Positivity and Convexity in Algebraic Geometry

虽然起源的积极性和凸性是发现在自然全序的真实的数字，这些基本结构出现在几个重要的和独特的方式在当代代数几何。 例如，代数闭域上的正规环面簇的理论在射影簇和有理凸多面体之间建立了一个强大的字典。 相反，当在一个真实的闭域上工作时，在任何真实的点的求值为正的线丛的截面形成一个凸锥，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丛的所有上同调群。 这个研究项目的广泛目标是了解积极性和凸性的这些不同体现之间的深刻和微妙的关系。 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、组合数学和凸几何，因此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的积极性的新框架，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成果和新的开源软件工具。 在短期内，我们将集中在以下三个问题：*（a）创建一个全面的字典之间的射影环面等变向量丛在一个完整的环面品种和适当的收集凸多面体。 * （B）给定一个真实的射影子簇上的任意2d次非负形式f，给出整数e的有效界，使得存在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，将复曲面簇上的相干层表示为算术自由向量丛的短复形（也称为线丛的直接和）。*研究生，博士后研究员和本科生谁有助于这个研究计划，不仅将获得宝贵的技术和科学技能，但他们也将成为称职的沟通者。 通过他们的培训，他们将在数学科学的各种职业中处于有利地位。**

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. **