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Extensions of matroid Hodge theory

拟阵霍奇理论的扩展

基本信息

批准号：
EP/X001229/1
负责人：
Alexander Fink
金额：
$ 42.75万
依托单位：
Queen Mary University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FX001229%2F1
关键词：
Extensions matroid Hodge theory

项目摘要

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.

组合数学是数学的一个分支，它包括对离散结构进行计数的问题。例如，给定一幅用轮廓线画出的地图，以及一些颜色，你可以用多少种方法给地图中的区域上色，使任何两个相邻的区域都是不同的颜色？组合数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（four colour theorem）--即有四种颜色时，着色数大于零--是一个著名的例子：它不再是一个未解决的问题，但它的解决方案在1976年是由一个长期的计算机搜索和许多数学家仍然希望一个更概念性的答案。拟议的研究是适用于这些问题的技术，从代数几何，这是对多项式方程组解的几何研究。2018年，Adiprasito、Huh和Katz在这方面取得了突破。他们使用代数几何中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年前的公开问题。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; Read的问题是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叫做拟阵。每个映射都有一个拟阵。拟阵带有一组对象，并挑选出这些对象中的某些较小的集合，使其相互兼容，或者用数学家的话说，相互独立。在地图的例子中，一组边界是独立的，如果没有办法在一个循环中穿过其中的一些边界，并在不穿过同一边界两次的情况下回到起点。其他拟阵来自矩阵。但也有所谓的不可表示拟阵，不是来自这些来源。当一个拟阵确实来自一个映射或矩阵时，我们可以用它来写出一个方程组，以进行几何研究。Adiprasito，Huh和Katz建立的是，即使对于不可表示的拟阵，方程组不存在，Hodge理论仍然像它确实存在一样工作。拟阵在数学优化、编码、物理学、统计学和生物学等领域有着广泛的应用。例如，我将要研究的一个尚未解决的问题是关于“交织多项式”，它是作为DNA链中打结和重组研究的一部分而发明的。