Keywords: astronomy, telescopes, observatories, space exploration, astrophysics, radio astronomy, cosmology, stellar observation, astronomical instruments, space science, galaxy discovery, Edwin Hubble, Mount Wilson Observatory, Parkes Radio Telescope, ALMA telescope, Jantar Mantar observatory, Leiden Observatory, ancient astronomy, radio telescopes, optical telescopes, celestial mechanics, star formation, pulsar detection, cosmic microwave background, deep space observation, astronomical history, telescope technology, space research, universe exploration, astronomical discoveries, stellar classification, galactic astronomy, infrared astronomy, millimeter astronomy, astronomical photography, constellation mapping, planetary observation, solar system exploration, exoplanet discovery, dark matter research, big bang theory, redshift measurement, spectroscopy, astronomical data analysis, observatory construction, telescope mirrors, radio interferometry, astronomical instruments history, space telescope technology, ground based telescopes, astronomical measurement, cosmic distance ladder, variable stars, supernova detection, neutron stars, black hole observation, interstellar medium, star clusters, nebulae observation, galactic structure, local group galaxies, cosmic evolution, astronomical catalogues, celestial coordinates, astronomical societies, observatory tours, stargazing equipment, amateur astronomy, professional astronomy, astronomical education, space science careers, telescope maintenance, observatory operations, astronomical software, sky surveys, astronomical databases, celestial photography, astrophotography techniques, telescope accessories, observatory locations, dark sky sites, light pollution effects, atmospheric seeing, telescope resolution, magnification limits, focal length calculations, aperture ratios, telescope mounts, equatorial tracking, alt-azimuth systems, computerized telescopes, goto telescopes, telescope reviews, observatory visits, astronomical events, meteor showers, lunar eclipses, solar eclipses, planetary alignments, comet observations, asteroid tracking, satellite tracking, international space station, space missions, NASA discoveries, ESA projects, space agencies, astronomical conferences, research publications, peer review process, scientific methodology, observational techniques, data reduction, statistical analysis, computational astronomy, theoretical models, cosmological parameters, hubble constant, cosmic inflation, dark energy research, gravitational waves, multi-messenger astronomy, time domain astronomy, transient events, gamma ray bursts, active galactic nuclei, quasar observations, galaxy clusters, large scale structure, cosmic web, redshift surveys, photometric systems, astrometry, proper motion, parallax measurements, stellar distances, luminosity functions, color magnitude diagrams, hertzsprung russell diagram, stellar evolution, main sequence stars, giant stars, white dwarf stars, brown dwarf detection, exoplanet atmospheres, habitable zones, astrobiology, SETI research, technosignatures, biosignatures, planetary science, comparative planetology, solar wind, magnetospheres, aurora phenomena, space weather, coronal mass ejections, solar flares, sunspot cycles, helioseismology, stellar seismology, gravitational lensing, weak lensing, strong lensing, microlensing, cosmological simulations, n body simulations, hydrodynamic codes, particle physics, quantum mechanics, general relativity, special relativity, spacetime curvature, event horizons, wormholes, time travel, multiverse theories, anthropic principle, fine tuning, cosmological constant, vacuum energy, virtual particles, hawking radiation, information paradox, string theory, loop quantum gravity, extra dimensions, supersymmetry, dark matter candidates, wimps, axions, sterile neutrinos, modified gravity, mond theory, primordial black holes, cosmic strings, domain walls, monopoles, inflation models, eternal inflation, cyclic cosmology, ekpyrotic scenario, big rip, heat death, vacuum decay, false vacuum, phase transitions, symmetry breaking, grand unified theories, theory of everything, quantum gravity, emergent gravity, holographic principle, ads cft correspondence, black hole thermodynamics, entropy bounds, information theory, computational