Extensions of matroid Hodge theory

组合学是数学的一个分支，它包括关于计算离散结构的问题。例如，给出一张用轮廓线绘制的地图和一定数量的颜色，你可以用多少种方法在地图上的区域中上色，从而使任何两个接壤的区域都是不同的颜色？组合学中许多悬而未决的问题都涉及到涉及这些计数的不等式。四色定理--有四种颜色，着色的数目大于零--是一个著名的例子：它不再是一个未解决的问题，但它在1976年的解决方案是通过长时间的计算机搜索，许多数学家仍然希望得到更概念性的答案。拟议的研究是将代数几何的技术应用于这些问题，代数几何是对多项式方程组的解的几何研究。2018年，Adiprasito、Huh和Katz在这方面取得了突破。他们利用代数几何中被称为霍奇理论的部分中的不等式来回答里德、威尔士和其他人提出的一个40年前的悬而未决的问题。在地图着色方面，着色的数量被证明是可用的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。这些问题实际上涉及的组合对象称为拟阵。每一张地图都有一个拟阵。拟阵带有一组对象，它挑选出某些较小的对象集作为彼此兼容的对象，或者正如数学家所说的那样，是独立的。在地图的例子中，一组边界是独立的，如果没有办法在循环中穿过其中一些边界并在不经过相同边界两次的情况下返回起点。其他拟阵来自于矩阵。但有一些所谓的不可代表的拟阵并不是来自这些来源。当拟阵确实来自映射或矩阵时，我们可以用它来写下一组方程，进行几何研究。Adiprasito、Huh和Katz建立的结论是，即使对于不可表示的拟阵，在方程系统不存在的情况下，霍奇理论仍然有效，就好像它确实存在一样。处理这些实际不存在的几何对象的阴影，增加了这些问题的难度。拟阵在数学优化、代码、物理、统计学和生物学中有许多应用。例如，我将致力于的一个仍未解决的问题是关于“交错多项式”的，它是作为DNA链中打结和重组研究的一部分而发明的。

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.