Speaker
Description
A binary function is a function $f:2^E\rightarrow\mathbb{C}$ for which $f(\emptyset)=1$, where $E$ is a finite ground set. Binary functions are closely related to quantum registers. They generalise binary matroids in the sense that any indicator function of a linear space over GF(2) is a $\{0,1\}$-valued binary function (using the natural correspondence between subsets of $E$ and their characteristic vectors in $\hbox{GF}(2)^E$); in fact, the author showed in 1993 that every matroid has an associated binary function, although it will not necessarily be just $\{0,1\}$-valued.
In a series of papers over 1993-2019, the author generalised some standard matroid transforms and operations — including rank, deletion, contraction, and duality (via the Hadamard transform) — to binary functions, and gave new versions of them that are parameterised by the complex numbers. In each of these settings, a theory of Tutte-Whitney polynomials was developed.
In this talk we review this work and discuss some recent results.
