Positivity and Convexity in Algebraic Geometry

虽然正性和凸性的起源是在实数的自然全序中发现的，但这些基本结构在当代代数几何中以几种重要而独特的方式出现。例如，代数闭域上的正规环变理论在射影变和有理凸多面体之间建立了一个鲁棒字典。相反，当在实闭场上工作时，线束的各部分在任何实点处的值都是正的，形成凸锥，这种凸锥很少是多面体。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束的所有上同调群。这项研究计划的主要目的是了解这些不同形式的正性和凸性之间深刻而微妙的关系。因为这些基本问题连接了数学的许多不同领域，包括代数几何、交换代数、优化、组合学和凸几何，这些进步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内部正性的新框架，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)在完全环型和凸多面体的适当集合上创建一个投影环型等变向量束之间的综合字典。***(b)给定实射影子簇上任意2d次的非负形式f，求出整数e上的有效界，使其存在e次形式的平方和g，使得乘积fg是2次形式的平方和（d+e）。***(c)发明新的同调机制，将环型上的相干束表示为不需要算术的向量束（也称为线束的直接和）的短复形。***为该研究项目做出贡献的研究生、博士后和本科生不仅将获得宝贵的技术和科学技能，而且还将成为称职的沟通者。通过他们的训练，他们将为数学科学领域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. **