I will describe new joint work with Evgeny Poletsky. We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient is a complex manifold with a local biholomorphism to $X$, except it need not be Hausdorff. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.

Many interesting natural phenomena are hard to predict. When modelled as a dynamical system, this unpredictability is often the result of rapid separation of nearby trajectories. Viewing the dynamics as acting on a probability measure, the mixing property states that two measurements (or random variables), evaluated at increasingly separated times, become independent in the time-separation limit. Thus, the later measurement becomes increasingly difficult to predict, given the outcome of the earlier measurement. If this approach to independence occurs exponentially quickly in time, one can profitably use linear operator tools to analyse the dynamics. I will give an overview of these techniques and show how they can be applied to answer mathematical questions, describe observed behaviour in fluid mixing, and analyse models of the ocean and atmosphere.

Lecture series by Rajesh Gopakumar (Harish-Chandra Research Institute). The lectures will be supplemented by talks by other invited speakers.

Space and time together constitute one of the most basic elements of physical reality. Since Einstein spacetime has become an active participant in the dynamics of the gravitational force. However, our notion of a quantum spacetime is still rudimentary. String theory, building upon hints provided from the physics of black holes, seems to be suggesting a very novel, "holographic" picture of what quantum spacetime might be. This relies on some very surprising connections of gravity with quantum field theories (which provide the framework for the description of the other fundamental interactions of nature). In this talk, I will try and convey some of the flavour of these connections as well as its significance.

Calabi-Yau manifolds are Riemannian manifolds with holonomy group SU(m). They are Ricci-flat and Kahler and admit a 2-parameter family of parallel spinors. In the talk we will discuss the Lorentzian conformal analogue of this situation. If on a manifold a class of conformally equivalent metrics [g] is given, then one can consider the holonomy group of the conformal manifold (M,[g]), which is a subgroup of O(p+1,q+1) if the metric g has signature (p,q). There is a close relation between algebraic properties of the conformal holonomy group and the existence of Einstein metrics in the conformal class as well as to the existence of conformal Killing spinors. In the talk I will explain classification results for conformal holonomy groups of Lorentzian manifolds. In particular, I will describe Lorentzian manifolds (M,g) with conformal holonomy group SU(1,m), which can be viewed as the conformal analogue of Calabi-Yau manifolds. Such Lorentzian metrics g, known as Fefferman metrics, appear on S^1-bundles over strictly pseudoconvex CR spin manifolds and admit a 2-parameter family of conformal Killing spinors.

Electricity demand forecasting plays an important role in short-term load allocation and long-term planning for future generation facilities and transmission augmentation. Planners must adopt a probabilistic view of potential peak demand levels, therefore density forecasts (providing estimates of the full probability distributions of the possible future values of the demand) are more helpful than point forecasts, and are necessary for utilities to evaluate and hedge the financial risk accrued by demand variability and forecasting uncertainty. Electricity demand in a given season is subject to a range of uncertainties, including underlying population growth, changing technology, economic conditions, prevailing weather conditions (and the timing of those conditions), as well as the general randomness inherent in individual usage. It is also subject to some known calendar effects due to the time of day, day of week, time of year, and public holidays. I will describe a comprehensive forecasting solution designed to take all the available information into account, and to provide forecast distributions from a few hours ahead to a few decades ahead. We use semi-parametric additive models to estimate the relationships between demand and the covariates, including temperatures, calendar effects and some demographic and economic variables. Then we forecast the demand distributions using a mixture of temperature simulation, assumed future economic scenarios, and residual bootstrapping. The temperature simulation is implemented through a new seasonal bootstrapping method with variable blocks. The model is being used by the state energy market operators and some electricity supply companies to forecast the probability distribution of electricity demand in various regions of Australia. It also underpinned the Victorian Vision 2030 energy strategy.

Lecture series by Jonathan Rosenberg (University of Maryland). There will be additional talks by other invited speakers.

The de Rham complex is of fundamental importance in differential geometry. After first introducing differential forms (in the familiar setting of Euclidean space), I will demonstrate how the de Rham complex elegantly encodes one half (in a sense which will become apparent) of the results from vector calculus. If there is time, I will indicate how results from the remaining half of the theory can be concisely expressed by a single, far more general theorem.

We've all heard about the grand challenges in mathematics. There was the Poincare Conjecture, which has now been resolved. There is the Riemann Hypothesis which many are seeking to prove. But one of the most intriguing is the so called "Navier-Stokes Equations" problem, intriguing because it not only involves some wickedly difficult mathematics but also involves questions about our deep understanding of nature as encountered in the flow of fluids. This talk will introduce the problem (without the wickedly difficult mathematics) and discuss some of the consequences of its resolution.

The term "overlapping" refers to a certain fairly simple type of piecewise continuous function from the unit interval to itself and also to a fairly simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used to: 1. find a necessary and sufficient condition for a fractal transformation from the attractor of one overlapping IFS to the attractor of another overlapping IFS to be a homeomorphism and 2. find a formula for the topological entropy of the dynamical system associated with an overlapping function. These results suggest a new method for analysing clocks, weather systems and prime numbers.

This is my PhD thesis submitted one month ago. Chapter 1 introduces the backgrounds of the research fields. Then each chapter is a published or an accepted paper. Chapter 2, to appear in Methodology and Computing in Applied Probability, establishes the theory of Backward Stochastic Difference Equations with respect to the single jump process in discrete time. Chapter 3, published in Stochastic Analysis and Applications, establishes the theory of Backward Stochastic Differential Equations with respect to the single jump process in continuous time. Chapter 2 and 3 consist of Part I Theory. Chapter 4, published in Expert Systems With Applications, gives some examples about how to measure financial risks by the theory established in Chapter 2. Chapter 5, accepted by Journal of Applied Probability, considers the question of an optimal transaction between two investors to minimize their risks. It's the applications of the theory established in Chapter 3. Chapter 4 and 5 consist of Part II Applications.

In this talk, I will describe a mechanism for determining instability of standing wave solutions to a class of inhomogeneous nonlinear Schrodinger (NLS) equations. The inhomogeneity in this case means that the equations will spatially alternate between NLS and the so-called Gross-Pitaevskii equation. Such equations are useful in 1-D models of Bose-Einstein Condensates (BECs). The mechanism is inherently topological and therefore robust, leading to its application to a number of different soliton solutions, such as gap solitons, surface gap solitons, and dark soliton among others.

The Faddeev-Mickelsson-Shatashvili anomaly arises in the quantisation of fermions interacting with external gauge potentials. Mathematically, it can be described as a certain lifting problem for an extension of groups. The theory of bundle gerbes is very useful for studying lifting problems, however it only applies in the case of a central extension whereas in the study of the FMS anomaly the relevant extension is non-central. In this talk I will explain how to describe this anomaly indirectly using bundle gerbes and how to use a generalisation of bundle gerbes to describe the (non-central) lifting problem directly. This is joint work with Pedram Hekmati, Michael Murray and Danny Stevenson.

Growing plant cells undergo rapid axial elongation with negligible radial expansion: high internal turgor pressure causes viscous stretching of the cell wall. We represent the cell wall as a thin fibre-reinforced viscous sheet, providing insight into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility and showing how either dynamical changes in material properties, achieved through changes in the cell-wall microstructure, or passive fibre reorientation may suppress cell elongation. We then investigate how the action of enzymes on the cell wall microstructure can lead to the required dynamic changes in macroscale wall material properties, and thus demonstrate a mechanism by which hormones may regulate plant growth.