Factorizations, Projective Duality, and Cycles

传统上，代数和几何是通过将方程的解视为形状而一起学习的。例如，圆、直线和抛物线由多项式公式定义。这是《代数几何》的基础。这个研究项目增加了第三个因素，灵感来自于高能理论物理学将爱因斯坦的引力理论与量子力学统一起来的伟大尝试。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已经在微观层面上以惊人的精度得到了验证。然而，在高能下，这两种理论是不成立的。统一这些理论的一种建议是，宇宙本身是由微小的振动弦(而不是无限小的点状粒子)组成的。虽然这还没有在现实世界中得到实验验证，但它对我们对几何的理解有着令人难以置信的影响。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独立的实体，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在处理的东西：水。当H20改变温度时，它会经历不同的阶段：水、冰、蒸汽。物理学家预测，时空也可能经历“相变”，从而改变其几何结构的各个方面。其中一些变化实际上是通过双曲几何很好地理解的，双曲几何是代数几何最经典的方面之一。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，例如在所谓的朗道-金兹堡模型中。在这个提议中，我们集中在高能物理预言的代数几何中的含义，例如对Landau-Ginzburg模型的“相变”的数学解释。具体地说，该项目的目的是为Landau-Ginzburg模型、代数循环和同调镜像对称的派生范畴创建和实现严格的数学机制。主要的新奇之处在于，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。

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.