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    Open questions / New possibilities

    Open problems

    8,961 open problems in the collection

    From classic conjectures to questions across disciplines, find the next problem worth pursuing.

    01 / Scientists

    Open questions from great scientists

    All scientists
    Shiing-Shen Chern

    Shiing-Shen Chern

    Differential geometry · Topology

    More related questions

    Can curvature determine global convexity?

    How much can local curvature tell us about the shape of a closed hypersurface? In even dimensions four and above, must everywhere nonnegative Gauss–Kronecker curvature force it to bound a convex body?

    How far can a negatively curved graph extend?

    If a higher-dimensional graph has scalar curvature bounded above by a fixed negative number, how far can it extend? Can that condition alone bound the radius of its domain and rule out a graph over the whole space?

    What shapes can higher-dimensional Dupin hypersurfaces take?

    On a Dupin hypersurface, each principal curvature stays constant along its corresponding curvature surfaces, but how many shapes does this allow in higher dimensions? Starting in five-dimensional Euclidean space, can we find all local models and distinguish them up to Lie sphere equivalence?

    David Hilbert

    David Hilbert

    Algebra · Geometry · Mathematical physics

    More related questions

    How small can an integer polynomial be on a long interval?

    With integer coefficients only, can the squared integral of a nonzero polynomial be made arbitrarily small on a given interval? Once the interval has length four or more, does arithmetic impose a positive lower bound, and can the interval’s position change the answer?

    How can the ovals of an octic curve fit together?

    A smooth real plane curve of degree eight can form up to twenty-two ovals, but how can they nest inside or sit apart from one another? Which seemingly possible arrangements can an octic equation actually draw, and which does its algebraic structure forbid?

    Can Hilbert’s gas expansion remain valid for all time?

    If we describe molecular motion with a Hilbert series, can the expansion keep pace with the gas for arbitrarily long times? Will it converge to the Boltzmann solution, or remain an approximation whose error stays under control?

    Terence Tao

    Terence Tao

    2006 Fields Medal

    Number theory · Analysis · PDEs

    More related questions

    Can a higher-dimensional torus tiling lose all sliding structure?

    If finitely many translates of one measurable tile cover a torus in dimension three or higher, must some proper nonempty part remain invariant under a continuous translation? Or can a completely locked tiling eliminate every such invariant sub-tiling?

    How does a nonlinear wave concentrate at blow-up?

    As a nonlinear wave concentrates toward finite-time blow-up, must at least a ground state’s worth of mass stay inside a shrinking region? Does that concentration persist all the way to blow-up, or can it repeatedly form and disperse?

    How far can a totient-monotone sequence exceed the primes?

    Among the integers up to a given bound, how much longer than the list of primes can a sequence with nondecreasing totient values be? Must the excess stay bounded, perhaps even settling at exactly sixty-four?

    Chen Ning Yang

    Chen Ning Yang

    1957 Nobel Prize in Physics

    Gauge theory · Statistical physics

    More related questions

    Which gauge fields can share a field strength?

    In a non-Abelian theory, knowing the field strength everywhere may still leave the gauge field undetermined, so what information is missing? For a sourceless field, when can its dual field strength arise from a gauge potential of its own?

    Can a source-free gauge field remain static?

    Can a globally regular, source-free gauge field remain static while decaying at infinity, or are such nontrivial configurations possible only for noncompact groups? If a noncompact group brings an indefinite energy, can its quantum theory still give physical states a nonnegative inner product?

    What else can Hubbard-model symmetry reveal?

    Which quantum states of the Hubbard model retain their form as the electron interaction strength changes? Can SO(4) symmetry reveal every such state and show where magnetic or pairing order can emerge?

    John Bardeen

    John Bardeen

    1956, 1972 Nobel Prize in Physics

    Superconductivity · Semiconductor physics

    More related questions

    What binds electrons in high-temperature superconductors?

    Why can electrons in cuprates pair up near one hundred kelvin and carry current without resistance? Is the attraction supplied by lattice vibrations, electronic or magnetic interactions, or a combination, and can BCS theory accommodate it all?

    What sets the barrier between a metal and a semiconductor?

    When a metal touches a semiconductor, how high an energy barrier must an electron cross, and why can that height change so little when the metal changes? Can the two materials and their atomic interface alone reveal how interface states set the barrier, before it is measured?

    Can excitons bind electrons into a superconductor?

    If a very thin metal layer touches a semiconductor, can virtual excitons create enough attraction between electrons to overcome their repulsion and produce superconductivity? Can we find a material structure that realizes this mechanism and distinguish exciton pairing from lattice vibrations or other effects?

    Paul A.M. Dirac

    Paul A.M. Dirac

    1933 Nobel Prize in Physics

    Quantum mechanics · Quantum field theory

    More related questions

    Do magnetic monopoles exist?

    Electric charges can exist on their own, so why have we not observed an isolated magnetic pole? If monopoles exist, can they explain the fixed units of electric charge and bring us closer to understanding the value of the elementary charge?

    Can gravitational energy be localized?

    We can discuss the total energy of a gravitating system, but can we say how much it stores at each spacetime point? Can a local definition be independent of coordinates, obey conservation and agree with the total energy measured at infinity?

