Classical algebraic geometry and modern combinatorics

枚举法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：它们有组合(离散)边和几何(连续)边。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，我们没有像这样的完整图景；事实上，我们知道所有部分是什么的实例相对较少，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，几乎不了解它们为什么起作用。对于如何解决新的相关问题，这类知识提供的线索很少。

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.