Harmonic maps can, in many cases, be viewed as analogues of harmonic 1-forms. As such, techniques from Hodge theory can often be carried over to this setting. This lecture will discuss some of the consequences of arguments of this type.
A harmonic map is an extremal for an energy functional on the space of maps from a Riemannian manifold $M$ into another Riemannian manifold $N$ (or, more generally, $N$ may be a metric space with suitable properties). In the case of a map $f\colon M\to N$ between manifolds, one defines the energy to be the $L^2$-norm of the differential $df$; the Euler-Lagrange equation satisfied by an extremal takes the form $D^*df=0$, where $D$ is the exterior derivative associated with the pullback of the Levi-Civita connection on $TN$, and $df$ is regarded as a 1-form with values in $f^*TN$. Thus, the condition that $f$ is harmonic is precisely the condition that its differential is a harmonic 1-form with values in $f^*TN$.
One of the basic facts of Hodge theory is that a harmonic 1-form on a compact Kähler manifold is the sum of a holomorphic 1-form and its complex conjugate. The analogous fact for harmonic maps was discovered by Siu: if $f\colon M\to N$ is a harmonic map from a compact Kähler manifold to a Riemannian manifold with nonpositive complex sectional curvature, then $\partial f$ is holomorphic in a suitable sense. Thi is one of the starting points for a nonabelian Hodge theory for compact Kähler manifolds, reported on by Simpson at the previous ICM. Recent work by A. Reznikov has used these ideas to prove a conjecture of Bloch which asserts that the Cheeger-Chern-Simons classes of fact vector bundles over smooth complex projective varieties are torsion.
In another direction, one can consider harmonic 1-forms on a compact locally symmetric space with nonpositive sectional curvature. Under certain hypotheses, Matsushima proved vanishing of such forms; analogoues for forms with values in certain bundles were proved by Weil, Calabi and others. Analogues of these results can be found for harmonic maps from appropriate symmetric spaces into Riemannian manifolds with nonpositive curvature in a suitably strong sense. The author was able to give a proof of superrigidity over the reals for lattices in the groups ${\rm Sp}(n,1)$ and $F^{-20}_4$ using similar ideas in a nonlinear setting, extending results of Margulis to these rank one groups. Gromov and Schoen developed a theory of harmonic maps into singular spaces, and used this to prove superrigidity in the $p$-adic case. As a consequence, they were able to prove arithmeticity of lattices in these groups. More recently, Mok-Siu-Yeung and Jost-Yau showed how to apply harmonic map techniques in the higher rank case as well, recovering many of Margulis' results in that case. These results also give geometric forms of superrigidity, and lead in particular to metric rigidity theorems for locally symmetric spaces. In the higher rank case, this leads to a new proof of Gromov's metric rigidity theorem. In yet another direction, Korevaar and Schoen have developed a vast generalization of the Gromov-Schoen theory of harmonic maps into singular spaces. Their theory allows one to treat harmonic maps into length spaces with nonpositive curvature, a category of spaces which includes infinite-dimensional Riemannian manifolds of nonpositive curvature, ${\bf R}$-buildings, and spaces of Riemannian metrics. In another direction, Simpson and the author have used the theory of harmonic maps into trees to study the question of when a representation of the fundamental group of a complex quasiprojective variety in ${\rm SL}(2,{\bf C})$ arises as a pullback by a holomorphic map into a Hilbert modular variety.