Derived categories, stability conditions and geometric applications.

几何学研究更高维的弯曲空间。我们可以用方程来描述这些空间，但我们唯一有希望使用它们进行计算的情况是当方程是多项式的时候。由此产生的空间是代数几何的对象，称为簇。虽然这些对象已经被研究了很长时间，但仍然有很多关键的开放问题：如果我们被赋予一个品种，我们是否可以将其嵌入到其他知名的品种中？例如，我们能找到一个包含给定曲线的“好的”曲面吗？如果是，有多少这样的曲面存在，我们能通过曲线的一些几何性质来描述它们吗？变种的几何信息可以编码在代数对象中，称为派生范畴。受弦论思想的启发，布里奇兰在派生范畴上引入了稳定性条件的概念。由于它与数学和物理的各个领域的联系，这一课题得到了高度的研究，并在该领域发展了许多思想和技术。现在是时候利用派生范畴和稳定性条件中的所有现代工具来解决迄今难以解决的几何问题。我最近的工作证明，稳定条件的变形和对象的稳定状态的变化(跨越墙现象)是解决长期存在的几何问题的强大的新技术，这些问题似乎不涉及派生范畴。令人惊讶的是，稳定条件和越界确实为研究这些问题提供了正确的背景。这项研究计划的主要目的是借鉴代数、几何和数学物理的思想和工具，从派生范畴和稳定性条件方面描述一些突出的几何问题，然后应用跨越墙的技巧来解决它们。

Geometry studies higher-dimensional curved spaces. We can describe these spaces by equations, but the only case where we have any hope to use them for calculation is when the equations are polynomials. The resulting spaces are the objects of algebraic geometry, which are called varieties. Although these objects have been studied for a long time, there are still lots of crucial open problems: If we are given a variety, can we embed it in other well-known varieties? For instance, can we find a "nice'' surface which contains a given curve? If yes, how many such surfaces exist, and can we characterise them via some of the geometrical properties of the curve? The geometric information of varieties can be encoded in algebraic objects, known as derived categories. Inspired by ideas in string theory, Bridgeland introduced the notion of stability conditions on derived categories. This topic has been highly studied due to its connections to various fields in mathematics and physics, and lots of ideas and techniques have been developed in the area. Now is the time to employ the whole spectrum of modern tools in derived categories and stability conditions to solve so far intractable geometrical problems. My recent work proves that deformation of stability conditions and varying stability status of an object (wall-crossing phenomenon) are powerful new techniques for solving long-standing geometrical problems, that do not appear to involve derived categories. Surprisingly, stability conditions and wall-crossing truly provide the right context for studying those problems. The main goal of this research programme is to draw on ideas and tools in algebra, geometry and mathematical physics to describe some outstanding geometrical problems in terms of derived categories and stability conditions, and then apply wall-crossing techniques to solve them.