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。

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.