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，优化，组合学和凸几何，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.