We have been developing algorithms for detecting "robust" zeros of multidimensional functions $f: X\to\Bbb R^n$ where $X$ is an $m$-dimensional domain in $\Bbb R^m$. More precisely, for a given $f$, we can compute an $r>0$ such that every continuous $g$, $\|g-f\|<r$ has a nonempty zero set in $X$ (under some quite weak conditions). This can be used for proving that the initial map $f$ has a zero even in cases when it is given only approximately, such as a list of function values in a multidimensional grid and a Lipschitz constant.
More involved robust properties of the zero set of $f$ could be analysed. For instance we can find "robust optima" over the zero set with the uncertainty when an additional objective function is given.
Can these algorithms have some natural applications in engineering and/or scientific computations? In some sense, it generalizes solving systems of equations to the case where the input is uncertain.
The mathematics behind the algorithm is nice and nontrivial (and has been the main motivating force for us so far), but the further development makes the best sense if we had specific problems to solve.