Homotopy Algebras in their Applications in Geometry and Physics

Homotopy algebras naturally appear in the study of different areas of modern mathematics and physics, such as deformation quantization, mirror symmetry, generalized complex geometry, vertex algebras and (string) field theory. In this talk I will discuss two types of such structures, $L_{\infty}$ and $A_{\infty}$ algebras, which are generalizations of Lie algebras and associative algebras correspondingly.