Relation between Algebraic and Convex Geometries

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I plan to work on very different theories (as I always do): on my Topological Galois Theory which explains why some equations could not be solved by explicit formulas; on "Tropical Geometry" which allows to solve algebraic problems by analyzing planar diagrams; on my "Fewnomials Theory" whose ideology is that "simple" not cumbersome systems of equations should define sets, having "simple" topology (this theory proved to be very powerful) and on other theories. But mainly I will concentrate on Theory of Newton-Okounkov Bodies. Recently with my former student K.Kaveh I have created this new surprising theory which connects algebra and geometry in a very unexpected and exiting way. This field grows very quickly. The main goal of my project is to complete our theory (including our version of the intersection theory), to unify different approaches, to make the theory simpler and to a large extent self-contained, to develop its local version, to study the case involving symmetries when the problem admits a group action. This huge program involves research in very different areas and suits perfectly to attract and to train young mathematicians. This research is of general interest for Pure Mathematics.