About geometry and physics

The field "Geometry and Physics" is a wide area of research within modern Mathematical Physics, which is concerned with problems of a geometric nature arising from the fundamental theories of physics. Subjects of interest include geometric mechanics, classical field theory, integrable systems, conformal field theory, quantum field theory on curved spacetime, general relativity, supergravity and string theory.

The wider field of Mathematical Physics also includes subjects which are not directly concerned with geometry, such as statistical mechanics of geometrically simple systems, rigorous quantum mechanics and quantum field theory in Minkowski and Euclidean spaces. Each can also be studied in geometrically nontrivial situations, as done for example in the statistical mechanics of theories with general phase spaces, geometric quantization or quantum field theory on curved spacetime.

It is impossible to understand any part of Geometry and Physics without a thorough background in the following areas of mathematics, which are minimal competencies required for working in the field :

1. Differential geometry and topology

2. Riemannian and semi-riemannian geometry

3. Lie groups, Lie algebras and their representations

4. Riemann surfaces  and some complex analysis.

5. Dynamical systems and geometric PDEs

6. Symplectic and Poisson geometry

Work on the mathematical foundations of String Theory also requires knowledge of:

7. Representation theory of infinite-dimensional Lie (super) algebras

8. Complex and Kahler geometry

9. Algebraic geometry and commutative algebra

Note: The first three of these subjects are generally viewed as minimal mathematics requirements for working in any area of gravitational physics, including physical cosmology.