Mather singularities

Singularities in Mather’s nice dimensions

Below we present a complete and repetition-free list of complex algebras corresponding to the singularities in Mather’s nice dimensions, for l≥0. Recall, that the Mather bond is 6l+8 for l=0,1,2,3, and it is 6l+7 if l>3.

The list contains 55 algebras.

For each algebra below there is a nonempty set of l’s for which that algebra appears corresponding to a singularity C^*->C^{*+l} in the nice dimension range.

Notation: d_i = dim_C(m^i/m^{i+1}) for i=1,2,…

: Thom-Boardman symbol

For a concrete l≥0, a subset of these algebras appear.

For example, for l=0 these.

For example, for l=1 these.

For l≥18, all but the last seven algebras appear.

Names. Some algebras have standard names: A_k, I_{a,b}=I_{ab}, III_{a,b}=III_{ab} (we suppress commas when convenient). For the rest we suggest a name in red below. In these names A, B, C, D, E refer to Thom-Boardman symbols . The first number after the letter refers to . The second number is simply the position in our list. For example C_{52}=C_{5,2}=C(5,2)=C52 is the 2nd algebra listed with , .