability Theory

Is the Medvedev lattice decidable, and if not, what is the precise complexity of its first-order theory?

The Medvedev lattice provides a way to rank the difficulty of mass problems, where a mass problem is a set of functions and solving it means computing any one member. The lattice has rich algebraic st...

Related: Ershov's theorem on the undecidability of the theory of the Turing degrees, Rogers semilattice theory, Sorbi's work on intermediate logics and Medvedev degrees

Probability Theory

Can optimal Wasserstein-2 CLT error rates be established for weakly dependent sequences exhibiting heavy-tailed marginal distributions or stable-like switching behavior?

The problem is open because Stein's method for Wasserstein-2 bounds relies critically on controlling second moment quantities and on smoothness of test functions, both of which break down under heavy-...

Related: Berry-Esseen theorem for stable distributions, Zolotarev ideal metrics and stable approximation, Ibragimov-Linnik conditions for CLT under heavy tails

Computable Analysis and Descriptive Set Theory

For which computable metric spaces does there exist an algorithm that, given a description of a bounded linear operator between Lp spaces, decides whether that operator admits a computable presentation?

The paper on computable presentations of separable Lebesgue spaces establishes conditions under which Lp spaces themselves can be given computable structure, but a deeper question remains untouched: c...

Related: Pour-El and Richards theorem on non-computable operators, Weihrauch reducibility framework for analysis, Effective descriptive set theory index set theorems

Probability Theory

For weakly dependent sequences with heavy tails or regime-switching dynamics, what is the optimal rate of convergence in the second order Wasserstein metric for the central limit theorem, and can Stein's method be extended to achieve it?

The recent work on optimal CLT error bounds via Stein's method achieves sharp convergence rates in the second order Wasserstein metric for weakly dependent sequences under finite moment assumptions co...

Related: Berry-Esseen theorem for mixing sequences, Stein-Chen method for Poisson approximation, Kantorovich-Rubinstein duality theorem

Computable Mathematics and Mathematical Logic

For which computable metric spaces does there exist a computable presentation of the corresponding L1 space of real-valued functions, and is there an algebraic or topological invariant that completely characterizes this property?

The paper on computable presentations of separable Lebesgue spaces raises a foundational question that remains unresolved: given an arbitrary computable metric space X equipped with a computable measu...

Related: Urysohn's theorem on computable Polish spaces, Pour-El and Richards theorem on computability in Banach spaces, Hoyrup and Rojas theorem on effective Borel measures

Probability Theory

Can optimal Wasserstein-2 CLT convergence rates be established for weakly dependent sequences exhibiting heavy-tailed marginal distributions or stable-like switching behavior?

Resolving this question would provide a comprehensive quantitative framework for approximation errors in CLTs applicable to a much broader class of models used in finance, physics, and communications,...

Related: Berry-Esseen theorem, Zolotarev ideal metrics conjecture, Ibragimov conjecture on CLTs for dependent sequences

Computable Analysis / Descriptive Set Theory

For which computable metric spaces does there exist an algorithm that, given a description of an Lp space over that space, produces a computable presentation of it?

The paper on computable presentations of separable Lebesgue spaces resolves which classical Lp spaces admit computable presentations, but a deeper structural question remains entirely open: can we cha...

Related: Banach-Mazur theorem on universal separable Banach spaces, Pour-El and Richards theorem on computability in analysis, Melnikov and Nies work on computable Polish groups

Mathematical Logic

Does every definable poset in an o-minimal structure that admits a partition into omega-chains also admit such a partition definably, without invoking the axiom of choice?

The problem asks whether the combinatorial structure of a poset that is definable in an o-minimal structure and satisfies the Dilworth-type criterion for partition into omega-chains can always be witn...

Related: Dilworth's theorem, o-minimal definable choice theorem, Shelah's partition calculus for linear orders

Probability / Statistical Mechanics

Does the mean-field limit of variance-feedback interacting diffusions admit a unique stationary measure when the noise coupling induces non-convex confinement, and if so, what is its phase structure?

Consider a large system of interacting particles where each particle is attracted toward the collective center of mass, but the intensity of random noise each particle experiences depends on the curre...

