Xavier Gómez-Mont (Guanajuato, Mexico)

Singularities of Vector Fields

Lecture 1: The Geometry of an Isolated Hypersurface Singularity
The Milnor Fibration of an Isolated Hypersurface Singularity;
Morse Type singular points and the Monodromy;
Vanishing Cycles, the Intersection Form and Monodromy.
Lecture 2: A Finite Dimensional Algebra associated to the Singularity
The Local Algebra of the Singularity and its non-singular bilinear form;
The non-degenerate duality and symmetry in the lattice of Ideals in the Local Algebra;
Real Hypersurfaces;
The Euler Characteristic of the Milnor Fibre and the signature of the bilinear form in the local Algebra.
Lecture 3: Algebraic Invariants of a Singularity of a Vector Field
Poincaré-Hopf index of a Vector Field;
A Finite Dimensional Algebra associated to the zero (singularity) of a vector field;
The bilinear form in the Algebra;
The signature of the bilinear form and the Poincaré-Hopf index of the vector field;
Index of a vector eld on an isolated hypersurface singularity.
Lecture 4: The Homological Index of a Vector Field and its Gobelin.
Differentials on Singular Varieties;
Koszul Complexes;
The Complex obtained by contracting differential forms with a vector field;
The Homological Index as an Euler Characteristic;
A Mechanism for doing computations: Double Complexes and Spectral Sequences;
A Richly Embroidered double complex (The Gobelin);
Residues and the multiplication map in the local algebra.


1. A. Dimca, Singularities and Topology of Hypersurfaces, Univesitexts, Springer Verlag, 1992, Chapter 3.
2. D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Math., 150, 1995, Chapter 17.