complexity, algorithmic information, kolmogorov complexity, cellular automata, digital physics, simulation hypothesis, consciousness, free will, determinism, chaos theory, complexity science, systems theory, network theory, graph theory, topology, differential geometry, tensor calculus, group theory, representation theory, lie algebras, fiber bundles, gauge theories, yang mills, standard model, beyond standard model, particle accelerators, collider experiments, neutrino detectors, gravitational wave detectors, ligo, virgo, kagra, lisa, cosmic ray detectors, gamma ray telescopes, x ray observatories, ultraviolet telescopes, infrared telescopes, submillimeter telescopes, radio telescope arrays, very long baseline interferometry, space based telescopes, hubble space telescope, james webb space telescope, spitzer space telescope, kepler mission, tess mission, gaia mission, planck satellite, wmap mission, euclid mission, roman space telescope, square kilometer array, thirty meter telescope, extremely large telescope, giant magellan telescope, vera rubin observatory, cherenkov telescope array, ice cube neutrino observatory, antarctic telescopes, south pole telescope, bicep experiment, cosmic microwave background polarization, baryon acoustic oscillations, type ia supernovae, cepheid variables, rr lyrae stars, surface brightness fluctuations, tully fisher relation, fundamental plane, sunyaev zeldovich effect, integrated sachs wolfe effect, gravitational redshift, time dilation, light travel time, lookback time, comoving distance, angular diameter distance, luminosity distance, distance modulus, absolute magnitude, apparent magnitude, bolometric magnitude, color indices, extinction correction, interstellar reddening, dust lanes, molecular clouds, bok globules, herbig haro objects, protoplanetary disks, circumstellar disks, debris disks, kuiper belt, oort cloud, asteroid belt, trojan asteroids, near earth objects, potentially hazardous asteroids, meteor showers, meteorites, carbonaceous chondrites, iron meteorites, stony iron meteorites, impact craters, mass extinctions, tunguska event, chicxulub crater, lunar craters, planetary geology, comparative geology, volcanism, tectonics, erosion, weathering, atmospheric dynamics, climate science, greenhouse effect, runaway greenhouse, snowball earth, habitable zone, goldilocks zone, tidal locking, synchronous rotation, orbital resonance, three body problem, restricted three body problem, lagrange points, hill sphere, roche limit, tidal forces, precession, nutation, obliquity, axial tilt, seasons, calendar systems, timekeeping, atomic clocks, coordinated universal time, leap seconds, time zones, equation of time, analemma, sundials, astrolabes, armillary spheres, quadrants, sextants, theodolites, transit instruments, meridian circles, photographic plates, ccd cameras, cmos sensors, photomultiplier tubes, spectrographs, echelle spectrographs, fiber optics, adaptive optics, laser guide stars, deformable mirrors, coronagraphs, interferometers, polarimeters, bolometers, calorimeters, transition edge sensors, kinetic inductance detectors, superconducting quantum interference devices, josephson junctions, dilution refrigerators, cryogenic systems, vacuum systems, optical coatings, anti reflection coatings, dichroic filters, neutral density filters, polarizing filters, bandpass filters, longpass filters, shortpass filters, beam splitters, prisms, gratings, mirrors, lenses, objectives, eyepieces, focal reducers, barlow lenses, field flatteners, corrector plates, schmidt correctors, maksutov correctors, cassegrain telescopes, newtonian telescopes, refractor telescopes, compound telescopes, catadioptric telescopes, ritchey chretien telescopes, dall kirkham telescopes, nasmyth focus, prime focus, coude focus, gregorian telescopes, keplerian telescopes, galilean telescopes, binoculars, spotting scopes, finder scopes, guide scopes, autoguiders, pier mounts, tripod mounts, dobsonian mounts, german equatorial mounts, fork mounts, strain wave gears, harmonic drives, stepper motors, servo motors, encoders, goto systems, computerized mounts, star alignment, polar alignment, drift alignment, autoguiding, periodic error correction, backlash compensation, flexure compensation, atmospheric dispersion correctors, field