In the first hour of the talk, we introduce the basic notions like Higgs bundles, flat connections, representations, harmonic maps, stability, and the moduli spaces. Then we give a brief introduction to the celebrated non-Abelian Hodge correspondence theory.

In the second hour of the talk, we study an algebraic inequality for nilpotent matrices and show some interesting geometric applications:

(i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface;

(ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi-Yau moduli.

If time permits, we also show a generalization of this work to n-Fuchsian fibers in the moduli space of Higgs bundles. Part of this talk is joint work with Song Dai (Tianjin University).