Extensions of matroid Hodge theory

点击翻译按钮获取中文摘要

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.