Classical algebraic geometry and modern combinatorics

枚举是对数学对象进行计数的研究。一些最重要的枚举问题起源于代数几何。这些“枚举代数几何”问题具有双重性质：它们有组合（离散）的一面和几何（连续）的一面。在某些情况下，代数可以在这两方面之间架起一座桥梁。然而，有许多计算问题是许多数学领域的基础，在这些领域我们没有这样的完整图景；事实上，我们很少知道所有的部分是什么，更不用说它们是如何组合在一起的了。在某些情况下，我们只有神秘的组合程序，似乎能给出正确的答案，就好像是偶然的，几乎不知道它们为什么会起作用。这种类型的知识对于如何解决新的相关问题提供的线索很少。

Enumeration is the study of counting mathematical objects. Some of the most important enumeration problems have their orgins in algebraic geometry. These "enumerative algebraic geometry" problems have a dual nature: they have a combinatorial (discrete) side and geometric (continuous) side. In some cases, algebra can provide a bridge between these two sides. However, there are many counting problems that are fundamental to a number of areas of mathematics, where we do not have a complete picture like this; in fact, there are relatively few instances where we know what all the pieces are, let alone how they fit together. In some cases we have only mysterious combinatorial procedures that seem to give the right answer, as if by accident, with little understanding of why they work. This type of knowledge provides few clues as to how to solve new, related problems.