    How should constrained systems be quantized?

    When a constrained classical system becomes a quantum theory, exchanging the order of two quantities can change the result, so how can the choices be made consistently? Can every such system be quantized, or are some constraints fundamentally incompatible with quantum rules?

    02 / Disciplines

    Research challenges across disciplines

    All disciplines

    Prime numbers seem irregularly scattered among the integers, yet the Riemann hypothesis connects their distribution to the zeros of a complex function. Questions about those zeros, function values and related sequences probe the structure behind that apparent disorder.

    1. Open analytical questions for Voros's discretized Keiper–Li sequence: transseries asymptotics, all-nn positivity, and remainder bounds
      Open for about 9 years
    2. The zeta spiral on the critical line: denseness of the values of ζ(1/2+it)\zeta(1/2+it) in C\mathbb{C} and the unconditional curvature sign
      Open for about 16 years
    3. The Alternative Hypothesis for zeta-zero pairs: determinacy of the limiting densities p_{k/2} under AH-Pairs alone
      Open for about 30 years
    More questions

    If a proposed answer can be checked quickly, can an answer also be found quickly? This deceptively simple distinction reaches into the limits of algorithms, cryptography and automated reasoning.

    1. P versus NP and the finite descriptive-complexity separations: NP versus co-NP (Σ11\Sigma^1_1 versus Π11\Pi^1_1) and P versus Δ11\Delta^1_1 over finite structures
      Open for about 55 years
    2. Fixed-Polynomial Circuit Lower Bounds for ⊕P\oplus P, Uniform MAMA, PNPP^{NP}, and NPNP
      Open for about 19 years
    3. Does the polynomial hierarchy collapse? (Is the polynomial-time hierarchy infinite?)
      Open for about 50 years
    More questions

    Can the equations for water and air evolve from smooth initial conditions into infinitely violent changes? Understanding regularity and the limit of vanishing viscosity connects physical intuition about fluids with rigorous mathematics.

    1. Global regularity versus finite-time blowup for the three-dimensional axisymmetric incompressible Navier–Stokes equations with swirl
      Open for about 58 years
    2. Global regularity versus finite-time blowup for the three-dimensional incompressible Navier-Stokes equations with arbitrary large data in the critical space BMO−1(R3)BMO^{-1}(\mathbb{R}^3)
      Open for about 92 years
    3. Inviscid limit of the three-dimensional inhomogeneous incompressible Navier–Stokes equations without the Kolmogorov spectral hypothesis
      Open for about 14 years
    More questions

    Why do electrons in cuprates form pairs that allow current to flow without resistance at low temperatures? Identifying the interactions behind pairing, and separating the roles of magnetism, lattice vibrations and other electronic behavior, is central to understanding high-temperature superconductivity.

    1. Relating the optically extracted electron-boson coupling function to the critical temperature of high-TcT_c cuprates: d-wave projection with pair-breaking and the underdoped pseudogap regime
      Open for about 17 years
    2. Pairing glue in electron-doped La2−xCexCuO4\mathrm{La}_{2-x}\mathrm{Ce}_x\mathrm{CuO}_4: direct verification of the spin-fluctuation assignment, the origin of the electronic bosonic spectrum, and the T†T^{\dagger} competing order
      Open for about 9 years
    3. Microscopic identification of the electronic bosonic 'glue' and its interplay with phonons in high-Tc cuprate superconductors
      Open for about 14 years
    More questions

    Quantum theory describes a superposition of possibilities, while an experiment leaves a definite record. How that record forms, and whether different observers can describe it consistently, reaches into our understanding of physical reality.

    1. Deciding between collapse for all observers and unitary evolution in isolated laboratories: the 1/121/12 versus 1/41/4 extended Wigner's-friend discrimination
      Open for about 41 years
    2. ψ-Ontology under weakened preparation independence: the status of the integral-past and exchangeability assumptions
      Open for about 11 years
    3. Quantum dilation of one-world hybrid classical-quantum dynamics: can every shared-noise classical-quantum description be reduced to an underlying fully quantum theory?
      Open for less than 1 year
    More questions

    Dark matter reveals itself through gravity, but its constituents remain unknown. If some dark-matter particles can annihilate, how can the resulting radiation be distinguished from ordinary astrophysical activity and tested across different wavelengths?

    1. Can dark matter annihilation jointly explain the Galactic Center and M31 gamma-ray excesses? Identification or exclusion under controlled systematics
      Open for about 5 years
    2. The survival of the dark-matter interpretation of the Fermi-LAT Galactic center excess under consolidated gamma-ray, antiproton, and positron constraints
      Open for about 5 years
    3. Unmeasured magnetic fields and electron transport in dwarf spheroidal galaxies: the limiting systematic for radio dark-matter searches in Reticulum II
      Open for about 13 years
    More questions

    A small amount of catalyst can change a reaction’s rate and pathway, yet the atomic sites doing the work often change during the reaction. Understanding how those sites form, transform and cooperate can guide more efficient, energy-saving chemistry.