rotators, instrument rotators, cable wraps, dome automation, observatory control systems, weather stations, cloud sensors, seeing monitors, sky quality meters, all sky cameras, meteor cameras, fireball networks, satellite tracking, space debris, kessler syndrome, space situational awareness, orbital mechanics, kepler laws, newton laws, vis viva equation, hohmann transfers, bi elliptic transfers, gravity assists, slingshot maneuvers, launch windows, synodic periods, opposition, conjunction, elongation, phase angle, albedo, bond albedo, geometric albedo, phase function, opposition effect, photometry, astrometry, spectroscopy, polarimetry, interferometry, occultations, transits, eclipses, limb darkening, gravitational microlensing, stellar aberration, parallax, proper motion, radial velocity, doppler shift, redshift, blueshift, relativistic beaming, time dilation, gravitational redshift, cosmological redshift, hubble law, hubble constant, hubble tension, distance ladder, standard candles, standard rulers, cosmic microwave background, big bang nucleosynthesis, recombination, decoupling, last scattering surface, acoustic peaks, silk damping, integrated sachs wolfe, matter power spectrum, transfer function, growth function, linear theory, nonlinear evolution, halo model, friends of friends, spherical overdensity, virial theorem, density profiles, navarro frenk white, einasto profile, burkert profile, isothermal sphere, king model, plummer model, hernquist model, jaffe model, dehnen model, sersic profile, exponential disk, bulge disk decomposition, spiral arms, density waves, bar instabilities, galactic dynamics, stellar kinematics, velocity dispersions, rotation curves, dark matter halos, substructure, via lactea, aquarius simulation, millennium simulation, illustris simulation, eagle simulation, horizon agn, magneticum simulation, flamingo simulation, colibre simulation, bahamas simulation, cosmo owls, owls simulation, reference simulation, recalibration, numerical methods, finite difference, finite element, finite volume, smoothed particle hydrodynamics, adaptive mesh refinement, tree codes, particle mesh, pp pm, gadget code, arepo code, ramses code, enzo code, flash code, athena code, pluto code, pencil code, phantom code, sphng code, mesa code, cloudy code, mappings code, starburst99, pegase code, galaxev code, fsps code, bpass code, slug code, pdrs code, radmc 3d, lime code, mollie code, cassis code, gildas package, casa software, miriad package, aips package, obit package, parseltongue, difmap, modelfit, uvmultifit, vis sample, galario, frank code, eddy code, bettermoments, diskjockey, gofish code, keplerian mask, moments maps, position velocity diagrams, channel maps, integrated intensity, peak intensity, velocity fields, velocity dispersions, linewidths, optical depths, excitation temperatures, column densities, abundance ratios, isotope ratios, chemical evolution, nucleosynthesis, stellar yields, galactic chemical evolution, supernova feedback, stellar winds, planetary nebulae, asymptotic giant branch, horizontal branch, red giant branch, main sequence turnoff, isochrones, color magnitude diagrams, synthetic photometry, stellar population synthesis, initial mass function, salpeter function, kroupa function, chabrier function, diet salpeter, top heavy, bottom heavy, stellar multiplicity, binary stars, contact binaries, detached binaries, semi detached binaries, common envelope, roche lobe overflow, mass transfer, accretion disks, cataclysmic variables, x ray binaries, millisecond pulsars, gravitational wave sources, compact objects, stellar remnants, planetary nebulae, white dwarfs, neutron stars, black holes, stellar mass black holes, intermediate mass black holes, supermassive black holes, active galactic nuclei, quasars, blazars, seyfert galaxies, radio galaxies, bll lacertae objects, optically violent variables, changing look quasars, tidal disruption events, jets, relativistic jets, superluminal motion, inverse compton scattering, synchrotron radiation, bremsstrahlung, thermal emission, non thermal emission, power law spectra, spectral indices, flux densities, luminosities, eddington limit, accretion rates, radiative efficiency, spin parameters, kerr black holes, schwarzschild black holes, reissner nordstrom, no hair theorem, black