Related: McKean-Vlasov propagation of chaos theorem, Dobrushin uniqueness criterion for Gibbs measures, Carrillo-McCann-Villani theorem on contractions in Wasserstein space

Model Theory / Combinatorial Logic

Can a full model-theoretic (p,q)-theorem be established for dependent but non-NIP structures, and if not, what is the precise combinatorial dividing line?

The local definable (p,q)-theorem establishes that in NIP theories, certain combinatorial covering properties analogous to the classical Helly-type theorems from convex geometry hold for definable fam...

Related: Helly theorem for NIP theories (Chernikov-Simon), VC theorem and NIP characterization (Shelah), Fractional Helly theorem (Katchalski-Liu)

Probability Theory

For which classes of exchangeable sequences does partial observation of k coordinates determine the de Finetti mixing measure uniquely, and what is the minimal k achieving this rigidity?

The de Finetti theorem tells us that an infinite exchangeable sequence of random variables can be represented as a mixture of independent identically distributed sequences, governed by a so-called mix...

Related: de Finetti representation theorem, Hamburger moment problem and its determinacy conditions, Aldous-Hoover theorem for exchangeable arrays

Mathematical Logic and Set Theory

Does the consistency strength ordering on natural extensions of ZFC form a well-order, and if so, what ordinal is its order type?

The consistency strength ordering places one mathematical theory below another if the first theory cannot prove the second is consistent but the second can prove the first is consistent. Researchers h...

Related: Cantor-Bendixson theorem on well-orders, Martin's conjecture on Turing degrees, Steel's conjecture on the mouse order

Probability Theory

Does the Fleming-Viot particle system with more than two particles admit an explicit stationary ancestral circulation formula analogous to the two-particle case?

The Fleming-Viot particle system is a finite population model where particles diffuse on a domain and, upon hitting the boundary (being killed), are instantly reborn at the location of a uniformly cho...

Related: Yaglom limit theorem for killed Markov processes, Kingman coalescent and its extensions, Moran model stationary distribution in population genetics

Probability

Does the ancestral circulation measure for Fleming-Viot particle systems on general graphs converge to a unique invariant measure as the number of particles tends to infinity, and if so at what rate?

Fleming-Viot particle systems describe populations of particles moving on a state space, where each particle that is killed is immediately replaced by a clone of a uniformly chosen surviving particle....

Related: Yaglom limit theorem for killed Markov chains, McKean-Vlasov mean-field limit theorems, Donsker-Varadhan large deviation theory for occupation measures

Probability Theory

Does the Fleming-Viot particle system with more than two particles admit a closed-form stationary distribution for the ancestral circulation on general graphs?

The Fleming-Viot particle system consists of a finite population of particles evolving on a graph or state space, where particles that are killed are instantaneously reborn at the location of a unifor...

Related: Yaglom limit theorem for absorbed Markov chains, Kolmogorov's criterion for reversibility of Markov chains, Donsker-Varadhan large deviation theory for occupation measures

Model Theory / Logic

Which pseudofinite abelian groups admit a complete axiomatization that also captures the asymptotic counting behavior of their finite approximations?

The paper on pseudofinite abelian groups studies which infinite abelian groups can be viewed as ultraproducts of finite abelian groups, meaning they share all first-order properties with some family o...

Related: Ax's theorem on pseudofinite fields, Hrushovski's theorem on one-dimensional asymptotic classes, the Prufer structure theorem for abelian groups

Probability

Does the sine beta process for general beta greater than zero satisfy a form of mixing or decay of correlations strong enough to imply a central limit theorem for linear statistics with a rate of convergence?

The sine beta process is a canonical random point process arising as the bulk scaling limit of eigenvalues from the Gaussian beta ensemble in random matrix theory. While the paper establishes that thi...

Related: Johansson central limit theorem for GUE linear statistics, Bufetov-Qiu rigidity for determinantal processes, Gustavsson variance asymptotics for beta ensembles

Mathematical Logic and Foundations

Does there exist a constructive proof of the independence of premise principle for intuitionistic ZF set theory that also generalizes to dependent type theories with universes?

The independence of premise principle says roughly that if a hypothesis P implies the existence of some object satisfying a property Q, and if P itself carries no computational content beyond its trut...

Related: Markov's principle in constructive mathematics, Church's thesis in intuitionistic arithmetic, Aczel's constructive set theory CZF

Probability Theory

Do uniform Bessel-type bounds for endpoint crossing probabilities in persistent random walks extend to higher-dimensional lattice analogs with memory?

