Morse theory is basically the toy case of a very general way of thinking about Lagrangians / Energy landscapes in physics. You can think of the Morse function as a kind of classical Hamiltonian, paths of steepest descent (gradient descent with respect to the morse function) can be identified as Instantons of supersymmetric quantum mechanics. This was the revolutionary insight of Witten in the 1980s ("Supersymmetry and Morse theory"). Another down to earth interpretation is to think of those paths as elastic rubber bands in high dimensional curved space.
Starting from this insight Witten and others identified sets of other Morse theory like constructions (Donaldson theory got condensed from ~1000s of pages to 40, Witten's treatment of Chern-Simons theory resulted in a fields medal).
> Morse theory is basically the toy case of a very general way of thinking about Lagrangians / Energy landscapes in physics. You can think of the Morse function as a kind of classical Hamiltonian, paths of steepest descent
are you sure that these sentences will be useful to a non-mathematician?
An easier concept to a layman may be the combinatorial properties of topographic maps. For example, the number of mountain peaks, lakes, and mountain passes on an island are not independent but must satisfy a strict numerical identity.
Starting from this insight Witten and others identified sets of other Morse theory like constructions (Donaldson theory got condensed from ~1000s of pages to 40, Witten's treatment of Chern-Simons theory resulted in a fields medal).