hole uniqueness, cosmic censorship, singularities, naked singularities, wormholes, traversable wormholes, closed timelike curves, causality violations, grandfather paradox, novikov self consistency, chronology protection, hawking radiation, black hole evaporation, information paradox, firewall paradox, complementarity, holographic principle, ads cft, gauge gravity duality, emergent spacetime, entropic gravity, quantum error correction, tensor networks, holographic entanglement entropy, bulk reconstruction, quantum teleportation, quantum entanglement, bell inequalities, epr paradox, hidden variables, many worlds interpretation, copenhagen interpretation, pilot wave theory, objective collapse, consciousness collapse, measurement problem, decoherence, einselection, quantum darwinism, quantum biology, quantum coherence, quantum computing, quantum algorithms, quantum supremacy, quantum advantage, quantum error correction, topological quantum computing, adiabatic quantum computing, quantum annealing, quantum machine learning, variational quantum eigensolver, quantum approximate optimization, quantum fourier transform, shor algorithm, grover algorithm, quantum teleportation, quantum cryptography, quantum key distribution, quantum internet, quantum sensors, atomic clocks, optical lattice clocks, ion traps, neutral atoms, superconducting qubits, transmon qubits, flux qubits, spin qubits, photonic qubits, squeezed light, non classical light, fock states, coherent states, thermal states, vacuum fluctuations, zero point energy, casimir effect, dynamical casimir effect, unruh effect, hawking unruh, equivalence principle, general covariance, diffeomorphism invariance, coordinate systems, reference frames, proper time, coordinate time, simultaneity, relativity of simultaneity, length contraction, time dilation, twin paradox, pole barn paradox, ladder paradox, relativistic velocity addition, four vectors, minkowski spacetime, lorentz transformations, poincare transformations, special relativity, general relativity, curved spacetime, metric tensor, christoffel symbols, riemann tensor, ricci tensor, einstein tensor, stress energy tensor, einstein field equations, cosmological constant, lambda cdm, concordance model, standard model of cosmology, inflation, cosmic inflation, eternal inflation, chaotic inflation, new inflation, hybrid inflation, natural inflation, axion inflation, string inflation, brane inflation, ekpyrotic cosmology, cyclic cosmology, bouncing cosmology, emergent cosmology, causal set theory, loop quantum cosmology, asymptotic safety, causal dynamical triangulation, spin foam models, group field theory, emergent gravity, induced gravity, sakharov gravity, thermodynamic gravity, entropic force, modified gravity, f r gravity, scalar tensor theories, brans dicke theory, jordan frame, einstein frame, conformal transformations, fifth force, equivalence principle tests, gravitational redshift tests, frame dragging, lense thirring effect, geodetic effect, shapiro delay, perihelion precession, gravitational lensing, weak lensing, strong lensing, microlensing, einstein rings, arcs, arclets, shear, convergence, magnification, time delays, fermat principle, lens equation, critical curves, caustics, fold caustics, cusp caustics, astroid caustics, diamond caustics, pseudocaustics, demagnification, multiple images, odd number theorem, magnification bias, lensing degeneracies, mass sheet degeneracy, source position transformation, lens modeling, parametric models, non parametric models, pixelated models, adaptive grid, free form, grale code, lenstool, glafic code, gravlens, lensmodel, fastlens, wl analysis, kids survey, des survey, hsc survey, euclid survey, lsst survey, roman survey, weak lensing mass mapping, cluster lensing, galaxy galaxy lensing, cosmic shear, intrinsic alignments, photo z errors, shear calibration, multiplicative bias, additive bias, selection bias, boost factors, blending, crowding, shape measurement, ellipticity, shear estimators, ksb method, regauss method, lensfit method, ngmix method, metacalibration, shear testing programme, great3 challenge, great10 challenge, step program, gravitational wave astronomy, ligo collaboration, virgo collaboration, kagra collaboration, et collaboration, cosmic explorer, lisa mission, taiji