A guide to tropical modifications

This paper surveys tropical modifications, which have already become folklore in tropical geometry. Tropical modifications are used in tropical intersection theory and in a study of singularities. They admit interpretations in various contexts such as hyperbolic geometry, Berkovich spaces, and non-standard analysis. Our main goal is to mention different points of view, to give references, and to demonstrate the abilities of tropical modifications. We assume that the reader had already met "tropical modifications" somewhere and wants to understand them better. There are novelties here: a new obstruction (Theorem 2.29) for realizability of non-transversal intersections is found and a tropical version of Weil's reciprocity law (Theorem 2.10) is proven via tropical Menelaus Theorem. A generalization of tropical momentum is given in Section 2.6

Born-Infeld inspired modifications of gravity

General Relativity has shown an outstanding observational success in the scales where it has been directly tested. However, modifications have been intensively explored in the regimes where it seems either incomplete or signals its own limit of validity. In particular, the breakdown of unitarity near the Planck scale strongly suggests that General Relativity needs to be modified at high energies and quantum gravity effects are expected to be important. This is related to the existence of spacetime singularities when the solutions of General Relativity are extrapolated to regimes where curvatures are large. In this sense, Born-Infeld inspired modifications of gravity have shown an extraordinary ability to regularise the gravitational dynamics, leading to non-singular cosmologies and regular black hole spacetimes in a very robust manner and without resorting to quantum gravity effects. This has boosted the interest in these theories in applications to stellar structure, compact objects, inflationary scenarios, cosmological singularities, and black hole and wormhole physics, among others. We review the motivations, various formulations, and main results achieved within these theories, including their observational viability, and provide an overview of current open problems and future research opportunities.Comment: 212 pages, Review under press at Physics Report