The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things as in differential geometry, algebraic topology and measure theory.
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces.
In algebraic topology, the pushforward of a continuous function $\displaystyle f : X\rightarrow Y$ between two topological spaces is a homomorphism $\displaystyle f_{{*}}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)$ between the homology groups for $\displaystyle n\geq 0$.
In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.