  Two seemingly unrelated problems are intimately connected.
  The first is the equsingularity problem in $\R^2$: For an analytic family
$f_t:(\R^2,0)\rar (\R,0)$, when should it be called an ``equisingular
deformation"? This amounts to finding a suitable trivialization condition (as
strong as possible) and, of course, a criterion.
  The second is on the Morse stability. We define $\R_*$, which is $\R$
"enriched" with a class of infinitesimals. How to generalize the Morse
Stability Theorem to polynomials over $\R_*$?

Algebraic Geometry