Convex Bodies, Fans and Algebraic Geometry

在我的研究项目中，我计划研究牛顿多面体理论，牛顿-奥肯科夫体理论和热带几何。这些理论将凸几何和扇形几何（分段线性几何的关键对象）与代数几何联系起来。这些联系非常深刻。它们甚至在计算几何体积和计算代数方程组的解的基本层次上也是有意义的 牛顿多面体理论开始于1975年，著名的伯恩斯坦-库什尼连科定理，它给出了一个公式的解决方案的数量n多项式方程在n个变量使用体积的牛顿多面体与方程。我发现了这个定理的许多证明和扩展，现在它通常被称为BKK定理，其中最后一个K代表我的名字。 我一直梦想着把BKK定理推广到任意的代数簇上，但当时还不清楚什么样的物体可以取代牛顿多面体在这种普遍性中的作用。这只是在A. Okounkov，适当的对象已被定义和命名为牛顿-Okounkov机构。牛顿-奥肯科夫体理论是在我与K。Kaveh（特别是我们发现了BKK定理的最一般版本）和独立的R。Lazarsfeld和M.穆斯塔塔从那时起，这一领域出现了大量的研究活动，出现了许多论文（并继续出现）。关于这一新问题，还举行了许多会议和讲习班。 热带几何提供了代数几何和分段线性几何之间的奇妙关系。热带几何学的一部重要著作是G. Mikhalkin，它解释了如何通过分析平面图来解决代数问题。这种方法的多维版本涉及代数几何与几何风扇。它可以被认为是BKK定理的一个推广（从完全交到环面上的一般子簇）。 我计划解决许多与牛顿多面体理论、牛顿-奥肯科夫体理论和热带几何有关的具体问题。其他一些主题，我计划的工作是我的拓扑伽罗瓦理论，解释了为什么许多方程不能解决明确的公式和我的“理论Fewnomials”的概念是，“简单”，而不是繁琐的系统方程应定义集“简单”拓扑（这一理论已被证明是一个非常强大的工具）。这个庞大的计划涉及非常不同的领域的研究，完全适合吸引年轻的数学家。这项研究是纯数学的普遍兴趣。

In my research project, I plan to work on Newton polyhedra theory, the theory of Newton-Okounkov bodies, and tropical geometry. These theories connect convex geometry and geometry of fans (key objects from piecewise liner geometry) with algebraic geometry. These connections are very deep. They make sense even on the basic level of calculating geometrical volumes and counting the solutions of systems of algebraic equations Newton polyhedral theory was started in 1975 with the celebrated Bernstein-Kushnirenko theorem, which gives a formula for the number of solutions of n polynomial equations in n variables using volumes of the Newton polyhedra associated with the equations. I found many proofs and extensions of this theorem, and nowadays it is often referred to as the BKK theorem where the last K stands for my name. I had always dreamed of extending the BKK theorem to arbitrary algebraic varieties, but it was not clear what kind of objects could replace the role of Newton polyhedra in such generality. It was only after a brilliant idea from A. Okounkov that appropriate objects have been defined and named Newton-Okounkov bodies. The theory of Newton-Okounkov bodies was systematically developed and generalized in my work with K. Kaveh (in particular we found the most general version of the BKK theorem) and independently by R. Lazarsfeld and M. Mustata. Since then there has been a burst of research activity in this area where many papers have appeared (and continue to appear). There have also been many conferences and workshops on this new subject. Tropical geometry provides a wonderful relation between algebraic geometry and piecewise linear geometry. One of the original works in tropical geometry is the celebrated work of G. Mikhalkin, which explains how to solve algebraic problems by analyzing a planar diagram. A multidimensional version of this approach relates algebraic geometry with the geometry of fans. It could be considered as an extension of the BKK theorem (from complete intersections to general subvarieties in the torus). I plan to address many concrete problems related to the Newton polyhedra theory, the theory of Newton-Okounkov bodies and tropical geometry. Some other subjects I plan to work on are my topological Galois theory which explains why many equations could not be solved by explicit formulas and my “theory of Fewnomials” whose concept is that “simple” and not cumbersome systems of equations should define sets with “simple” topology (this theory has proved to be a very powerful tool). This huge program involves research in very different areas and suits perfectly to attract young mathematicians. This research is of general interest for Pure Mathematics.