School of Mathematics

There is an assumption throughout much of the English-speaking world that the transition from secondary school study in mathematics to first year university should be made a smooth as possible to maximise learning. On the basis of that assumption foundation courses and other mechanisms have been established to address deficits students may have. Based on my work with Lovric (McMaster) this talk argues for a different model of transition that is as least as well justified and may be more effective and satisfying for students.

Classical Physics has a long and intimate relation with Geometry, going back to Galileo and Newton, in which force bends (or curves) the motion of a particle. This broad idea carries over to Maxwell’s Electromagnetism and Einstein’s General Relativity. However in the 20th century quantum mechanics altered the picture, but at the same time geometers widened their horizons by taking up topology. I will try to explain how the force- curvature link extends to a quantum-topology one. Interestingly a prime example of a topological problem is that of distinguishing knots, and Kelvin in the 1870’s suggested that knots might explain the structure of atoms. Although, with the advent of quantum mechanics, Kelvin’s theory was discarded, it was too beautiful an idea to waste. In a sense Kelvin’s basic idea has survived but applied at the subatomic level. The new understanding of the Quantum –Topology link has had a profound effect on both mathematics and theoretical physics, as I hope to indicate.

The central topic of the talk will be the mathematical models of the evolution of biological populations (such as plankton). The space distribution of the particles can be described by the reaction-diffusion (or KPP) equations, perhaps with the negative feedback (in the spirit of the classical works by Fisher and Kolmogorov-Petrovskii-Piskunov). The distribution of the masses of particles is described by a non-standard differential-functional equation (which must catch the mitosis processes) coupled with the KPP equation. The talk will present several analytic results and limit theorems on the space-mass distribution of the particles.

In the talk I will describe a development project aiming at furthering independent student work and developing mathematical competencies.

For a long time there have been two kinds of mathematical computation: symbolic and numerical. Symbolic computing manipulates algebraic expressions exactly, but it is unworkable for many applications since the space and time requirements tend to grow combinatorially. Numerical computing avoids the combinatorial explosion by rounding to 16 digits at each step, but it works just with individual numbers, not algebraic expressions. This talk will describe a new kind of computing that aims to combine the feel of symbolics with the speed of numerics. The idea is to represent functions by Chebyshev expansions whose length is determined adaptively to maintain an accuracy of close to machine precision. Our chebfun system is implemented in object-oriented Matlab, with familiar vector operations such as sum and diff being overloaded to analogues for functions such as integration and differentiation. The system is surprisingly effective, and a demonstration will be given together with a discussion of the underlying mathematics and of the prospects for the future. The chebfun system is a joint project with Zachary Battles, Ricardo Pachon, Rodrigo Platte, and Toby Driscoll.

We study limiting behavior of rescaled size distributions in several models of clustering or coagulation dynamics, `solvable' in the sense that the Laplace transform converts them into nonlinear PDE. The scaling analysis that emerges has many connections with the classical limit theorems of probability theory, and a surprising application to the study of shock clustering in the inviscid Burgers equation with random-walk initial data. I'll focus on recent progress regarding a `min-driven' clustering model related to domain coarsening dynamics in the Allen-Cahn equation.

Soil moisture provides the physical link between soil, climate and vegetation. It increases via the infiltration of rainfall and decreases through evapotranspiration, run-off and leakage, all these effects being dependent on the existing soil moisture level. During wet periods, soil moisture tends largely to be driven by the topography, while evapotranspiration has more influence in dry periods. In this talk, I will describe models for soil moisture dynamics in which marked Poisson processes are used to model the temporal process of rainfall input to the soil moisture dynamics, and storms are allowed to have both spatial and temporal extents. Losses due to evapotranspiration depend on vegetation cover and the models allow for variable, and possibly random, vegetation processes. Under arid/semi-arid conditions, many transient and equilibrium properties of these models can be determined analytically and used for comparison with data on soil moisture dynamics.

This talk will be about the work of OCIAM and its various international imitators, and the new mathematical challenges posed by industrial questions.

Abstract: In this talk we look at the life, labours and legacy of Leonhard Euler (1707-1783), the most prolific mathematician of all time. The talk will be preceded by tea at 4pm and followed by a wine reception, both in the level 9 foyer of the School of Mathematics.