Cho-Matsumoto and Kita-Yoshida introduced intersection forms on the twisted (co)homology of hyperplane arrangement complements. Recently, Thomas Lam gave a closed form of the pairing in terms of the matroid of the arrangement. We compute the Bergman fan of some arrangements of interest in cosmology in order to apply Lam's formula in this setting.

Tangent categories are special categories that generalize the structure of the category of smooth manifolds. Originally defined for smooth manifolds by Rosický in 1984, tangent categories were rediscovered and generalized to arbitrary categories by Cockett and Cruttwell in 2014. In the twelve years following this paper's publication, the theory of tangent categories has undergone rapid development, and tangent categories have been studied in a wide range of settings, including (synthetic) differential geometry, algebraic geometry, differential linear logic, and machine learning.

We will give a concrete explanation of how the theory of tangent categories can be developed from scratch using basic tools from differential geometry (manifolds and tangent bundles) and category theory (categories, functors, and natural transformations). We'll also provide some examples to illustrate the relationship between tangent categories and other mathematical disciplines like differential/algebraic geometry.