Classical algebraic geometry and modern combinatorics

枚举是对数学对象进行计数的研究。一些最重要的计数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：它们具有组合（离散）方面和几何（连续）方面。在某些情况下，代数可以在这两个方面之间提供桥梁。然而，有许多计数问题是数学的许多领域的基础，我们没有这样一个完整的画面;事实上，我们知道所有碎片是什么的情况相对较少，更不用说它们如何组合在一起了。在某些情况下，我们只有神秘的组合程序，似乎是偶然地给出了正确的答案，而对它们为什么起作用却知之甚少。这类知识对于如何解决新的、相关的问题提供了很少的线索。

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.