Prerequisites
Advanced Linear Algebra, Abstract Algebra I, Analysis on Manifolds
Curriculum:
- Critical and noncritical points. Implicit function theorem.
- Morse lemma. Examples of non-Morse functions.
- Typical singularities of mappings \(\mathbb{R}^2 \to \mathbb{R}^2\) and \(\mathbb{R}^2 \to \mathbb{R}^3\). Equivalence of singularities. Stable singularities.
- Thom-Boardman classification.
- Index of a singular point of a mapping \(\mathbb{R}^n \to \mathbb{R}^n\) and of a smooth function. Global index and Euler characteristic.
- Calculus of jets. Transversality theorems. Whitney (weak) embedding and immersion theorems.
- Multiplicity of a smooth map \(\mathbb{C}^n \to \mathbb{C}^n\). Milnor number of a smooth function (geometrical definitions).
- Rings of polynomials and power series. Their ideals. Local algebra of a singularity. Sufficient jet theorem. The Milnor number as the dimension of the local algebra.
- Newton polyhedra and their applications.
- Classification of singular points of functions and their normal forms. Simple singularities.
- Milnor fiber and its topology. Monodromy operator.
- Braid groups. Deformations of singularities, monodromy groups. Galois group of a general polynomial equation of degree \(n\). Applications to complexity theory. Applications to problems of integral geometry. Simple singularities and reflection groups.
- Resolution of singularities of algebraic varieties.