Relation between Algebraic and Convex Geometries

代数几何和凸几何之间的关系

• 批准号: 156833-2012
• 负责人: Khovanskii, Askold
• 金额: $ 2.55万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569509
• 关键词: Relation; between; Algebraic; Convex; Geometries

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.

我计划研究非常不同的理论（就像我一直做的那样）：关于我的拓扑伽罗瓦理论，它解释了为什么某些方程不能通过显式公式求解； “热带几何”允许通过分析平面图来解决代数问题；关于我的“少数项理论”，其思想是“简单”而不繁琐的方程组应该定义集合，具有“简单”拓扑（该理论被证明非常强大）以及其他理论。 但我将主要关注牛顿-奥孔科夫体理论。最近，我和我以前的学生 K.Kaveh 一起创造了这个令人惊讶的新理论，它以一种非常出乎意料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善我们的理论（包括我们版本的相交理论），统一不同的方法，使理论更简单并且在很大程度上是独立的，开发其本地版本，研究当问题承认群体行为时涉及对称性的情况。这个庞大的计划涉及非常不同领域的研究，非常适合吸引和培养年轻数学家。这项研究引起了纯数学界的普遍兴趣。