The following colloquia will take place in Room 340 (9 Rainforest Walk, Monash University Clayton). Please email me at if you would like to speak in our colloquium series.

 24/02/16 11:00 am Wednesday Caroline Bardini (The University of Melbourne) Secondary and university mathematics: Do the speak the same language? In mathematics at university, students encounter new symbols, familiar symbols used in new ways, increasingly more complex symbolic syntax, and greater expectation of mathematical rigour. Our ARC Discovery Project examines students' understanding of mathematical notation at university and explore how this differs from their experience at school, with the aim of investigate the impact on students’ confidence to continue studying subjects with high mathematical content at university. In 2015, pilot data from our Project suggested that what may appear to be careless errors actually follow identifiable patterns indicating misinterpretation of symbols and expressions. Strengthening and expanding these findings will have implications for both secondary and tertiary mathematics teaching. At this seminar we will share findings to date and discuss the involvement of Monash tutors and students in the 2016 data collection.

COLLOQUIUM ARCHIVE

 03/12/15 03:00 pm Thursday Bruce Reed (McGill University) How to determine if a random graph with a fixed degree sequence has a giant component Many 21st century networks are formed via procedures, such as preferential attachment, which make some nodes much more likely to have attachments than others. For example, we observe empirically in the web that certain authoritative pages, called hubs, will have many more links entering them. Thus it is of interest to determine, for a fixed degree sequence D = {d_1, …, d_n}, the probability that a uniformly chosen (simple) graph on nodes {1, …, n} with the given degree sequence (i.e. such that node i is linked to d_i other nodes) has a giant component. A heuristic argument suggests that a giant component will almost surely exist provided the sum of the squares of the degrees is at least twice the sum of the degrees. In 1996, Molloy and Reed essentially proved this to be the case provided the degree sequence under consideration satisfied certain technical conditions. This work has attracted considerable attention and has been applied to random models of a wide range of complicated 21st century networks such as the web or biological networks operating at a sub-molecular level. Many authors have obtained related results which prove this result under a different set of technical conditions or tie down the size of the giant component. In this paper we characterize when a random graph with a fixed degree seqeunce almost surely contains a giant component, essentially imposing no conditions. This is joint work with Felix Joos, Guillem Perarnau, and Dieter Rautenbach. 05/11/15 03:00 pm Thursday Marston Conder (University of Auckland) Discrete objects with maximum possible symmetry Symmetry is pervasive in both nature and human culture. The notion of chirality (or 'handedness') is similarly pervasive, but less well understood. In this lecture, I will talk about a number of situations involving discrete objects that have maximum possible symmetry in their class, or maximum possible rotational symmetry while being chiral. Examples include geometric solids, combinatorial graphs (networks), maps on surfaces, dessins d'enfants, abstract polytopes, and even compact Riemann surfaces (from a certain perspective). I will describe some recent discoveries about such objects with maximum symmetry, illustrated by pictures as much as possible. 15/10/15 03:00 pm Thursday Yihong Du (University of New England) Reaction-diffusion equations and spreading of species In this talk, I will start by reviewing some classical works (of Fisher, Kolmogorov-Petrovsky-Piscunov, and Skellam) on traveling waves and constant spreading speed. I will then look at the theory of Aronson-Weinberger that describes the spreading by suitable Cauchy problems. Finally I will describe some recent theory obtained with my collaborators based on free boundary value problems, and compare it with results arising from the Cauchy problem along the lines of Aronson-Weinberger. 22/04/15 04:00 pm Wednesday VENUE Rm 442 Bldg 28 Brett Parker (Australian National University) What classical mechanics can and can't do A beautiful formulation of classical mechanics goes by the name of Hamiltonian mechanics. I shall sketch why any volume preserving transformation of phase space may be approximated (in some weak sense) by a Hamiltonian mechanical system. After working hard to understand that Hamiltonian mechanics is clearly very flexible, we will work towards understanding the opposite: that on some deep level, Hamiltonian mechanics exhibits some subtle `rigidity' that is detected by holomorphic curves. I shall sketch the idea of Gromov's nonsqueezing theorem which implies that it is impossible to design a clever (classical) measuring device to circumvent the quantum uncertainty principle. 19/02/15 03:00 pm Thursday VENUE Rm S13 Bldg 29 Persi Diaconis (Stanford University) Adding numbers and shuffling cards When a list of numbers are added in the usual way, "carries" occur. These carries turn out to form a Markov chain with an "amazing" transition matrix. The same matrix appears in group theory, sections of power series and elsewhere. These results are in turn closely related to the usual method of shuffling cards (as in "seven shuffles suffice"). I will try to explain all this to a general mathematical audience in "English". 22/01/15 03:00 pm Thursday Michel Chipot (University of Zurich) Asymptotic issues We would like to present some results on the asymptotic behaviour of different problems set in cylindrical domains of the type $\ell \omega_1 \times \omega_2$ when $\ell \to \infty$. For $i = 1, 2$ $\omega_i$ are two bounded open subsets in $\mathbb{R}^{d_i}$. To fix the ideas on a simple example consider for instance $\omega_1 = \omega_2 = (-1, 1)$ and $u_\ell$ the solution to $- \Delta u_\ell = f \text{ in } \Omega_\ell = (-\ell, \ell) \times (-1, 1), \qquad u_\ell = 0 \text{ on } \partial \Omega_\ell.$ It is more or less clear that, when $\ell \to \infty$, $u_\ell$ will converge toward $u_\infty$ solution to $- \Delta u_\infty = f \text{ in } \Omega_\infty = (-\infty, \infty) \times (-1, 1), \qquad u_\infty = 0 \text{ on } \partial \Omega_\infty.$ However this problem has infinitely many solutions since for every integer $k$ $\exp(k \pi x_1) \sin(k \pi x_2)$ is solution of the corresponding homogeneous problem. Our goal is to explain the selection process of the solution for different problems of this type when $\ell \to \infty$. 