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，如果我们能从一个空间连续变形到另一个空间。代数簇是由多项式方程定义的空间；研究其同伦理论意味着试图辨别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地变形为由多项式定义的映射。如果定义簇的多项式是有理数(即分式)，这自动地给出簇的一组对称，称为伽罗瓦群。尽管这些对称性不是连续的，但它们表现得足够好，足以保留该种类的许多拓扑特征。我的很多工作都涉及研究伽罗瓦群如何与拓扑相互作用。我还研究有限特征的代数簇。后者是通过选择素数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。这种结构倾向于在具有很大对称性的对象的参照点上表现不佳。为了避免这一困难，代数几何仪使用模数堆栈，模数堆栈的参数表示对称性和对象。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。部分受到理论物理思想的启发，人们猜测模数堆栈应该扩展到派生的模数堆栈，它具有预期的维度，但有些维度只是虚拟的。延伸到这些虚拟维度也消除了奇点，这是一种被称为隐藏平滑的现象。不同的分类问题可以具有相同的模堆栈，但不同的派生模堆栈。我的部分工作将是尝试为一大类问题构造派生模堆栈。

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.