School of Mathematics and Statistics Te Kura Mātai Tatauranga

Model theory studies formal logical languages, such as propositional logic or first-order logic, as well as the interaction of these languages with the objects that they describe.

Matroids are abstractions of discrete geometrical configurations of points. They arise from matrices and graphs.

This talk will be an introduction to both model theory and matroid theory. The fundamental question here concerns properties of matroids, and whether or not we can describe those properties with sentences in a given logical language. This type of investigation originated in 1978, but was neglected for many years. Recently I have been working with Mike Newman and Geoff Whittle on the so-called "missing axiom" of matroid theory, with some interesting results.

I have encountered a wide variety of interesting problems during my research as an applied mathematician. The time has come to talk of many things; steaming volcanic bombs and frozen Southern seas, cooking crispy cereals, tall tapered feeders, and fast heavy fruit, to mention but a few. Mathematics is what binds these themes together and I will talk about how math has helped me to understand what is going on, and sometimes to solve the problem.

Digital image synthesis has undergone a major paradigm shift in recent years. By simulating the flow of light based on the laws of nature new levels of photorealism, consistency and predictability could be achieved and workflows be simplified. Computing the solution to the fundamental equation of light transport is conceptually simple, it is just an integration problem that is commonly solved by a class of algorithms called path-tracing. However, our natural environment is full of visually interesting phenomena that are very difficult to integrate as they are effects of inherently geometric nature of varying scales. We give an overview of the challenges this imposes, some of the algorithmic tools that are available and which problems remain difficult to solve.