Algebraic combinatorics, matrix integrals and algebraic geometry

组合数学是研究有限对象集合的排列以及它们之间的关系。组合学的研究主要是由其他数学和物理科学以及工程学的应用所推动的。许多这样的应用程序只需要确定一个集合中对象的数量，称为“计数”集合。在代数组合学中，我们广泛使用代数的结果和方法进行研究。最基本的组合对象之一被称为置换，它对有限集合的元素进行重新排序。转置是一种非常简单的排列，它只是简单地交换（“转置”）两个元素的顺序。置换的乘积是通过连续的重新排序得到的，在代数中，具有该乘积的集合的所有置换的集合被称为对称群。

Combinatorics is the study of arrangements of sets of finite objects and relationships between them. Research in combinatorics is centrally motivated by applications in other mathematical and physical sciences, and in engineering. Many such applications require only that one determine the number of objects in a set, called "counting" the set. In algebraic combinatorics we make extensive use of results and methods from algebra in this study. One of the most fundamental combinatorial objects is called a permutation, which reorders the elements of a finite set. A transposition is a very simple permutation, that simply interchanges the order of ("transposes") two of the elements. A product of permutations is obtained by successive reordering, and in algebra, the set of all permutations of a set with this product is called the symmetric group.