  We explain the effective renormalization method of quantum field theory in
the Batalin-Vilkovisky formalism and illustrate its mathematical applications
by three geometric examples: (1) Topological quantum mechanics and algebraic
index, (2) Elliptic curve and higher genus mirror symmetry, (3) Calabi-Yau
geometry and integrable hierarchy. This note is an expansion of author's talk
at ICCM 2016.
