Homological methods in algebra and geometry

没有两片枫叶是相同的，也没有两片雪花是相同的。然而，没有人会把前者误认为后者。这个简单的比喻是数学结构变形和分类理论的基础。给定一个由公理、结构数据或方程指定的代数或几何对象，如果一个人（稍微）改变定义参数，那么结构还剩下什么？哪些方面会发生变化？数学家们用来回答这些问题的工具和技术都包含在不断扩大的同态代数工具箱中，同态代数努力对形状或形式相等的东西进行分类。“同源的”。粗略的分类通常是通过离散不变量获得的，比如一个维度可能对应于叶子的穗数；一个更好的可能需要连续变化的模，这些模可以再次成为数字或通过更简单的结构表现出来。这里的隐喻类比可以是树叶的颜色。一个深刻的自然事实是，雪花有六个尖状物，任何七倍对称的东西都不可能是雪花。同调代数的目标是类似的基本结果：寻找不变量和模使用高度抽象的概念，如三角化范畴，变形群，或控制微分梯度李代数，导致结构具有简单的一般定义，但很难在任何给定的具体情况下利用。因此，人们会寻找更易于管理的次级不变量，其中包括上同调群、阻碍理论或具体的几何对象，如判别式或Frobenius结构。应用范围从代数中的分类理论到几何和物理中的基本问题。

No two maple leaves are alike and neither are two snow flakes. However, nobody will mistake the former for the latter. This simple metaphor is at the basis of the deformation and classification theory of mathematical structures. Given an algebraic or geometric object that is specified by axioms, structural data, or equations, what remains of the structure and which aspects change if one (slightly) varies the defining parameters? The instruments and techniques mathematicians employ to answer these questions are contained in the ever expanding toolbox of homological algebra, which strives to classify that which is of equal shape or form, i.e. ``homological''.A coarse classification is usually obtained through discrete invariants, such as a dimension that may correspond to the number of spikes of a leaf; a finer one may necessitate continuously varying moduli that can be numbers again or manifest themselves through simpler structures. The metaphorical analogue here could be the shade of colour of a leaf. A deep fact of nature is that snow flakes have six spikes and that nothing with sevenfold symmetry can ever be a snow flake. Homological Algebra aims at similar fundamental results: the hunt for invariants and moduli employs highly abstract concepts such as triangulated categories, deformation groupoids, or controlling differential graded Lie algebras, leading to structures that have a simple general definition, but are difficult to exploit in any given concrete case. One thus looks for more manageable secondary invariants, among them cohomology groups, obstruction theories or concrete geometric objects such as discriminants or Frobenius structures. Applications range from classification theory in Algebra to fundamental questions in Geometry and Physics.