Jyväskylä, Finland

Resolution of Singularities

when 15 August 2016 - 19 August 2016
language English
duration 1 week
credits 2 EC

In 1964, Heisuke Hironaka published his famous work on Resolution of Singularities, for which he was awarded the Fields Medal. This beautiful theorem in complex algebraic geometry has surprisingly powerful applications to analysis. The course will be an introduction to this theorem, with an application to computing the analytic index of singularities of certain functions.

We will begin by studying hypersurfaces. A hypersurface is the zero set of one polynomial in several variables. As you see in the example of a cone defined by the zero set of x^2+y^2-z^2, most points of a hypersurface are smooth, but there can be singularities, such as the vertex of the cone (the origin). Hironaka's amazing theorem tells us that these singularities can be resolved: we can construct a smooth hypersurface mapping properly onto the singular one, isomorphically except at the singular points.

Course leader

Lecturer: Professor Karen Smith (University of Michigan, Ann Arbor, United States)

Coordinator: Prof. Daniel Meyer (Jacobs University, Bremen, Germany & University of Jyväskylä)

Target group

Basic mathematical maturity expected of a beginning PhD student in pure mathematics. The course should be accessible to advanced undergraduates with strong mathematical background. Some geometry (the language of manifolds, charts) will make it easier.

The Summer School annually offers courses for advanced master’s students, graduate students, and post-docs in the various fields of science and information technology.

Course aim

The most important aims of the Summer School are to develop post-graduates scientific readiness and to offer students the possibility to study in a modern, scientific environment and to create connections to the international science community. The Summer School offers an excellent pathway to develop international collaboration in post-graduate research.

Credits info

2 EC
Grading: Pass/fail

Fee info

EUR 0: Participating the Summer School is free of charge, but student have to cover the costs of own travel, accommodation and meals at Jyväskylä.


The 26th Jyväskylä Summer School is not able to grant any Summer School students financial support.