Topological methods in algebraic geometry

在同伦理论中，拓扑空间（即形状）被认为是相同的，如果我们可以从一个连续变形到另一个。代数簇是由多项式方程定义的空间;研究他们的同伦理论意味着试图告诉哪些拓扑空间可以连续变形以得到代数簇，或者代数簇之间的连续映射何时可以连续变形为多项式定义的映射。（即分数），这就自动地给了这个变种一个对称群，称为伽罗瓦群。虽然这些对称性不是连续的，但它们的行为足以保持该簇的许多拓扑特征。我的大部分工作涉及调查伽罗瓦集团如何与拓扑结构相互作用。我也研究有限特征的代数簇。后者是通过选择一个素数p并将其设置为零来修改算术规则的宇宙。例如，在特征3中，等式1+1+1=0成立。拓扑和几何的品种仍然有意义的有限特征，在那里我们获得了许多宝贵的洞察行为的伽罗瓦群。模空间参数化类的几何对象，并可以自己经常被赋予几何结构，类似于代数簇。这种结构往往会在具有大量对称性的参数化对象的点处表现不佳。为了克服这一困难，代数几何学家使用模堆栈，它将对称性和对象参数化。通常，模栈的维数可以通过简单地计算它所参数化的几何对象的自由度来计算。然而，空间通常包含奇点（空间不光滑的点）和不同维度的区域。部分受到理论物理学思想的启发，已经证明模栈应该扩展到导出模栈，导出模栈具有预期的维度，但其中一些维度只是虚拟的。扩展到这些虚拟维度也消除了奇异性，这种现象称为隐藏平滑。不同的分类问题可以有相同的模栈，但不同的派生模栈。我的工作的一部分将是尝试为一大类问题构造导出模栈。

In homotopy theory, topological spaces (i.e. shapes) are regarded as being the same if we can deform continuously from one to the other. Algebraic varieties are spaces defined by polynomial equations; studying their homotopy theory means trying to tell which topological spaces can be deformed continuously to get algebraic varieties, or when a continuous map between algebraic varieties can be continuously deformed to a map defined by polynomials.If the polynomials defining a variety are rational numbers (i.e. fractions), this automatically gives the variety a group of symmetries, called the Galois group. Although these symmetries are not continuous, they behave well enough to preserve many of the topological features of the variety. Much of my work involves investigating how the Galois group interacts with the topology. I also study algebraic varieties in finite characteristics. These latter are universes in which the rules of arithmetic are modified by choosing a prime number p, and setting it to zero. For instance, in characteristic 3 the equation 1+1+1=0 holds. Topology and geometry of varieties still make sense in finite characteristics, where we gain much valuable insight into the behaviour of the Galois group.Moduli spaces parametrise classes of geometric objects, and can themselves often be given geometric structures, similar to algebraic varieties. This structure tends to misbehave at points parametrising objects with a lot of symmetry. To obviate this difficulty, algebraic geometers work with moduli stacks, which parametrise the symmetries as well as the objects. Sometimes the symmetries can have symmetries and so on, giving rise to infinity stacks.Usually, the dimension of a moduli stack can be calculated by naively counting the degrees of freedom in defining the geometric object it parametrises. However, the space usually contains singularities (points where the space is not smooth), and regions of different dimensions. Partially inspired by ideas from theoretical physics, it has been conjectured that moduli stacks should be extended to derived moduli stacks, which have the expected dimension, but with some of the dimensions only virtual. Extending to these virtual dimensions also removes the singularities, a phenomenon known as hidden smoothness''. Different classification problems can have the same moduli stack, but different derived moduli stacks. Part of my work will be to try to construct derived moduli stacks for a large class of problems.