**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;

Morsification;

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.

*References:*

2. D. Eisenbud,