Positivity and Convexity in Algebraic Geometry

代数几何中的正性和凸性

• 批准号: RGPIN-2015-04776
• 负责人: Smith, Gregory
• 金额: $ 1.24万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675846
• 关键词: Positivity; Convexity; Algebraic; Geometry

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. **

尽管在实际数字上的自然总顺序中发现了积极性和凸度的起源，但这些基本结构以当代代数几何形状以几种重要和不同的方式出现。例如，在代数封闭场上的正常曲曲面理论建立了投射品种和理性凸多属性之间的强大词典。相反，当在实际闭合场上工作时，线束的部分（在任何实际点的评估都是正面的），形成很少是多面体的凸锥。作为第三个例子，Boij-Söderberg理论表征了通过凸线上投影空间上矢量束的所有共同体组。该研究计划的广泛目的是了解这些积极和凸性的各种化身之间的深厚和微妙的关系。因为这些基本问题将数学的许多不同领域连接起来，包括代数几何，交换代数，优化，组合学和凸几何形状，进步可能会影响和影响大型社区。该研究将产生新的数学结果和新的开源软件工具。在短期内，我们将集中精力于以下三个问题：***（a）在完整的折叠品种和适当的凸多物质集合中创建一个综合词典。 ***（b）在实际的投影亚动物上给出了2d的任何非负形式f，在整数e上建立了有效的界限，而整数e的平方g g a形式e e的总和e，使得乘积fg是2级的正方形（d+e）。 ***（c）发明了新的同源机制，用于表示曲折变化的连贯的滑轮作为短的算术载体捆绑包（也称为直接的线条捆绑包）。通过他们的培训，他们将在数学科学的各种职业中都有很好的位置。**