Resolving this problem would unify persistent random walk theory with the broader literature on correlated random walks on lattices, with direct applications to the study of reinforced walks, directio...

Related: Bessel function representation of simple random walk return probabilities, Goldstein-Kac telegraph process and its higher-dimensional extensions, Donsker invariance principle for correlated random walks

Mathematical Logic / Model Theory

Does Zilber's quasiminimality conjecture hold for the complex exponential field when restricted to specific definable subsets arising from entire functions of finite order?

Resolving this question would clarify exactly which analytic properties of a function are responsible for logical tameness in the sense of quasiminimality. A positive result for finite order entire fu...

Related: Zilber's quasiminimality conjecture for the complex exponential, Hadamard factorization theorem for entire functions of finite order, Ax-Schanuel theorem

Probability Theory / Interacting Particle Systems

For the contact process on graphs with social cluster structure, does the critical infection rate exhibit a sharp phase transition that depends analytically on the cluster size distribution?

The contact process is a classical model for epidemic spreading on graphs, where each node is either infected or healthy, and infection spreads across edges at some rate while recovery happens indepen...

Related: Bezuidenhout-Grimmett theorem on critical contact process survival, Liggett theorem on contact process phase transition on homogeneous trees, Durrett conjecture on epidemic thresholds in random graph models

Logic and Combinatorics

Does every finite tournament embed into a finite symmetric group in a way that extends all partial permutations compatible with the tournament structure?

The central problem asks whether finite tournaments, which are complete directed graphs where every pair of vertices has exactly one directed edge between them, have a strong extension property with r...

Related: Hrushovski extension property, sofic group conjecture, Fraisse amalgamation theorem

Domain Theory and Topology

Does every countable meet-continuous dcpo that is not sober admit a computable structural obstruction to sobriety that can be detected from its Scott topology alone?

Sobriety is a topological property asking that every irreducible closed set is the closure of a unique point. For dcpos equipped with the Scott topology, sobriety is deeply connected to the order-theo...

Related: Hofmann-Mislove theorem, Isbell's non-sober complete lattice construction, Johnstone's theorem on sobriety of continuous dcpos

Set Theory / Cardinal Characteristics of the Continuum

Does the equality of the dominating number and omega_1 imply that the almost disjointness number also equals omega_1 in all models of ZFC where the continuum is strictly greater than omega_1?

The paper establishes that the dominating number d equaling omega_1 forces the almost disjointness number a to equal omega_1. A natural and unresolved question is whether this implication is part of a...

Related: Cichoń's diagram, Martin's Axiom and its consequences for cardinal characteristics, Shelah's theorem on the relationship between the splitting number and other characteristics

Probability / Mathematical Physics

Does the variance of the free energy in the Sherrington-Kirkpatrick model at the critical temperature obey a universal scaling law that interpolates smoothly between the high-temperature and low-temperature regimes?

The Sherrington-Kirkpatrick spin glass model has a critical temperature below which the system enters a complex disordered phase. Near this critical point, the free energy variance is known to exhibit...

Related: Parisi formula for the SK free energy, Guerra-Toninelli interpolation theorem, Tracy-Widom distribution for GOE fluctuations

Probability and Statistical Mechanics

Does the free energy variance in the Sherrington-Kirkpatrick model at criticality obey a universal scaling law that interpolates continuously between the replica-symmetric and full replica symmetry breaking regimes?

The Sherrington-Kirkpatrick spin glass model exhibits a phase transition at its critical temperature, and near this transition the variance of the free energy is expected to capture deep information a...

Related: Parisi Formula (Talagrand-Panchenko theorem), Guerra-Toninelli interpolation inequality, Wigner semicircle law and Tracy-Widom universality at spectral edges

Set Theory / Mathematical Logic

Does the dominating number equaling omega_1 imply that the almost disjointness number also equals omega_1, and if so, can this implication be reversed or shown independent of ZFC?

Resolving whether a equals omega_1 implies d equals omega_1 would clarify a significant portion of the diagram of cardinal characteristics below the continuum. If the implication is one-directional on...

Related: Cichon's diagram, Blass-Shelah theorem on cardinal characteristics, Malliaris-Shelah theorem on p equals t

Proof Theory / Modal Logic

Does non-contingency logic admit a cut-free sequent calculus that is both sound and complete without requiring translation into standard modal systems?

