Windows and Mirror Symmetry

多项式方程的解集形成几何形状(例如直线、平面、曲线)。代数几何研究这些多项式方程的解和这些形状的几何之间的联系。辛几何也研究几何形状，但忽略了它们可能具有的任何代数性质，而是保留了一种测量内部面积的方法。20世纪90年代初，高能理论物理学家发现，辛几何中的问题可以用代数几何来回答，反之亦然。这两个数学领域之间的这种联系被称为“镜像对称”。1994年，马克西姆·康采维奇在他的菲尔兹奖章演讲中提出了镜像对称的一个广泛的数学框架，通过范畴理论将代数几何和辛几何联系起来，范畴理论是一种现代数学语言，可以用来描述大多数数学主题。康采维奇的镜面对称公式是完全革命性的，在数学上是没有先例的。在过去的25年里，它迅速发展成为一个庞大的研究领域，吸引了来自代数几何、辛几何和高能理论物理的不同国际研究领域的研究人员。作为他的程序的一个重要部分，Kontsevich将镜像对称的代数几何方面提炼成一个数学概念，称为派生范畴。这项研究计划旨在提供派生范畴的基本描述，特别是当它们出现在高能理论物理的模型中时。它将派生类别分解成更简单的片段，对它们进行比较，并旨在证明关于它们行为的一些最大问题。它还在康采维奇的框架内构建了镜像对称性的新版本。最后，将得到的理论应用于解决代数几何中的经典问题。

The set of solutions to polynomial equations form geometric shapes (e.g. lines, planes, curves). Algebraic geometry studies the connection between the solutions to these polynomial equations and the geometry of these shapes. Symplectic geometry also studies geometric shapes but disregards any algebraic natural they may have, retaining instead a way to measure the area inside. In the early 1990s, high-energy theoretical physicists found that questions in symplectic geometry can be answered using algebraic geometry and vice versa. This connection between these two areas of mathematics is called "mirror symmetry". In 1994, during his Fields Medal address, Maxim Kontsevich proposed a vast mathematical framework for mirror symmetry, connecting algebraic geometry and symplectic geometry through category theory, a modern mathematical language which can be used to describe most mathematical topics. Kontsevich's formulation of mirror symmetry was completely revolutionary and without any precedent in mathematics. In the last 25 years, it has burgeoned into a massive field of study attracting a diverse international field of researchers from algebraic geometry, symplectic geometry, and high-energy theoretical physics. As an essential part of his program, Kontsevich distilled the algebro-geometric side of mirror symmetry into a mathematical concept known as the derived category. This research proposal aims to provide a fundamental description of derived categories, especially as they appear in models from high-energy theoretical physics. It decomposes derived categories into simpler pieces, makes comparisons between them, and aims to prove some of the biggest questions remaining about their behavior. It also constructs new versions of mirror symmetry within Kontsevich's framework. Finally, it aims to apply the resulting theory to solve classical problems in algebraic geometry.