mission, tianqin mission, decigo mission, big bang observer, ultimate deci hertz, atom interferometer, terrestrial detectors, space based detectors, laser interferometry, michelson interferometer, fabry perot cavities, power recycling, signal recycling, squeezed light injection, quantum noise, thermal noise, seismic noise, newtonian noise, suspension systems, test masses, mirror coatings, vacuum systems, vibration isolation, active isolation, passive isolation, feedback control, lock acquisition, length sensing, angular sensing, photodiodes, quadrant photodiodes, balanced homodyne, rf sidebands, pound drever hall, schnupp modulation, arm cavity, mode cleaner, input mode cleaner, output mode cleaner, prestabilized laser, laser frequency stabilization, temperature stabilization, seismic isolation stacks, multiple pendulum suspensions, silica fibers, fused silica, sapphire test masses, silicon test masses, cryogenic operation, maraging steel, invar, ultra high vacuum, turbo molecular pumps, ion pumps, titanium sublimation pumps, residual gas analyzers, leak detection, bakeout procedures, outgassing rates, partial pressures, contamination control, clean room protocols, particle counting, surface analysis, atomic force microscopy, scanning electron microscopy, x ray photoelectron spectroscopy, secondary ion mass spectrometry, optical profilers, interferometric microscopy, white light interferometry, stylus profilometry, contact angle measurements, ellipsometry, spectroscopic ellipsometry, variable angle, wavelength scanning, real time monitoring, in situ measurements, ex situ characterization, thin film deposition, sputtering, evaporation, molecular beam epitaxy, chemical vapor deposition, atomic layer deposition, ion beam assisted deposition, pulsed laser deposition, electron beam evaporation, thermal evaporation, resistance heating, ion beam sputtering, magnetron sputtering, reactive sputtering, co sputtering, substrate heating, substrate biasing, plasma assisted processes, rf plasma, dc plasma, inductively coupled plasma, electron cyclotron resonance, helicon plasma, plasma diagnostics, langmuir probes, optical emission spectroscopy, mass spectrometry, residual gas analysis, quartz crystal microbalance, film thickness monitoring, rate monitoring, uniformity control, substrate rotation, planetary rotation, substrate holders, load locks, transfer chambers, cluster tools, multi chamber systems, process control, recipe development, parameter optimization, design of experiments, statistical process control, six sigma, lean manufacturing, total quality management, iso certification, clean room classification, particle contamination, molecular contamination, electrostatic discharge, humidity control, temperature control, air flow patterns, laminar flow, turbulent flow, hepa filters, ulpa filters, activated carbon filters, chemical filters, air handling units, fan filter units, minienvironments, isolators, glove boxes, biosafety cabinets, fume hoods, chemical storage, waste disposal, hazardous materials, safety data sheets, risk assessment, hazard analysis, failure mode effects analysis, fault tree analysis, event tree analysis, probabilistic risk assessment, monte carlo methods, latin hypercube sampling, importance sampling, markov chain monte carlo, gibbs sampling, metropolis hastings algorithm, hamiltonian monte carlo, no u turn sampler, variational inference, expectation maximization, maximum likelihood estimation, method of moments, generalized method of moments, instrumental variables, two stage least squares, three stage least squares, seemingly unrelated regression, panel data models, fixed effects, random effects, hausman test, durbin wu hausman test, sargan test, hansen j test, weak instruments, strong instruments, cragg donald statistic, stock yogo critical values, anderson rubin test, conditional likelihood ratio, lagrange multiplier test, wald test, likelihood ratio test, akaike information criterion, bayesian information criterion, hannan quinn criterion, cross validation, leave one out, k fold, bootstrap, jackknife, permutation tests, randomization tests, exact tests, asymptotic tests, finite sample properties, consistency, unbiasedness, efficiency, asymptotic normality, central limit theorem, law of large numbers, slutsky