List of Figures ............................................... vii
Unfolding CR singularities ...................................... 1
1. Introduction ................................................. 1
2. Topological considerations ................................... 3
2.1. A grassmannian construction ............................. 3
2.2. Global index sums ....................................... 4
3. Local defining equations and transformations ................. 7
3.1. Defining equations for m-submanifolds in  n ............. 7
3.2. Parametrized families of submanifolds ................... 8
3.3. The transformation group for unfoldings ................. 9
4. A complexification construction ............................. 10
5. Real surfaces in 2 ......................................... 13
5.1. Normal forms ........................................... 13
5.2. Unfolding CR singularities of surfaces ................. 17
5.3. Visualization .......................................... 26
6. Real m-submanifolds in n, m < n ............................ 37
6.1. Normal forms ........................................... 37
6.2. Unfolding CR singularities of m-submanifolds ........... 43
6.3. The geometry of the algebraic unfolding ................ 46
7. Rapid convergence proof of the main theorem ................. 48
7.1. A functional equation .................................. 48
7.2. A solution of a linearized equation .................... 50
7.3. The new defining equations and some estimates .......... 76
7.4. Composition of approximate solutions ................... 84
8. Some other directions ....................................... 87
Bibliography ................................................... 89
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