Relation between Algebraic and Convex Geometries

我计划研究完全不同的理论（我总是这样做）：在我的拓扑伽罗瓦理论，解释了为什么有些方程不能解决明确的公式;在“热带几何”，允许解决代数问题，通过分析平面图;我的“少项理论”的思想是，“简单”而不是繁琐的方程组应该定义集合，具有“简单”拓扑（这个理论被证明是非常强大的）和其他理论。 但我将主要集中在牛顿-奥肯科夫体理论。最近，我和我以前的学生K.Kaveh一起创造了这个新的令人惊讶的理论，它以一种非常意想不到和令人兴奋的方式将代数和几何联系起来。这个领域发展得非常快。我的项目的主要目标是完成我们的理论（包括我们的交集理论），统一不同的方法，使理论更简单，并在很大程度上自成一体，发展其本地版本，研究当问题允许群作用时涉及对称性的情况。这个庞大的计划涉及研究在非常不同的领域，完全适合吸引和培养年轻的数学家。这项研究是纯数学的普遍兴趣。

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.