Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a projective manifold has a pluri-harmonic metric. Moreover, Simpson generalized many important theorems for polarizable variation of Hodge structures, such as Hard Lefschetz Theorem, to the context of harmonic bundles.
To adapt the theory of harmonic bundles to the context of meromorphic flat bundles, or more generally holonomic D-modules, we need to understand the wild singularity of harmonic bundles, the irregular singularity of meromorphic flat bundles, etc. In this talk, we shall explain some of fundamental theorems in these topics.