His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis . Hopf's work is also of the greatest importance to the hydrodynamics, theory of turbulence and radiative transfer theory.
Kanyang interes at mga punong-guro ay nagawa sa larangan ng bahagyang at ordinarydifferential equation, calculus ng mga pagkakaiba-iba, ergodic teorya, topological dinamika, integral equation, kaugalian geometry, kumplikadong function teorya at functional analysis. Hopf ng trabaho ay din ng mga pinakamalaking kahalagahan sa haydrodinamika, teorya ng gulo at namamanaag transfer theory.