  Given a reduced local algebra $T$ over a suitable ring or field $k$ we study
the question of whether there are nontrivial algebra surjections $R\to T$ which
induce isomorphisms $\Omega_{R/k}\otimes T \to \Omega_{T/k}$. Such maps, called
evolutions, arise naturally in the study of Hecke algebras, as they implicitly
do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the
existence of non-trivial evolutions of an algebra $T$ can be characterized in
terms of the symbolic square of an ideal defining $T$. We give a
characterization of the symbolic square in terms of Fitting ideals. Using this
and other techniques we show that certain classes of reduced algebras --
codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in
general, and some others -- admit no nontrivial evolutions. On the other hand
we give examples showing that non-trivial evolutions of reduced Cohen-Macaulay
algebras of codimension 3 do exist in every positive characteristic.