theorem, continuous mapping theorem, delta method, cramer wold theorem, portmanteau theorem, lindeberg feller theorem, lyapunov theorem, martingale central limit theorem, functional central limit theorem, invariance principle, brownian motion, wiener process, levy processes, poisson processes, compound poisson, jump diffusion, stochastic differential equations, ito calculus, stratonovich calculus, girsanov theorem, feynman kac theorem, fokker planck equation, kolmogorov equations, dynkin formula, generator, semigroup, markov property, strong markov property, stopping times, optional stopping, doob decomposition, martingale representation, predictable quadratic variation, local martingales, semimartingales, cadlag processes, skorohod topology, weak convergence, tightness, prokhorov theorem, continuous mapping, portmanteau lemma, levy metric, wasserstein distance, kantorovich rubinstein, optimal transport, monge problem, kantorovich problem, cyclical monotonicity, c transform, convex duality, brenier theorem, mccann theorem, displacement interpolation, wasserstein geodesics, otto calculus, jordan kinderlehrer otto, gradient flows, heat equation, porous medium equation, fast diffusion, logarithmic sobolev inequality, poincare inequality, bakry emery criterion, ricci curvature, sectional curvature, scalar curvature, riemann curvature tensor, bianchi identities, gauss bonnet theorem, chern gauss bonnet, atiyah singer index theorem, hirzebruch riemann roch, grothendieck riemann roch, serre duality, kodaira vanishing, kawamata viehweg vanishing, nadel vanishing, inversion of adjunction, multiplier ideals, test ideals, tight closure, frobenius splitting, f rationality, f regularity, f purity, cohen macaulay, gorenstein, canonical module, dualizing complex, local cohomology, depth, dimension, krull dimension, transcendence degree, noether normalization, hilbert nullstellensatz, zariski topology, constructible sets, irreducible components, primary decomposition, associated primes, minimal primes, embedded primes, support, annihilator, radical, jacobson radical, nilradical, spectrum, maximal spectrum, real spectrum, etale cohomology, galois cohomology, group cohomology, lie algebra cohomology, de rham cohomology, dolbeault cohomology, cech cohomology, sheaf cohomology, derived functors, ext functor, tor functor, homological dimension, global dimension, weak dimension, finitistic dimension, little finitistic dimension, representation type, tame type, wild type, auslander reiten quiver, almost split sequences, irreducible morphisms, indecomposable modules, krull schmidt theorem, fitting lemma, nakayama lemma, artinian rings, noetherian rings, principal ideal domains, unique factorization domains, dedekind domains, discrete valuation rings, complete local rings, henselian rings, strict henselization, completion, adic topology, inverse limit, direct limit, flat modules, faithfully flat, descent theory, fpqc topology, etale topology, nisnevich topology, zariski topology, site, topos, grothendieck topology, sheafification, stalks, germs, presheaves, sheaves, flasque sheaves, soft sheaves, fine sheaves, acyclic sheaves, godement resolution, injective resolution, projective resolution, free resolution, minimal resolution, betti numbers, castelnuovo mumford regularity, hilbert function, hilbert polynomial, hilbert series, poincare series, growth function, gelfand kirillov dimension, bernstein inequality, weyl algebra, universal enveloping algebra, poincare birkhoff witt theorem, highest weight modules, verma modules, category o, kazhdan lusztig polynomials, hecke algebras, iwahori hecke algebras, affine hecke algebras, double affine hecke algebras, cherednik algebras, rational cherednik algebras, symplectic reflection algebras, quiver varieties, nakajima quiver varieties, moduli spaces, hilbert schemes, quot schemes, picard schemes, jacobian varieties, albanese varieties, chow varieties, barlet spaces, douady spaces, parameter spaces, deformation theory, infinitesimal deformations, obstruction theory, derived deformation theory, l infinity algebras, dg lie algebras, operads, props, colored operads, symmetric operads, non symmetric operads, cyclic operads, modular operads, swiss cheese operad, little disks operad, associahedron, permutohedron, cyclohedron, multiplihedron, compactifications, wonderful