Base Exchange and Extremal Properties of Matroids

拟阵的概念最早由惠特尼于1932年提出。 拟阵是由集合上的公理定义的代数结构。 许多组合结构、代数结构和几何结构具有共同的性质，而拟阵在某种意义上是一个代表这三者的抽象对象。 拟阵已被证明是重要的组合数学领域，如图论，组合优化和编码理论。 越来越多地，他们被认为在数学的其他领域，特别是代数中发挥着重要作用。这个建议的目的是双重的：首先，我们将研究处理拟阵类的所谓“增长率”的具体问题，也就是说，在这个类中拟阵的最大大小（即最大元素数）如何作为秩的函数增长。 我们知道拟阵的次闭类，它们的增长率分为四类：线性增长、二次增长、指数增长或无界增长。在许多情况下，我们对它们的增长率没有“严格”的界限。 这个建议的目的是提高这些界限的各类拟阵。 对于“图”拟阵，这样的问题已经在图论领域得到了广泛的研究。 但对拟阵的一般性质知之甚少。 图和拟阵之间的联系越来越引起人们的兴趣，最近的进展似乎表明，许多已知的结果图有一个相应的结果拟阵。 我的工作图和经常拟阵给了我一个独特的见解如何可能攻击问题的较大类的拟阵，如二进制拟阵。

The concept of a matroid was first introduced by Whitney in 1932. A matroid is an algebraic structure which is defined by axioms on sets. Many combinatorial, algebraic and geometric structures share common properties and a matroid is, in some sense, an abstract object representing all three. Matroids have proven to be important in a number of areas of combinatorics such as graph theory, combinatorial optimization, and coding theory. Increasingly, they are seen to play an important role in other areas of mathematics, especially algebra. The aims of this proposal are two-fold: first, we shall look at specific problems dealing with the so-called "growth-rates" of matroid classes, that is, how the maximum size (i.e. maximum number of elements) of matroids in this class grow as a function of rank. We know that minor-closed classes of matroids , their growth rates fall into four categories: either linear, quadratic, exponential, or unbounded growth. In many cases, we do not have "sharp" bounds for their growth rates. This proposal aims to improve these bounds for various classes of matroids. For "graphic" matroids, such problems have already been studied extensively in the field of graph theory. But relatively little is known for matroids in general. There is a growing interest in the connections between graphs and matroids, and recent progress seems to indicate that many known results for graphs have a corresponding result for matroids.  My work on graphs and regular matroids has given me a unique insight into how one might attack problems for larger classes of matroids such as binary matroids.