In many areas of differential equations topological methods can provide important qualitative information. This particularly applies to complicated problems where the number of variables is large. Examples will be given illustrating the scope and power of topological ideas.
For constant coefficient linear hyperbolic systems the fundamental solution is supported in the forward “light-cone”. For general systems this cone has many compartments and, in some of these (called lacunas) the fundamental solution may vanish. The determination of lacunas is a topological problem (solved originally by Petrovsky) involving the topology of multi-dimensional contour integrals in the complex domain.
For elliptic boundary value problems of Cauchy–Riemann or Dirac type the index, measuring the difference between the number of solutions of the problem and its adjoint, is a topological invariant which can be computed from the geometrical data. This has important applications to models in elementary particle physics.
For a periodic family of self-adjoint elliptic operators the spectral flow, counting the net number of eigenvalues which change sign over a period, is a topological invariant. This can be computed from the geometrical data and can be used to derive sharp bounds on gaps in the spectrum.
In non-linear PDE, “solitons” may have a topological origin, and soliton-interaction can be related to underlying topological features. This is illustrated by models in 2 and 3 spatial dimensions, following original ideas of Skyrme.