Factorizations, Projective Duality, and Cycles

因式分解、射影对偶性和循环

• 批准号: RGPIN-2014-03848
• 负责人: Favero, David
• 金额: $ 1.17万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566130
• 关键词: Factorizations; Projective; Duality; Cycles

Traditionally, algebra and geometry are studied together by thinking of solutions to equations as shapes. For example circles, lines, and parabolas are defined by polynomial equations. This is the basis of Algebraic Geometry. This research program adds a third ingredient, inspired by the great attempt in High-Energy Theoretical Physics to unify Einstein's Theory of Gravity with Quantum Mechanics. Einstein's Theory of Gravity is incredibly accurate when predicting the movement of planets, comets, and stars while the Standard Model of Quantum Mechanics has been verified with amazing precision at the microscopic level. However, at high energy these two theories are incapable. One proposal to unify these theories is that the universe itself is made up of tiny little vibrating strings (as opposed to infinitesimal point-like particles). While this has not been experimentally verified in the real world, it it has incredible implications towards our understanding of geometry. An important aspect of Einstein's Theory of Gravity is that he viewed space and time not as separate entities but as one continuous "4-dimensional" geometry - spacetime. Now, let's treat spacetime like something we all deal with daily: H2O. As H20 changes temperature it transitions through various phases: water, ice, steam. Physicists predict that spacetime can also undergo "phase-transitions" which change aspects of its geometry. Some of these changes are actually well understood through birational geometry, one of the most classical aspects of Algebraic Geometry. On the other hand, physicists often incorporate additional data into their study of spacetime for example in so-called Landau-Ginzburg models. In this proposal, we concentrate on the implications in Algebraic Geometry of predictions from High-Energy Physics such as a mathematical interpretation of "phase-change" for Landau-Ginzburg models. Specifically, the purpose of the project is to create and implement rigorous mathematical machinery for derived categories of Landau-Ginzburg models, algebraic cycles, and Homological Mirror Symmetry. The main novelty is that, by using physical interpretations of space and time, we can produce new and unexpected results about algebraic equations and geometric shapes.

传统上，代数和几何是一起学习的，通过思考方程的解作为形状。例如，圆、直线和抛物线由多项式方程定义。这是代数几何的基础。这项研究计划增加了第三个因素，灵感来自高能理论物理学的伟大尝试，统一爱因斯坦的引力理论与量子力学。爱因斯坦的引力理论在预测行星、彗星和恒星的运动时非常准确，而量子力学的标准模型在微观层面上已经得到了惊人的精确验证。然而，在高能量下，这两个理论是不可能的。统一这些理论的一个建议是，宇宙本身是由微小的振动弦组成的（而不是无限小的点状粒子）。虽然这还没有在真实的世界中得到实验验证，但它对我们理解几何有着难以置信的影响。爱因斯坦引力理论的一个重要方面是，他认为空间和时间不是分开的实体，而是一个连续的“四维”几何-时空。现在，让我们把时空当作我们每天都要处理的东西：水。当H2O改变温度时，它通过各种相转变：水，冰，蒸汽。物理学家预测时空也可以经历“相变”，从而改变其几何形状。其中一些变化实际上是很好地理解通过双有理几何，其中一个最经典的方面代数几何。另一方面，物理学家经常在他们的时空研究中加入额外的数据，例如所谓的朗道-金兹伯格模型。在这个建议中，我们集中在代数几何的影响，从高能物理的预测，如数学解释的“相变”的朗道-金兹伯格模型。具体来说，该项目的目的是创建和实施严格的数学机器的衍生类别的朗道-金兹伯格模型，代数循环，同调镜像对称。主要的新奇在于，通过使用对空间和时间的物理解释，我们可以产生关于代数方程和几何形状的新的和意想不到的结果。