Non-contingency logic replaces the standard necessity operator with an operator expressing that a statement has the same truth value in all accessible worlds, meaning it is either necessarily true or ...

Related: Belnap display calculus for modal logics, Sahlqvist correspondence theory, cut-elimination theorem for S5

Probability and Stochastic Analysis

Can the two-scale sampling algorithm for local free energy minimizers be proven to achieve dimension-free convergence rates when the energy landscape satisfies a uniform Poincare inequality at the local level?

The two-scale approach for sampling local free energy minimizers constructs an algorithm that separates fast local fluctuations from slow transitions between metastable states. A core open question is...

Related: Holley-Stroock perturbation lemma for log-Sobolev inequalities, Cheeger inequality for continuous Markov chains, Bovier-Eckhoff-Gayrard-Klein metastability theory

Computability Theory / Mathematical Logic

Does pseudo-hyperjump inversion hold for any natural class of Turing degrees strictly between the computable degrees and the hyperdegrees?

The pseudo-hyperjump operator is a generalization of the classical hyperjump, mapping Turing degrees to higher degrees in a structured way. The paper establishes that full inversion fails for Turing d...

Related: Friedberg Jump Inversion Theorem, Gandy-Kreisel-Tait Theorem on hyperarithmetical sets, Shore and Slaman definability results in the Turing degrees

Set Theory / Mathematical Logic

Does the existence of ultraexacting cardinals remain consistent after arbitrary class forcing, and if so, which forcing notions preserve their defining reflection properties?

Ultraexacting cardinals are among the most recently introduced large cardinal axioms, sitting near the upper reaches of the large cardinal hierarchy and defined by extremely strong elementary embeddin...

Related: Laver's theorem on supercompact cardinals and indestructibility, Woodin's results on the consistency strength of forcing axioms, Bagaria and Magidor work on large cardinal preservation under class forcing

Probability Theory

For Hermite rank k greater than two, do explicit closed-form constants exist in the weighted law of the iterated logarithm under long-range dependence, and if so, what is their general formula?

Resolving this would provide a complete quantitative picture of extremal behavior for long-range dependent non-linear functionals of Gaussian sequences, unifying the central and non-central regimes un...

Related: Breuer-Major theorem, Non-central limit theorem for Hermite processes, Chung-Fuchs law of the iterated logarithm

Mathematical Logic and Computational Complexity

Is there a constructive proof system in which the solvability of a problem in polynomial time is equivalent to the existence of a constructive polynomial-time certificate, and if so does this system separate P from NP?

The core problem asks whether one can build a formal constructive proof framework, grounded in intuitionistic or constructive logic, in which a decision problem being in P is provably equivalent to th...

Related: Cook-Levin Theorem, Curry-Howard Correspondence, Krajicek-Pudlak Theorem on proof complexity

Mathematical Logic / Algorithmic Randomness

Does every neutral measure on Cantor space admit a computable randomness test that characterizes exactly the sequences that are random with respect to it?

A neutral measure is a probability distribution on infinite binary sequences that assigns positive measure to every open set, meaning no finite initial segment is ruled out as impossible. The central ...

Related: Martin-Lof randomness theorem, Levin-Schnorr theorem on randomness and compression, Vaught's Conjecture

Descriptive Set Theory and Combinatorial Topology

Can the topological equivalence classes of countable trees always be characterized by a single cardinal invariant, or are there trees whose class size escapes all standard cardinal arithmetic?

Resolving this question would clarify the boundary between combinatorial and set-theoretic phenomena in topological classification problems. If the trichotomy holds in full generality across all set-t...

Related: Laver's theorem on scattered linear orders, Borel reducibility of countable tree isomorphism, Shelah's classification theory for countable models

Probability

Does the directed distance exponent in spanning-tree-decorated planar maps remain universal across all critical weighted planar map families in the same universality class?

The paper on spanning-tree-decorated planar maps establishes an exact exponent and scaling limit for directed distances in a particular model, and claims universality within a certain class. However, ...

