Three topics shall be briefly covered here.

1. The Gauss-Bonnet Theorem and generalizations.
2. Hopf Maps: applications of Hopf’s Little Map, 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).

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 Compute Science, the main present application of Tensor Networks is Artificial Intelligence (AI).

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.