Extensions of matroid Hodge theory

组合学是数学的一个分支，包括关于离散结构计数的问题。例如，给定一张用轮廓画出的地图和一些颜色，你可以用多少种方法给地图上的区域上色，使任意两个相邻区域都是不同的颜色？组合学中许多未解决的问题都涉及到这些计数的不等式。四色定理是一个著名的例子，即有四种颜色时，颜色的数量大于零。这个定理不再是一个未解决的问题，但1976年，它的答案是通过长时间的计算机搜索得到的，许多数学家仍然希望得到一个更概念化的答案。提出的研究是将代数几何中的技术应用于这些问题，代数几何是多项式方程系统解的几何研究。2018年，Adiprasito、Huh和Katz在这方面取得了突破。他们使用了代数几何中被称为霍奇理论的不等式来回答由里德、威尔士和其他人提出的一个40年前的开放问题。在地图着色方面，着色的数量是可用颜色数量的多项式函数；里德的问题是关于这个多项式的系数如何增长。我将扩展这些技巧。问题所涉及的组合对象实际上被称为拟阵。每个地图都有一个矩阵。矩阵由一组对象组成，并从中挑选出一些较小的对象集合，这些对象是相互兼容的，或者用数学家的话说，是独立的。在地图的例子中，一组边界是独立的，如果你无法循环穿越其中的一些边界并回到你开始的地方而不两次穿越相同的边界。其他拟阵来自于矩阵。但也有一些所谓的不可代表的类母体，它们并非来自这些来源。当一个矩阵确实来自于一个映射或矩阵时，我们可以用它来写一个方程组来进行几何研究。Adiprasito， Huh和Katz所建立的是，即使对于不存在方程组的不可表示的拟阵，Hodge理论仍然有效，就好像它确实存在一样。处理这些实际上并不存在的几何物体的阴影，增加了这些问题的难度。拟阵有许多应用，包括数学优化、代码、物理学、统计学和生物学。例如，我将致力于解决的一个尚未解决的问题是“交错多项式”，它是作为研究DNA链打结和重组的一部分而发明的。

Combinatorics is the branch of mathematics that includes questions about counting discrete structures. For example, given a map drawn in outline, and some number of colours, in how many ways can you colour in the regions in the map so that any two bordering regions are different colours? Many unsolved problems in combinatorics ask about inequalities involving these counts. The four colour theorem -- that with four colours, the number of colourings is greater than zero -- is a famous example: it is no longer an unsolved problem, but its solution in 1976 was by a long computer search and many mathematicians would still like a more conceptual answer.The proposed research is to apply to these problems techniques from algebraic geometry, which is the geometric study of solutions to systems of polynomial equations. A breakthrough in this respect was made in 2018 by Adiprasito, Huh and Katz. They used inequalities from the part of algebraic geometry called Hodge theory to answer a 40-year-old open problem posed by Read, Welsh and others. In terms of map colouring, the number of colourings turns out to be a polynomial function of the number of colours available; Read's problem was about how the coefficients of this polynomial grow. I will be extending those techniques.The combinatorial objects the questions are actually about are called matroids. For every map there is a matroid. A matroid comes with a set of objects, and picks out certain smaller sets of those objects as being compatible with each other, or as mathematicians say, independent. In the example of maps, a set of borders is independent if there's no way to walk in a loop through some of those borders and arrive back where you started without crossing the same one twice. Other matroids come from matrices. But there are so-called unrepresentable matroids that don't come from these sources. When a matroid does come from a map or a matrix, we can use that to write down a system of equations to study geometrically. What Adiprasito, Huh and Katz established is that even for unrepresentable matroids, where the system of equations does not exist, the Hodge theory still works as if it did exist. Working with these shadows of a geometric object that doesn't actually exist, as it were, contributes to the difficulty of these problems.Matroids have many applications including in mathematical optimisation, codes, physics, statistics, and biology. For example, one of the still unsolved problems I will work on is about the "interlace polynomial", which was invented as part of the study of knotting and recombination in DNA strands.