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-奥库科夫天体理论和热带几何。这些理论将凸几何和扇形几何(来自分段线性几何的关键对象)与代数几何联系起来。这些联系非常深厚。即使在计算几何体积和计算代数方程组的解的基本水平上，它们也是有意义的 牛顿多面体理论始于1975年著名的Bernstein-Kushnirenko定理，该定理利用与方程相关的牛顿多面体的体积给出了n元n个多项式方程的解的个数的公式。我发现了这个定理的许多证明和推广，现在它经常被称为BKK定理，其中最后的K代表我的名字。 我一直梦想着将BKK定理扩展到任意代数变体，但不清楚在这种普遍性中，什么样的物体可以取代牛顿多面体的作用。只是在A.Okounkov提出了一个绝妙的想法后，才定义了合适的物体，并将其命名为牛顿-奥孔科夫天体。在我与K.Kaveh(特别是我们发现了BKK定理的最一般版本)的工作中，R.Lazarsfeld和M.Mustata独立地系统地发展和推广了牛顿-Okounkov天体理论。从那时起，这一领域的研究活动激增，出现了许多论文(并将继续发表)。还举行了许多关于这一新主题的会议和讲习班。 热带几何在代数几何和分段线性几何之间提供了一种奇妙的联系。热带几何学的原创著作之一是G.米哈尔金的著名著作，它解释了如何通过分析平面图来解决代数问题。这种方法的一个多维版本将代数几何与风扇的几何联系起来。它可以被认为是BKK定理的推广(从环面上的完全交到一般的子簇)。 我计划解决与牛顿多面体理论、牛顿-奥库科夫天体理论和热带几何有关的许多具体问题。我计划从事的其他一些主题是我的拓扑伽罗瓦理论，它解释了为什么许多方程不能用显式求解，以及我的“有限项理论”，它的概念是“简单”而不是繁琐的方程系统应该定义具有“简单”拓扑的集合(该理论已被证明是一个非常强大的工具)。这个庞大的项目涉及非常不同领域的研究，非常适合吸引年轻的数学家。这项研究对《纯数学》具有普遍意义。

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.