compactification, de concini procesi, bott samelson, hansen, springer resolution, grothendieck springer, steinberg variety, flag varieties, grassmannians, schubert varieties, richardson varieties, opposite cells, moment graphs, gkm theory, torus actions, equivariant cohomology, localization theorem, atiyah bott localization, kirwan surjectivity, martin theorem, duistermaat heckman, symplectic reduction, marsden weinstein, geometric invariant theory, mumford conjecture, hilbert mumford criterion, moment maps, hamiltonian actions, symplectic quotients, kahler quotients, hyperkahler quotients, nahm equations, bpst instantons, adhm construction, twistor spaces, penrose transform, ward correspondence, seiberg witten, gromov witten, donaldson thomas, pandharipande thomas, stable maps, quantum cohomology, mirror symmetry, homological mirror symmetry, syz conjecture, strominger yau zaslow, large complex structure limit, landau ginzburg models, matrix factorizations, fukaya categories, wrapped fukaya, microlocal sheaves, constructible sheaves, perverse sheaves, intersection cohomology, ique singularities, rational singularities, log canonical singularities, kawamata log terminal, divisorial log terminal, purely log terminal, minimal model program, abundance conjecture, termination conjecture, flip conjecture, cone theorem, contraction theorem, base point free theorem, rationality theorem, bend and break, rationally connected, uniruled, kodaira dimension, iitaka dimension, plurigenera, canonical ring, log canonical ring, cox ring, total coordinate ring, toric varieties, fans, cones, lattices, polytopes, newton polytopes, secondary polytopes, fiber polytopes, cyclic polytopes, birkhoff polytopes, matroid polytopes, root polytopes, weight polytopes, string polytopes, canonical bases, crystal bases, tropicalization, tropical geometry, amoebas, coamoebas, ronkin function, logarithmic limit sets, valuations, berkovich spaces, rigid analytic geometry, non archimedean geometry, perfectoid spaces, tilting equivalence, almost mathematics, diamonds, prismatic cohomology, motivic cohomology, algebraic k theory, topological cyclic homology, trace methods, cyclotomic spectra, equivariant stable homotopy, genuine equivariant, RO graded, mackey functors, tambara functors, green functors, burnside rings, representation rings, character theory, induced representations, restriction, frobenius reciprocity, mackey decomposition, induction functor, restriction functor, conjugation, centralizer, normalizer, sylow theorems, hall theorems, schur zassenhaus, jordan holder, composition series, solvable groups, nilpotent groups, p groups, abelian groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, general linear groups, special linear groups, orthogonal groups, symplectic groups, unitary groups, exceptional groups, lie groups, algebraic groups, linear algebraic groups, reductive groups, parabolic subgroups, borel subgroups, maximal tori, weyl groups, root systems, weight lattices, dominant weights, alcove, affine weyl group, extended affine weyl group, double affine weyl group, braid groups, artin groups, coxeter groups, buildings, simplicial complexes, cell complexes, cw complexes, simplicial sets, kan complexes, quasicategories, infinity categories, model categories, quillen equivalences, homotopy limits, homotopy colimits, derived functors, triangulated categories, stable infinity categories, spectra, ring spectra, module spectra, symmetric monoidal categories, braided monoidal categories, ribbon categories, modular tensor categories, fusion categories, vertex operator algebras, conformal field theory, chiral algebras, factorization algebras, costello gwilliam, quantum field theory, feynman diagrams, renormalization, regularization, dimensional regularization, pauli villars, point splitting, zeta function regularization, heat kernel regularization, effective field theory, wilsonian renormalization, exact renormalization group, functional renormalization group, wetterich equation, polchinski equation, noncommutative geometry, spectral triples, dirac operators, fredholm modules, cyclic cohomology, entire cyclic cohomology, periodic cyclic cohomology, connes chern character, local index formula, dixmier trace, noncommutative