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RESEARCH |
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OPEN QUESTIONS |
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June 14th, 2008 |
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GENERAL |
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Lists of unsolved problems |
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Science magazine 125 big questions |
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MATHEMATICS (PHYSICIST'S PERSPECTIVE) |
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Sir Michael Atiyah's Fields Lecture (.ps)
Areas long to learn: quantum groups, motivic cohomology, local and micro local analysis of large finite groups
Exotic areas: infinite Banach spaces, large and inaccessible cardinals
Some recent links between mathematics and physics
Number theory and physics
Conjectured links between the Riemann zeta function and chaotic quantum-mechanical systems
Deep and relatively recent ideas in mathematics and physics
Standard model and mathematics:
Gauge field or connection
Dirac operators or fundamental classes in K-theory (Atiyah-Singer index theorem)
String theory and mathematics:
Mirror symmetry
Conformal field theory
Mathematics behind supersymmetry
Mathematics of M-Theory
Chern-Simons theory
Higher gauge theory
Geometric Langlands Program |
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Unified theory: Langlands Program, Witten on Langlands,
Theory of "motives" |
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Lists of unsolved problems
ABC Conjecture
Lang Conjecture
Long standing open problems PRICE
P versus NP
The Hodge Conjecture
The Poincaré Conjecture (solved)
The Riemann Hypothesis
Yang-Mills Existence and Mass Gap
Navier-Stokes Existence and Smoothness
The Birch and Swinnerton-Dyer Conjecture
Mathworld list
Mathematical challenges of the 21st century including moduli spaces and borderland physics
Goldbach conjecture
Normality of pi digits in an integer base
Unsolved problems and difficult to understand areas |
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PRICES
Fields Medal and Rolf Nevanlinna Prize
Abel Prize |
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PHYSICS |
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Important unsolved problems in physics
Quantum gravity
Explaining high-Tc superconductors
Complete theory of the nucleus
Realizing the potential of fusion energy
Climate prediction
Turbulence
Glass physics
Solar magnetic field
Complexity, catastrophe and physics
Consciousness |
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Required mathematics |
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Peter Woit's list
Riemannian geometry
More general geometry of principal and vector bundles: connection, curvature, etc.
Spinor geometry
Lie groups and representation theory
deRham cohomology |
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Required physics |
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Another list from the European Journal of Physics |
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Learn it all in one fell swoop |
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Physics Today (NRC) |
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Learn it all on the web |
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John Baez's list |
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David Gross list |
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Nature's greatest puzzles at SLAC |
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PRICES
Nobel Prize for physics
Wolff |
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COSMOLOGY AND ASTROPHYSICS |
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Eleven key questions about the universe |
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Inflation |
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Survey Of The Universe
Linde-Vilenkin, inflation (horizon, flatness, density-fluctuation) Photons, ordinary visible matter, ordinary nonluminous matter, MACHOs, exotic dark matter WIMPs, dark energy, Standard model, supersymmetry, technicolor, string theory, M-theory, Multiverses (eternal inflation, Smolin, ekpyrosis) |
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Brane cosmology |
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Ekpyrotic Universe |
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Randall-Sudrum |
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Black holes and nonlocality |
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QUANTUM GRAVITY |
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What they look like:
Schrodinger's equation
Dirac's equation
Einstein's equation
Superstring action
Connes-Chamseddine spectral action
Area eigenvalues |
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Survey of quantum gravity |
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Problem of continuous approaches: parametrization of the dynamical degrees of freedom in a diffeomorphism invariant way |
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M-theory |
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Problem of infinity of vacua |
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Top questions in M-theory |
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Review article |
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Top ten string theory questions |
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String theory |
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Ten physics problems for the next millenium from the Strings 2000 Conference
Are all the (measurable) dimensionless paramters that characterize the physical universe calculable in principle or are some merely determined by historical or quantum mechanical accident and uncalculable?
How can quantum gravity help explain the origin of the universe?
What is the lifetime of the proton and how do we understand it?
Is nature supersymmetric and if so, how is supersymmetry broken?
Why does the universe appear to have one time and three space dimensions?
Why does the cosmological constant have the value that it has, is it zero and is it really constant?
What are the fundamental degrees of freedom of M-theory (the theory whose low-energy limit is eleven-dimensional supergravity and which subsumes the five consistent superstring theories) and does the theory describe nature?
What is the resolution of the black hole information paradox?
What physics explains the disparity between the gravitational scale and the typical mass scale of the elementary particles?
Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap? |
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Timeline |
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Holographic principle |
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AdS/CFT |
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Supergravity |
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Twistor correspondence |
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What are twistors? |
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Twistor theory |
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Witten's article |
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Problems: construct a quantum theory of gravity from some basic principles assuming noncommutative geometry (John Madore, ...) or express some sector or limit of an underlying theory in terms of the language of noncommutative geometry |
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Noncommutative geometry |
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Euclidean quantum gravity |
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Stephen Hawking |
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Problem: show the classical limit of smooth space-time can be recovered |
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Discrete approaches |
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Lorentzian |
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Regge calculus |
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a variant: causal dynamical calculations |
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Causal sets (Rafael Sorkin, ...) |
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Problem: cannot mimic general relativity at large scales |
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Loop quantum gravity |
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Hamiltonian or spin network approach |
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Lagrgangian approach or spin foam models |
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Topos theory |
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Emerging properties (Sakharov induced gravity, ...) |
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CONDENSED MATTER |
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List of open questions including condensed matter problems |
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PARTICLE PHYSICS |
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Standard model open questions |
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Areas of research |
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Technicolour: the unifying symmetry is a scaled-up version of the strong force |
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Unnaturalness problem: original calculation in which the introduction of the Higgs boson in the standard model gives it and the Z and two W infinite mass |
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Supersymmetry: for every fermion in the standard model, there is a corresponding supersymmetric boson, and vice versa |
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Failure to account for gravity |
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Theories of extra dimensions: there are at least five dimensions and a single new particle, a gravitational boson called a graviton |
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Flavour problem: why are they three and only three generations of fermions and why do the particles in each generation have the masses that they do? |
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Neutrino oscillation |
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Hierarchy problem: why do the different forces operate at such different energies, are they all manifestations of the same underlying phenomenon, and if they are, can they be united mathematically? |
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COMPLEX AND CHAOTIC SYSTEMS |
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Non-linear dynamics
Evolutionary dynamics
Cellular automata
Self-organising systems
Networks |
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PARADOXES |
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Monty Hall, Gamow-Stern, Kruskal, Cantor, Banach-Tarski
List of paradoxes |
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PHYSICS OF FINANCE |
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Future of econometrics |
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Open questions in financial econometrics
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Evolutionary finance |
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Nonparametric estimation |
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Forecasting Economic and Financial Time Series Using Nonlinear Methods |
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Scientific study of financial data |
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UNSOLVED PROBLEMS IN BIOLOGY |
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SEVENTY OPEN QUESTIONS IN ARCHEOLOGY |
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NINE UNDECIPHERED LANGUAGES |
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