Beijing Institute of Mathematical Sciences and Applications

Spin geometry sits at the vibrant intersection of differential geometry, global analysis, and topology, offering profound insights into the structure of manifolds through the lens of spinors and Dirac operators. This course explores how spin structures—refinements of Riemannian metrics enabling the definition of spinor bundles—unlock powerful tools for studying geometric and topological invariants.

Central to our journey is the Dirac operator, a first-order elliptic operator whose analytical properties encode deep topological information. We will investigate its role in the landmark Atiyah-Singer Index Theorem, which relates the index of Dirac-type operators to characteristic classes of the underlying manifold. This theorem not only unifies disparate results in geometry but also fuels applications across mathematical physics, including gauge theory and supersymmetry.

Drawing on foundational texts like Lawson and Michelsohn’s seminal Spin Geometry and Bär’s modern treatment, we will cover:

Spin and Spin^c structures on vector bundles,

Clifford algebras and spinor representations,

Spectral theory of Dirac operators,

Index theory (Dai, Nicolaescu) and its geometric consequences,

Seiberg-Witten theory (Salamon) as a contemporary application.

The course assumes familiarity with Riemannian geometry, de Rham cohomology, and fiber bundles. By synthesizing perspectives from Hanke, Figueroa-O’Farrill, Walpuski, and Jiang’s notes, we aim to reveal spin geometry as both a unifying framework and an active research frontier, where analysis, algebra, and topology converge in elegant and unexpected ways.