北京雁栖湖应用数学研究院

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, the theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and
technical sciences. In this lecture, we develop the relevant techniques, the basic theory of complex spaces and their germs and sheaves on them, including the key ingredients - the Weierstraß preparation theorem and its other forms (division theorem and finiteness theorem), and the finite coherence theorem.
Then we pass to the main object of study, isolated hypersurface and plane curve singularities. Isolated hypersurface singularities and especially plane curve singularities form a classical research area which still is in the centre of current research. In many aspects they are simpler than general singularities,
but on the other hand they are much richer in ideas, applications, and links to other branches of mathematics. Furthermore, they provide an ideal introduction to the general singularity theory. Particularly, we treat in detail the classical topological and analytic invariants, finite determinacy, resolution of singularities, and classification of simple singularities.