tori, theta functions, modular forms, eisenstein series, cusp forms, hecke operators, newforms, oldforms, twist, congruences, galois representations, modular galois representations, serre conjecture, fontaine mazur conjecture, buzzard taylor, diamond taylor, kisin, emerton gee, calegari geraghty, thorne, newton slopes, hodge tate weights, de rham representations, crystalline representations, semistable representations, potentially semistable, geometric galois representations, compatible systems, monodromy, inertia, frobenius, verschiebung, cartier operator, hasse witt matrix, a numbers, newton polygons, ekedahl oort, supersingular, ordinary, p divisible groups, barsotti tate groups, dieudonne theory, formal groups, honda tate theory, complex multiplication, main theorem of complex multiplication, kronecker jugendtraum, hilbert class field, ray class field, artin reciprocity, chebotarev density, effective chebotarev, siegel zeros, exceptional zeros, landau siegel zeros, deuring heilbronn phenomenon, linnik constant, least quadratic nonresidue, artin primitive root conjecture, hooley conjecture, elliott halberstam conjecture, bateman horn conjecture, hardy littlewood conjecture, goldbach conjecture, twin prime conjecture, green tao theorem, zhang theorem, maynard theorem, polymath projects, bounded gaps, admissible tuples, prime gaps, cramers conjecture, bertrand postulate, prime number theorem, riemann hypothesis, generalized riemann hypothesis, extended riemann hypothesis, grand riemann hypothesis, selberg class, selberg orthogonality, explicit formulas, weil explicit formula, trace formulas, arthur selberg trace formula, kuznetsov trace formula, petersson trace formula, relative trace formulas, beyond endoscopy, functoriality conjecture, langlands program, automorphic forms, modular forms, maass forms, eisenstein series, poincare series, theta series, siegel modular forms, hilbert modular forms, bianchi modular forms, adelic automorphic forms, cuspidal representations, discrete series, holomorphic discrete series, limits of discrete series, complementary series, principal series, degenerate principal series, induced representations, parabolic induction, jacquet modules, whittaker models, kirillov model, gelfand graev model, multiplicity one, strong multiplicity one, local langlands correspondence, global langlands correspondence, functoriality, base change, symmetric power, exterior power, rankin selberg, euler products, dirichlet series, completed l functions, functional equations, gamma factors, local factors, unramified factors, ramified factors, conductor, artin conductor, swan conductor, different, discriminant, ramification groups, higher ramification groups, herbrand theorem, hasse arf theorem, local class field theory, lubin tate theory, formal group laws, honda theory, gross hopkins, morava k theory, morava e theory, johnson wilson theories, brown peterson cohomology, steenrod algebra, adem relations, serre cartan basis, milnor basis, dual steenrod algebra, hopf algebroids, cobar construction, adams spectral sequence, adams novikov spectral sequence, chromatic spectral sequence, slice spectral sequence, motivic spectral sequence, atiyah hirzebruch spectral sequence, leray serre spectral sequence, grothendieck spectral sequence, lyndon hochschild serre spectral sequence, bar construction, classifying spaces, eilenberg maclane spaces, infinite loop spaces, group completion, plus construction, quillen plus construction, algebraic k theory, waldhausen k theory, hermitian k theory, l theory, surgery theory, s cobordism theorem, h cobordism theorem, poincare conjecture, geometrization conjecture, ricci flow, hamilton program, perelman entropy, noncollapsing, canonical neighborhoods, finite extinction time, ancient solutions, eternal solutions, immortal solutions, type i, type ii, type iii, kappa solutions, gradient shrinking solitons, gradient expanding solitons, gradient steady solitons, bryant solitons, cigar soliton, king rosenau solution, angenent solution, daskalopoulos hamilton, daskalopoulos sesum, brendle schoen, bohm wilking, margerin, ni wallach, petersen wei, sesum tian, chen zhu, kleiner lott, morgan tian, bessières besson maillot, cao zhu, milnor conjecture, bloch kato conjecture, quillen lichtenbaum conjecture