PhD Notes
This page contains the background notes I have made in order to aid my PhD work.
Toric Geometry
These notes aim to expand on the Complex Manifolds notes previously written. While those notes focues on Calabi-Yau manifolds from a differential geometry perspective, these notes we introduce the algebraic geometry techniques in order to study them. We show how to construct what are known as toric varieties from so-called fans. This approach is almost a trivial combinatorics game, and provides a very powerful and useful way to construct Calabi-Yau manifolds with desired properties.
Complex Manifolds
(Calabi-Yau)
Geometry has found immense use in the study of mathematical physics, and often provides a much more intuitive explanation to difficult physical problems. Perhaps the most obvi- ous/prominent example of this is general relativity. This is built on the mathematical con- struction of real manifolds and their associated structures. However, of course the natural extension of such tools would be to consider the complex counterpart, complex manifolds. These notes aim to do just that, giving a somewhat smooth transition from a real manifold to a complex manifold, in a hopefully pedagogical manner. The main goal is the construction of Calabi-Yau manifolds as hypersurfaces in complex projective spaces.
