Open Problems In Mathematics And Physics

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OPEN QUESTIONS

GENERAL

MATHEMATICS (PHYSICIST'S PERSPECTIVE)

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

Unified theory: Langlands Program, Witten on Langlands,
Theory of "motives"

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 centuryincluding moduli spacesand borderland physics
Goldbach conjecture
Normality of pi digits in an integer base
Unsolved problems and difficult to understand areas

PRICES
Fields Medal and Rolf Nevanlinna Prize
Abel Prize

PHYSICS

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

Required mathematics

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

Required physics

Another list from the European Journal of Physics

Learn it all in one fell swoop

Physics Today (NRC)

Learn it all on the web

John Baez's list

David Gross list

Nature's greatest puzzles at SLAC

PRICES
Nobel Prize for physics
Wolff

COSMOLOGY AND ASTROPHYSICS

Eleven key questions about the universe

Inflation

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)

Brane cosmology

Ekpyrotic Universe

Randall-Sudrum

Black holes and nonlocality

QUANTUM GRAVITY

What they look like:
Schrodinger's equation
Dirac's equation
Einstein's equation
Superstring action
Connes-Chamseddine spectral action
Area eigenvalues

Survey of quantum gravity

Problem of continuous approaches: parametrization of the dynamical degrees of freedom in a diffeomorphism invariant way

M-theory

Problem of infinity of vacua

Top ten string theory questions

String theory

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?

Timeline

Holographic principle

AdS/CFT

Supergravity

Twistor correspondence

What are twistors?

Twistor theory

Witten's article

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

Noncommutative geometry

Euclidean quantum gravity

Stephen Hawking

Problem: show the classical limit of smooth space-time can be recovered

Discrete approaches

Lorentzian

Regge calculus

a variant: causal dynamical calculations

Causal sets (Rafael Sorkin, ...)

Problem: cannot mimic general relativity at large scales

Loop quantum gravity

Hamiltonian or spin networkapproach

Lagrgangian approach or spin foam models

Topos theory

Emerging properties (Sakharov induced gravity, ...)

CONDENSED MATTER

List of open questions including condensed matter problems

PARTICLE PHYSICS

Standard model open questions

Areas of research

Technicolour: the unifying symmetry is a scaled-up version of the strong force

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

Supersymmetry: for every fermion in the standard model, there is a corresponding supersymmetric boson, and vice versa

Failure to account for gravity

Theories of extra dimensions: there are at least five dimensions and a single new particle, a gravitational boson called a graviton

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?

Neutrino oscillation

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?

COMPLEX AND CHAOTIC SYSTEMS

Non-linear dynamics

Evolutionary dynamics
Cellular automata
Self-organising systems
Networks

PARADOXES

Monty Hall, Gamow-Stern, Kruskal, Cantor, Banach-Tarski
List of paradoxes

PHYSICS OF FINANCE

Future of econometrics

Open questions in financial econometrics

Evolutionary finance

Nonparametric estimation

Forecasting Economic and Financial Time Series Using Nonlinear Methods

Scientific study of financial data

UNSOLVED PROBLEMS IN BIOLOGY