I want to ask if the following purely mathematical problem (that I'm working on) might have some applications to physics.

The problem in a nutshell: describe properties of solution sets of real systems of equations that don't change under perturbations of the system.

More formally: consider a system of $n$ real equations in $m$ variables (nonlinear in general); I assume that the domain is compact, for example $0\leq x_i\leq 1$ for each variable. Such system can be formalized as $f(x)=0$ for $f: K\to\mathbb{R}^n$ for an $m$-dimensional domain $K$, $f$ continuous.

For $r>0$ (the "error" in our knowledge of the system) I want to compute properties of the infinite family of sets $$ Z_r(f):=\{g^{-1}(0)|\,\,g:K\to\mathbb{R}^n\,\,\mathrm{continuous,}\,\|g-f\|_\infty<r\} $$ If $m=n$, the generic solution of $g(x)=0$ is discrete and the description of $Z_r(f)$ can be reduced to the computation of topological degree. For $n<m<2n-2$, some properties of $Z_r(f)$ can still be algorithmically computed. Surprisingly, for $m\geq 2n-2$ and $f$ a piece-wise linear function on a simplicial complex, the question whether $\emptyset\in Z_r(f)$ is already algorithmically undecidable (it can be reduced to undecidable problems in unstable homotopy theory).

Are there natural instances of physical problems where one is interested in topological properties of solution sets of systems of equations robust wrt. $r$-perturbations of the system?

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  • $\begingroup$ Would solitons qualify? In wave phenomena, they can propagate through other waves without being affected. I would imagine that is similar to experiencing no effects from a perturbation. $\endgroup$ – honeste_vivere Jun 23 '16 at 19:16
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    $\begingroup$ I suspect that many users on Physics SE would (like me) be non-plussed by the mathematical formalism of your question. Is it possible for you to translate into language which non-mathematicians can understand? It might help if you could give an example, even if only in a (simple) mathematical context. $\endgroup$ – sammy gerbil Jul 23 '16 at 13:43

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