The Theory of Aperiodic Solids: Physics and Mathematics

Solid State Physics: an overview
A historical viewpoint starting with the 1900-Drude's paper
Quantum Theory, Bloch Theorem, band theory
Statistical Mechanics of Fermions, Fermi level, conduction and valence bands.
Metal, insulators, semiconductors
Band calculation, symmetries and group theory (the gruppenpest),
Crystal Electrons in a uniform magnetic field: from Landau and Peierls, to magnetic oscillations.
Semiconductors at very low temperature, the Anderson transition.

C*-algebra: an apology
A historical viewpoint starting with Fourier (1807).
Fourier transform: Bessel->Parseval->Pontryagin->Gelfand & Tanaka.
Axioms for C*-algebra: the norm has purely algebraic origin
Groupoids in the work of Heisenberg, quantization and commutation rules
Groupoids C*-algebras: from Mackey's program, to the books of Connes and Renault.
Noncommutative measure theory, probability, topology, geometry and foundation of Quantum Mechanics.
C*-algebra: the right tool to extend Bloch theory to study and describe aperiodic solids.

Some Important Materials
Carbon, diamond structure, elements in the column IV
Semiconductors, GaAs, other compounds.
Pure metals: copper, gold,
Metallic alloys: steel, bronze,

Aperiodic media
Bloch electrons in a uniform magnetic field.
Time dependent Hamiltonians:
  (i) periodic case, the kicked rotor
  (ii) non periodic time dependence
Lightly doped semi-conductors
Bulk Metallic Glasses

The Noncommutative Brillouin Zone
Hull, Rd-action, Transversal, Groupoids
C*-algebra of the Hull, of the transversal. The periodic case
Calculus: differentiating, integrating
Integrated Density of State, Density of States
Current-current correlations
Numerical approximations

Gap Labelling and K-Theory
Hamiltonians with Cantor spectrum
K-theory: the abstract gap labelling theorem
1D Hamiltonians, automatic potentials,
Gap labelling for finite type tilings (including quasicrystals)

Semi-classical Analysis
Harper like models, Hofstadter butterfly, braiding
Fast rotating molecules
Canonical formalism, the quantum phase space
Arnold-Liouville theorem for quantum integrable systems
Perturbation theory uniform in Planck's constant

Applications of Noncommutative Geometry
Topological Invariants, cyclic cocycles, Bott map
The integer quantum Hall effect (IQHE)
Chern numbers in rotation bands
Topological Insulators, Z2-invariant, electro-magnetic response, higher Chern numbers
Noncommutative Fermi surfaces

Transport theory
Localization: localization length, pure point spectra, KAM method
Localization: the multiscale analysis approach, overview and results
Anomalous diffusion, transport and spectral exponents, anomalous Drude formula
Dissipation, random Hamiltonian processes, the relaxation-time approximation, Kubo's formula
Variable range hopping, the Mott-Efros-Schklovsky theory, application to the IQHE
Transport in quasicrystals

Mathematical Description of the Structure of Aperiodic Solids
Uniformly discrete sets, Hull, Transversal
Delone sets, definition and use for describing liquids and solids
Voronoi tiling, Delone graphs, Voronoi points, contiguousness
Repetitivity and minimality of the Rd-action on the Hull
Finite Local Complexity, the Anderson-Putnam Complex
Meyer sets and quasicrystals, cut-and-project method, Hull and Transversal

Wannier Transform and Bloch Theory for FLC Aperiodic Media
Toolbox, the Lagarias group, the labeling group, address map, lattice representation of FLC Delone sets.
Fourier analysis, Wannier transform
The C*-algebra of an Anderson Putnam complex
Finite volume approximation, Anderson-Putnam complex, Bloch boundary conditions, band spectrum.
Continuous field of C*-algebras associated with a decreasing sequence of Delone sets: continuity
Renormalization, convergences Theorems, convergence estimates.

Ground States
Is the ground state of a solid a crystal ? Densest packing of identical hard balls
Building glasses: densest packing of non-identical hard balls
Are alloys polycristals or glassy at zero temperature ?
Delone sets as describing ground states
Geometrical constraints induced by chemical bonds

Microscopic Approach to Continuum Mechanics (in progress)
Delone Graphs, genericity and almost sure properties
Curved space theory, local cluster theory
The graph of contiguousness, Markov process
The anankeon theory, thermodynamic and dissipative dynamics of liquids
Microscopic description of the anankeon-phonon interaction.
Viscosity and Green-Kubo formula
From the atomic to the macroscopic scale: the STZ theory and the STZ equations.
Computations and numerical simulations: the elasticity, plasticity curve, description of fractures.
Liquid-glass transition: a dynamical effect or a phase transition ?