Algebra and geometry of matroids

拟阵是组合数学中的基本结构之一，与图和偏序集一样。因此，它们出现在组合学和相关领域，包括优化，运筹学，编码和信息论，计算机科学，代数几何，并与生物学，物理学和统计学有关。拟阵是建立在一组对象上的，并挑选出这些对象的某些较小的集合，这些集合以某种方式表示_independence_。(For例如，如果对象是向量，其中两个和为第三个，则这三个并不独立。大约在过去的十年里，通过从代数和几何技术中提取的新方法，在与拟阵结构有关的老问题上取得了很大进展：不是从头开始考虑拟阵的结构，而是将其与多项式方程的集合相关联;或者不只是记住上面的独立关系，也许，做一个更丰富的结构，记住更多的原始对象集的代数。我的项目包括试图系统化，统一和扩展这些方法，并在上述进展的基础上进行。

Matroids are one of the basic structures in combinatorics, like graphs and partially ordered sets. As such they appear throughout combinatorics and associated areas, including optimization, operations research, coding and information theory, computer science, algebraic geometry, and have connections to biology, physics, and statistics. A matroid is built on a set of objects, and picks out certain smaller collections of those objects as being somehow compatible in a way that represents _independence_. (For example, if the objects are vectors and two of them sum to a third, the three together are not independent.)In roughly the last decade, much progress has been made on old questions pertaining to the structure of matroids by new methods drawn from the techniques of algebra and geometry: instead of just considering the structure of a matroid from scratch, start by associating it to a collection of polynomial equations; or instead of just remembering the independence relations as above, perhaps, make a richer structure that remembers more algebra of the original set of objects. My projects consist of attempts to systematise, unify, and extend these approaches, and in so doing build on the aforementioned progress.