Positivity and Convexity in Algebraic Geometry

虽然正性和凸性的起源是在实数上的自然全序中找到的，但这些基本结构在当代代数几何中以几种重要和独特的方式出现。例如，代数闭域上的正规环面簇理论在射影簇和有理凸多面体之间建立了一本稳健的词典。相反，当在实闭域上工作时，在任何实点上求值为正的线丛的截面形成一个很少是多面体的凸锥。作为第三个例子，Boij-Söderberg理论通过凸几何刻画了射影空间上向量丛的所有上同调群。这个研究项目的广泛目标是了解这些不同的正性和凸性化身之间的深层次和微妙的关系。由于这些基本问题连接了许多不同的数学领域，包括代数几何、交换代数、最优化、组合学和凸几何，这些进展可能会影响和影响一个大的社区。*长期目标是发现代数几何中正性的新框架，并完善我们对代数几何中出现的特定凸锥的理解。这项研究将产生新的数学结果和新的开源软件工具。在短期内，我们将集中于以下三个问题：*(A)在完全环面簇上的射影环面等变向量丛和适当的凸多面体集合之间建立一个全面的词典。*(B)给定实投射子簇上任何2d次非负形式f，在存在e次形式平方和g的整数e上建立有效界，使得乘积fg是2次形式平方和(d+e)。*(C)发明新的同调机制，用于将环状变种上的凝聚层表示为无算术运算的向量丛(也称为线丛的直和)的短复形。*为该研究项目做出贡献的研究生、博士后研究员和本科生不仅将获得宝贵的技术和科学技能，而且还将成为有能力的沟通者。通过他们的培训，他们将为数学科学中的各种职业做好准备。

Although the origins of positivity and convexity are found in the natural total ordering on the real numbers, these basic structures emerge in several important and distinct ways within contemporary algebraic geometry. For instance, the theory of normal toric varieties over an algebraically closed field builds a robust dictionary between projective varieties and rational convex polytopes. In contrast, when working over a real closed field, the sections of a line bundle, whose evaluation at any real point is positive, form a convex cone that is rarely polyhedral. As a third example, Boij-Söderberg theory characterizes all of the cohomology groups for vector bundles on projective space via convex geometry. The broad aim of this research program is to understand the deep and subtle relations between these various incarnations of positivity and convexity. Because these fundamental problems connect many different areas of mathematics including algebraic geometry, commutative algebra, optimization, combinatorics, and convex geometry, advances will likely impact and influence a large community.****The long-term goals are to discover new frameworks for positivity inside algebraic geometry and to refine our understanding of specific convex cones appearing in algebraic geometry. The research will produce new mathematical results and new open-source software tools. In the short-term, we will concentrate on the following three problems: ***(a) Create a comprehensive dictionary between projectivized torus-equivariant vector bundles over a complete toric variety and appropriate collections of convex polytopes. ***(b) Given any nonnegative form f of degree 2d on a real projective subvariety, develop effective bounds on the integers e for which there exists a sum of squares g of forms degree e such that the product fg is a sum of squares of forms of degree 2(d+e). ***(c) Invent new homological mechanisms for representing coherent sheaves on toric varieties as short complexes of arithmetically-free vector bundles (also known as direct sums of line bundles).***The graduate students, postdoctoral fellows, and undergraduate students who contribute to this research program, will not only obtain valuable technical and scientific skills, but they will also become competent communicators.  With their training, they will be well-positioned for a variety of careers in the mathematical sciences. **