When electrons in two dimensions are exposed to a high magnetic field, their kinetic energy is suppressed, spin frozen, and only the Coulomb interaction term survives in the Hamiltonian. What could be simpler? We have learned during the past three decades that this system generates some of the most beautiful emergent structures in physics. It has spawned topological particles that have fractional charge and fractional statistics, and it has produced exotic states, such as fractional quantum Hall effect, emergent Fermi sea, topological superconductivity and Wigner crystals. I shall begin with a pedagogical introduction to the understanding of the essential phenomenology in terms of emergent particles called composite fermions. I shall then report on recent progress on two topics: competition between the liquid the crystal phases, and the mechanism of fractional quantum Hall effect in the second Landau level.
This colloquium is organized in the framework of the PSI2 program "Topological phases of matter: from the quantum Hall effect to spin liquids" (Topmat)