  The Hubbard model is a "highly oversimplified model" for electrons in a solid
which interact with each other through extremely short ranged repulsive
(Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two
pieces; H_hop which describes quantum mechanical hopping of electrons, and Hint
which describes nonlinear repulsive interaction. Either H_hop or H_int alone is
easy to analyze, and does not favor any specific order. But their sum
H=H_hop+H_int is believed to exhibit various nontrivial phenomena including
metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism,
Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can
find various interesting "universality classes" of strongly interacting
electron systems by studying the idealized Hubbard model.
  In the present article we review some mathematically rigorous results on the
Hubbard model which shed light on "physics" of this fascinating model. We
mainly concentrate on magnetic properties of the model at its ground states. We
discuss Lieb-Mattis theorem on the absence of ferromagnetism in one dimension,
Koma-Tasaki bounds on decay of correlations at finite temperatures in
two-dimensions, Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in
one-dimension, Lieb's important theorem for half-filled model on a bipartite
lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities
of half-filled models at finite temperatures, and three rigorous examples of
saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to
make the article accessible to nonexperts by describing basic definitions and
elementary materials in detail.
