Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. In this talk we will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.

Discrete symplectic mappings are not called discrete Hamiltonian systems by the cognoscenti for a variety of reasons, and my aim with this project was to explore the problems involved. In this talk I first show how conservation laws for high order Lagrangian systems transfer to their equivalent Hamiltonian systems with a strikingly beautiful formula. This formula transfers mutatis mutandis to the discrete case. However, analogues of other results do not transfer. I present a series of examples that illustrate the various difficulties, and end with some conjectures and possible ways forward that could involve specialist analytic techniques. The talk assumes no specialist knowledge of the topic.

Pascal's matrices are constructed from Pascal's triangle and they are encountered frequently in probability theory. In this talk we will discuss some of their striking properties and their application to polynomials.

(Note: lecture will start at 5:15)

Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. In this talk we discuss moving frames and some applications. We then show how what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains, offers significant computational advantages over a single moving frame for some applications, in particular to discrete integrable systems. The talk assumes no specialist knowledge.