"Applications of Homological Algebra in Algebra, Geometry, and Physics"

没有两片枫叶是一样的，两片雪花也是一样的。然而，没有人会把前者误认为后者。这个简单的隐喻是数学结构变形和分类理论的基础。给出一个由公理、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，结构的哪些剩余部分和哪些方面会发生变化？数学家用来回答这些问题的工具和技术包含在不断扩大的同调代数工具箱中，该工具箱致力于对具有相同形状或形式的东西进行分类，即“同调”。 粗略的分类通常通过离散的不变量来获得，例如可以对应于叶子的穗数的维度；更精细的分类可能需要连续变化的模数，这些模数可以是数字，或者通过更简单的结构表现出来。这里比喻的比喻可能是树叶的颜色。自然界的一个深刻事实是，雪花有六个尖刺，七重对称的东西永远不可能成为雪花。同调代数的目标是类似的基本结果：寻找不变量和模使用了高度抽象的概念，如三角范畴、变形群胚或控制微分分次李代数，导致结构具有简单的一般定义，但在任何给定的具体情况下都很难利用。因此，人们寻找更容易管理的次级不变量，其中包括上同调群、障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.