Given a finite collection of points in space, we can record which sets of points are collinear, which sets of points are coplanar, and so on. This combinatorial data is called a matroid. For example, here is a visualization of a famous matroid:
To a matroid \(M\), we can associate a rational function in one variable, \(Z_M(s)\), called the topological zeta function of \(M\). While some is known about these functions, they remain mysterious in many ways. For instance, we know that \(Z_M(0) = 1\) and \(Z_M'(0)\) is the number of points in \(M\). However, there is no known combinatorial interpretation of the higher derivatives at zero.
The goal of this project is to find a conjectural combinatorial interpretation for (at least) \(Z_M''(0)\). We will do this by computing \(Z_M(s)\) for a sufficient number of matroids \(M\).
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