Related: KPZ universality conjecture for random planar maps, Brownian map universality theorem of Le Gall and Miermont, mating-of-trees theorem of Duplantier-Miller-Sheffield

Mathematical Logic / Descriptive Set Theory

Does there exist a complete, linearly ordered hierarchy of Borel-Tukey degrees for Ramsey homogeneity relations on countable structures, or are there incomparable degrees at every level of complexity?

The problem asks whether the partial order formed by Borel-Tukey morphisms between Ramsey homogeneity relations can be refined into a total linear order, or whether incomparability is unavoidable and ...

Related: Galvin-Prikry theorem, Ellentuck theorem, Tukey order on ultrafilters

Probability

For the repeated averages process on an arbitrary connected graph, does the mixing time always exhibit a cutoff phenomenon, and if so, can the cutoff time be characterized purely in terms of the graph's spectral gap?

The repeated averages process on a graph works as follows: assign a real number to each node, then repeatedly pick a random edge and replace both endpoint values with their common average. The questio...

Related: Aldous-Diaconis cutoff conjecture for Markov chains, Peres cutoff conjecture for random walks on groups, Chatterjee-Diaconis theory of cutoff via information theory

Domain Theory and Topology

For which classes of meet-continuous domains does the Scott topology coincide with the Lawson topology, and can this coincidence be characterized purely in terms of order-theoretic properties without invoking topological ones?

Meet-continuous domains generalize continuous domains and arise naturally in theoretical computer science and logic as semantic models of computation. The Scott topology on such a domain captures the ...

Related: Lawson duality theorem for continuous lattices, Hofmann-Mislove theorem, Rudin lemma for dcpos

Probability and Abstract Harmonic Analysis

For which non-abelian locally compact groups does an analogue of the Skitovich-Darmois theorem hold, characterizing Gaussian measures via independence of two group-valued linear combinations?

The classical Skitovich-Darmois theorem characterizes Gaussian distributions on the real line by the property that two distinct linear combinations of independent random variables are themselves indep...

Related: Skitovich-Darmois theorem, Cramer decomposition theorem for Gaussian measures, Heyer characterization of Gaussian semigroups on locally compact groups

Logic and Order Theory

Does every uncountable meet-continuous lattice that satisfies a natural completeness condition admit a Scott sober topology without requiring the axiom of choice?

The recent paper establishes that every countable meet-continuous lattice is Scott sober, a result connecting order theory and topology in a clean and foundational way. The open problem is whether thi...

Related: Hofmann-Mislove theorem, Scott topology sobriety conjecture for dcpos, independence of sobriety results from choice axioms

Probability and Abstract Harmonic Analysis

For which non-abelian locally compact groups does an analogue of the Skitovich-Darmois theorem hold, characterizing Gaussian measures via independence of two group-valued linear forms?

The classical Skitovich-Darmois theorem characterizes Gaussian distributions on the real line by the property that two distinct linear combinations of independent random variables are themselves indep...

Related: Skitovich-Darmois theorem, Heyer-Hazod theorem on Gaussian semigroups on groups, Ghurye-Olkin theorem on matrix-variate normality

Computability Theory / Mathematical Logic

Does every introenumerable set of natural numbers contain a uniformly introreducible subset that is also introenumerable?

The paper establishes that every introenumerable set contains a uniformly introreducible subset, but it leaves open whether this subset can always be chosen to itself be introenumerable. Introenumerab...

Related: Sacks Splitting Theorem, Friedberg Completeness Criterion, Low Basis Theorem

Probability and Information Theory

Can the higher-order interaction measures on partition lattices be characterized by a complete set of axioms analogous to those that uniquely determine Shannon entropy for pairwise dependencies?

Classical Shannon entropy and mutual information satisfy elegant axiomatic characterizations that make them essentially unique measures of pairwise statistical dependency. When one tries to generalize...

Related: Shannon uniqueness theorem for entropy, Mobius inversion on lattices, Partial Information Decomposition conjecture of Williams and Beer

Universal Algebra and Logic

Does there exist a finitely axiomatizable equational theory that captures exactly the equations over the natural numbers involving addition, multiplication, exponentiation, and the constant 1?

Resolving this question would have broad consequences. A positive answer would give logicians and algebraists a concrete finite basis for reasoning about number-theoretic identities involving exponent...

Related: Gurevic theorem on non-finitely-based exponentiation identities, Tarski high school algebra problem, Birkhoff completeness theorem for equational logic

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