Three topics shall be briefly covered here.

1. The Gauss-Bonnet Theorem and generalizations. Spherical triangle precursors. Notions of curvature (mostly extrinsic). The Euler characteristic. Gauss Bonnet proper. Poincaré index and the Gauss-Bonnet-Poincaré theorem. Fibre bundles and the Gauss-Bonnet-Chern theorem. Further generalizations of the Gauss-Bonnet theorem: higher-d, further bundles and characteristic classes, index theorems.
2. Hopf Maps. Visualizing Hopf’s little map. Applications of Hopf’s Little Map to the triangleland sphere, the space of qubits, the Dirac monopole, and topological excitations of matter. Introduction to Hopf’s generalized map and its minimum nontrivial realization: from the 5-sphere S^5 to the complex-projective space CP^2.
3. Tensor Networks (Computer Science name) alias Penrose Birdtracks (Theoretical Physics name): Nonrelativistic and special-relativistic tensors. A few morsels from Differential Geometry and Lie Theory.

With an aim to finish producing the review on Hopf’s little map. To produce a new review on Gauss-Bonnet. And to kick-start the Online Encyclopedia of Tensors and other Multi-Linear Arrays, which, in good part, shall be written up in Tensor Networks notation. As well as explaining variants of this, by comparing Penrose, Cvitanovic and the more recent open-source Tensor Networks package. In Theoretical Computer Science, the main present application of Tensor Networks is Machine Learning.

Most of the material in this summer school will be at the level of beginning Graduate School. A core of the material presented will be suitable for people who have completed 2 years of study in a Mathematics, Computer Science or Physical Natural Sciences undergraduate degree.