For several decades, gauge theory has been one of the driving forces of Geometry and Analysis. Its problems and methods have been foundational for many other fields. It is also an important topic in Physics. This course will offer a brief introduction to the mathematical point of view on gauge theory and to some of its relationships to other parts of geometry. It should be of interest to students of both Geometry and Analysis.

Basic knowledge of differential geometry and of complex analysis. An understanding of (i) Riemannian Hodge theory and (ii) Riemann surfaces, line bundles and the first Chern class would also be useful.

The course will attempt to cover the following topics. Some of these will be discussed only briefly and could be the topic of further self-study, in view of the final exam. - Smooth vector bundles, connections, curvature. The Yang-Mills functional. - Flat bundles and connections. Holomorphic vector bundles. - Overview of stability and of the Narasimhan-Seshadri theorem. - ASD connections. Relationship to 4-dimensional topology. - Moduli spaces of Yang-Mills connections. Compactness and bubbling phenomena.

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Presentazione orale

P.D.1-1 - Gennaio