The Riemann-Roch Theorem and Dirac Operators

The Riemann-Roch Theorem is one of the most classic results on Riemann surfaces, originally stated as the relation between the spaces of meromorphic functions on the surface and the algebraic and topological properties of the underlying surface. The development of index theory has provided it with a modern interpretation using Dirac operators. We start by investigating classical points of view for the theorem, together with the relevant cohomology theory. Then we move on to the modern generalisation for general dimensions through index theory for Dirac operators, also considering concrete examples.

Tiernan Cartwright

University of Sydney

Tiernan Cartwright is a University of Sydney student focusing on mathematics, although he has also completed a minor in statistics. Tiernan enjoys many areas of maths. He has recently been particularly interested in differential geometry, due to its rich interaction between theory and examples, and because it provides a setting in which to witness interactions between other fundamental areas of maths. Tiernan is looking forward to using his AMSI project both to learn more about differential geometry and to gain an introduction to ideas from complex geometry and algebraic topology. He also anticipates that this project will be useful for further study; next he will undertake honors in 2023.

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