    1. Steady-state active-site dynamics and mechanism of Au/CeO2 catalysts in room-temperature CO oxidation beyond the activation period
    2. Active-site nature and reaction mechanism of supported chromium oxide catalysts in non-oxidative propane dehydrogenation: valence speciation, dynamic evolution, and the desorption/regeneration bottleneck
    3. Generalizing single-atom-catalysis fluorescence correlation spectroscopy (SAC-FCS) beyond two-site-class resolution and a single fluorogenic probe chemistry
    More questions

    Many molecules come in mirror-image forms, like left and right hands, yet life strongly favors one form of key building blocks. Where did the initial imbalance come from, and how was it amplified into a shared feature of life’s chemistry?

    1. The initiating mechanism of life's homochirality: whether metal-assisted parity-violating autocatalysis dominated, and whether its geochemical prerequisites held on the early Earth
    2. Energy thresholds for directed chiral symmetry breaking in autocatalytic self-replication and the viability of parity-violating seeds of homochirality
    3. Origin of the initial enantiomeric excess behind biological homochirality: which pool — amino acids, sugars, or neither — was enantioenriched first on the early Earth?
    More questions

    Hydrogen carries much energy per unit mass, but storing it compactly, safely and economically is difficult. How can storage materials, cryogenic vessels and complete systems balance capacity, reliability and reversible release?

    1. Experimental verification of pressure-tunable reversible hydrogen storage in Ca-intercalated COF-1
    2. Decade-scale vacuum endurance of automotive cryogenic pressure vessels for hydrogen storage: insulation degradation tolerance and in-service vacuum regeneration
    3. Hydride-storage-specific equipment reliability data and verification of the inherent-safety ranking of hydrogen storage technologies
    More questions

    Proteins are moving molecular machines rather than static models, continually shifting among different shapes. How motions on different timescales control recognition, catalysis and long-range regulation is crucial to connecting structure with function.

    1. The timescale dichotomy of protein dynamics: how fast conformational fluctuations couple to slow functional cycles
    2. Establishing or refuting whether true reaction coordinates from the generalized work functional determine protein biological activity: testable confirmation, generality, and theoretical grounding of the structure-function roadmap
    3. Causal mechanism of long-range allostery in the lac repressor: dynamics-mediated switching versus two-state conformational pre-equilibrium
    More questions

    Starting from a small group of cells, how does an organ grow to the right size, bend in the right places and stop growing? Connecting cellular signals, tissue mechanics and developmental history is essential to understanding form and regeneration.

    1. Molecular implementation of morphogen-gradient size scaling and size-matched growth control during axolotl limb regeneration
    2. Molecular control of differential extracellular-matrix growth anisotropy between tissue layers, and completion of the mechanical model, in the developing Drosophila wing disc
    3. What mechanism limits pancreas size by embryonic progenitor cell number, and how general is progenitor-number-limited versus regulative organ size control?
    More questions

    An active microbial world exists in rocks and groundwater beyond the reach of sunlight. Its energy sources, dispersal routes and global extent challenge our understanding of life’s limits and Earth’s material cycles.

    1. Global prevalence and quantitative significance of the geologically mediated microbial dispersal loop in the deep biosphere
    2. Origins, in situ rates, and global extent of hydrogen and dark oxygen fueling high microbial productivity in ancient groundwaters
    3. Functional characterization of microbial life in ultradeep crystalline bedrock: active metabolism, energy sources, and drivers of community composition below 2 km (Otaniemi–Pyhäsalmi, Fennoscandian Shield)
    More questions

    Applied economics

    How much does an extra year of schooling increase earnings? Ability, family background and schooling choices also shape income, so how can we separate the effect of education itself and understand why its returns differ across people?

    1. Identification and interpretation of twins-based returns to education in urban China: residual within-pair ability bias, nonclassical measurement error in schooling reports, and the screening interpretation of low academic-high-school returns
    2. Identification and feasible inference for marginal treatment effects in the returns-to-education selection framework under weaker assumptions
    3. What drove the post-1980 rise in ability bias in the U.S. return to education?
    More questions

    Computer science

    Prediction error need not keep rising as a model becomes more complex: near the point where it can fit the training data, error may rise and then fall again as the model grows. What causes this curve, and can it help us choose models more reliably?

    1. Extending the effective-parameter account of double descent to deep learning: test-time complexity axes and label-free model selection
    2. Non-asymptotic and dynamical completion of the Hessian-based theory of double descent in finite-width neural networks
    3. A rigorous quantitative theory of the three-factor mechanism for test-error divergence at the interpolation threshold
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    Earth science

    When did the Yangtze and Yellow Rivers cut through mountain barriers and connect separate basins into the great rivers we see today? Rocks, sediments and river terraces preserve different timelines; how can we reconcile them and distinguish the roles of uplift, climate change and erosion?

    1. Timing and mechanism of the entrenchment of the Jinshaan Gorge and the formation of the Yellow River's Ordos bend
    2. Timing and mechanism of Three Gorges valley incision and integration of the Yangtze River through the gorges
    3. Origin age and incision mechanism of the upper Yellow River valley across the northeastern Tibetan Plateau
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    More open questions are being explored.