Geometric Analysis consists in applying PDE tools in order to prove deep theorems in Geometry. It is one of the areas mathematics with more spectacular results in the last half a century.
The school is primarily aimed for Master and beginning Ph. D. students, and hence prerequisites will be kept to a minimum. The objective is to develop the tools and some working knowledge in Geometric Analysis, that is the interaction of manifold theory with PDE analysis. We hope to bring together students in analysis and geometry and establish a fruitful interaction between them.
We have chosen as a guiding goal of the school, to develop the tools and explain a proof of one of the most classic and striking theorems in geometric analysis, namely the Atiyah-Singer Index Theorem. Besides the importance of the theorem, we feel that developing the tools, assembling a proof and explaining some applications (Hirzebruch-Riemann-Roch and Hirzebruch signature theorems) will help the students in learning useful techniques in geometry and analysis, and learning how they can be combined in order to prove deep results.
The school will last for three weeks. The first two weeks will be devoted to develop basic techniques in geometry and analysis of PDE’s, which are needed for the proof of the theorem, but which are completely central by themselves. In the third one Atiyah-Singer index theorem will be proved, some applications will be derived, and in addition some topics of modern geometric analysis will be surveyed, in order to let the participants taste some topics of current research.