University of Melbourne

The complexity of biological systems necessitates that we develop mathematical models to further our understanding of these systems. Mathematical models of these systems are generally based on heterogeneous sets of experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system, and to reveal underlying design principles, we therefore need to understand how the different models are related to each other with a view to obtaining a unified mathematical description. This goal is complicated by the number of distinct mathematical formalisms that may be employed to represent the same system, making direct comparison of the models very difficult. In this talk I will discuss two general methodologies, namely comparison by distance and comparison by equivalence, that allow us to compare model structures in a systematic way by representing models as labelled simplicial complexes. The distance can be obtained either directly from the simplicial complexes, or from the persistence intervals obtained by employing persistent homology with a flat filtration. Model equivalence is used to determine the conceptual similarity of models and can be automated by using group actions on the simplicial complexes. We apply our methodology for model comparison to demonstrate a particular equivalence between a positional-information model and a Turing-pattern model from developmental biology, which constitutes a novel observation for two classes of models that were previously regarded as unrelated. We also discuss an alternative framework for model comparison by representing models as groups, which allows for the application of group-theoretic techniques within our model comparison methodology.

Dark matter is known to exist, but no-one knows what it is or where it came
from.  We describe a new production mechanism of particle dark matter, which
hinges on a first-order cosmological phase transition.  We then show that
this mechanism can be slightly modified to produce primordial black holes.

While solar mass and supermassive black holes are now known to exist,
primordial black holes have not yet been seen but could solve a number of
problems in cosmology.  Finally, we demonstrate that if an evaporating
primordial black hole is observed, it will provide a unique window onto
Beyond the Standard Model physics.

It was implicitly conjectured by Hambly-Lyons in 2010, which was made explicit by Chang-Lyons-Ni in 2018, that the length of a tree-reduced path with bounded variation can be recovered from its signature asymptotics. Apart from its intrinsic elegance, understanding such a phenomenon is also important for the study of signature lower bounds and may shed light on more general signature inversion properties. In this talk, we discuss how the idea of path development onto suitably chosen Lie groups can be used to study this problem as well as its rough path analogue.

It is well known that a rough path is uniquely determined by its signature (the collection of global iterated path integrals) up to tree-like pieces. However, the proof the uniqueness theorem is non-constructive and does not give us information about how quantitative properties of the path can be explicitly recovered from its signature. In this talk, we examine the quantitative relationship between the local p-variation of a rough path and the tail asymptotics of its signature for the simplest type of rough paths ("line segments"). What lies at the core of the work a novel technique based on the representation theory of complex semisimple Lie algebras. 

This talk is based on joint work with Horatio Boedihardjo and Nikolaos Souris

Many industrial optimisation problems involve the challenging task of efficiently searching for optimal decisions from a huge set of possible combinations. The optimal solution is the one that best optimises a set of objectives or goals, such as maximising productivity while minimising costs. If we have a nice mathematical equation for how each objective depends on the decisions we make, then we can usually employ standard mathematical approaches, such as calculus, to find the optimal solution. But what do we do when we have no idea how our decisions affect the objectives, and thus no equations? What if all we have is a small set of experiments, where we have tried to measure the effect of some decisions? How do we make use of this limited information to try to find the best decisions?

This talk will present a common industrial optimisation problem, known as expensive black box optimisation, through a case study from the manufacturing sector. For problems like this, calculus can’t help, and trial and error is not an option! We will introduce some methods and tools for tackling expensive black-box optimisation. Finally, we will discuss new methodologies for assessing the strengths and weaknesses of optimisation methods, to ensure the right method is selected for the right problem.