09/10/14 03:00 pm Thursday Markus Hegland (Australian National University) Solving multidimensional partial differential equations on emerging high performance computers --- the challenges of large scale parallelism and high dimensionality Since the 1950s a variety of numerical techniques and state-of-the-art computers have been used successfully to solve partial differential equations arising mostly from industrial and environmental problems. Prediction of weather and tsunamis, the production of computer chips and cars, the design of drugs and and even the determination of option prices would be unthinkable without numerical techniques. Computers increased their speed by a factor of around 1000 every 10 years. This continuous increase in speed allowed the solution of increasingly complex problems. As such, computational power is a major driver in technological and scientific advances. Originally this speed increase was obtained by increasing the speed of computational components. In the recent 20 years, a major driver of this speed increase was the simultaneous deployment of increasingly large numbers of identical components. Many new parallel numerical algorithms were developed which allow the effective usage of highly parallel computers. Increasing the number of computational components at an exponential rate poses some new challenges. These relate to energy costs and in addition hardware reliability becomes a concern. These challenges require new algorithmic solutions. In addition to these hardware-related challenges there are computational problems which cannot be solved using standard techniques by even the next generation of computers. These include multidimensional and in particular high-dimensional problems which occur in computational quantum chemistry, finance, machine learning, uncertainty quantification and in plasma physics. Traditional numerical techniques applied to these problems all suffer under the curse of dimensionality. In this talk I will provide some background on the computational challenges mentioned and some approaches on how to deal with them. 28/08/14 03:00 pm Thursday Carla Cederbaum (Universität Tübingen) Mass in Newtonian gravity and general relativity Isolated gravitating systems such as stars, black holes, and galaxies play an important role both in Newton's theory of gravity and in Einstein's theory of general relativity. While the definition of mass and center of mass via the mass density is straightforward in Newtonian gravity, there is no definitive corresponding notion in general relativity. Instead, there are several alternative approaches in general relativity to defining the center of mass of an isolated system. We will discuss these different approaches and present some explicit examples. Moreover, we will introduce the notion of Newtonian limit and use it to relate the Newtonian and the relativistic centers in the case of static systems. 14/08/14 03:00 pm Thursday Ole Warnaar (The University of Queensland) Generalised Rogers-Ramanujan identities and arithmetics The Rogers-Ramanujan q-series play an important role in many areas of mathematics, such as combinatorics, number theory, statistical mechanics and representation theory. In this talk I will try to give a gentle introduction to the subject, as well as describe some recent new developments concerning the arithmetic properties of generalised Rogers-Ramanujan series arising from the representation theory of affine Lie algebras. 10/04/14 03:00 pm Thursday Paul Zinn-Justin (Université Pierre et Marie Curie) Combinatorics and loop models: a review We shall review recent developments in the combinatorial interpretation of certain models coming from two-dimensional statistical mechanics, mainly loop models. Among the subjects we shall discuss are the Razumov-Stroganov correspondence, the enumeration of Alternating Sign Matrices and the geometry of Orbital Varieties. 13/02/14 03:00 pm Thursday Luca Chiantini (University of Siena) Symmetric tensors and the geometry of sets of points We illustrate a link between two fields of Mathematics: the multilinear algebra of tensors and the algebraic geometry of sets of points in projective space. We show how tools such as the Hilbert function of finite sets can be used to analyze properties of (symmetric) tensors, like the complexity or the identifiability. 07/11/13 01:00 pm Thursday Kai Zuber (Dresden University of Technology) Neutrinos: The X-files of  particle physics Neutrinos are still the big mystery of particle physics. However, in the last decade dramatic progress was made by establishing a non-vanishing rest mass of the neutrino not foreseen in the Standard Model of Particle Physics. The experimental observations leading to this breakthrough will be presented and the current picture of neutrinos masses and mixing will be explained. Also the astrophysical aspect of neutrino physics will be presented. Based on that the major open questions will be discussed including in outlook into future activities. 07/10/13 02:00 pm Monday Akshay Venkatesh (Stanford University) Dynamics and the geometry of numbers It was understood by Minkowski that one could prove interesting results in number theory by considering the geometry of lattices in R^n. (A lattice is simply a grid of points.) This technique is called the "geometry of numbers." We now understand much more about analysis and dynamics on the space of all lattices, and this has led to a deeper understanding of classical questions. I will review some of these ideas, with emphasis on the dynamical aspects. 26/09/13 03:00 pm Thursday Sommer Gentry (United States Naval Academy) Optimization, ethics, and organs: mathematical methods for rationing transplantation The notion of rationing healthcare is taboo: people naturally feel no one should limit the resources spent extending human life, particularly theirs or their loved ones'. Transplantation can transform the lives of organ recipients, but must be rationed by access to the far-too-small supply of donated organs, so it is a microcosm of ethical dilemmas in rationing healthcare. Operations research techniques can maximize the number of life years gained from transplantation, or redistrict geographic allocation units to distribute organs more fairly across large countries like the United States. For example, kidney paired donation matches a living kidney donor who is incompatible with his intended receipient with another incompatible pair for an exchange. Kidney paired donation uses graph algorithms for maximum weight matching to select the best combination of exchanges. Beyond the sophistication of methods, the real challenge is to help decision-makers scrutinize how "fair" and "optimal" can be defined.  I will share my experiences as a mathematician in the transplant community. 05/09/13 03:00 pm Thursday Tim Trudgian (Australian National University) The Riemann hypothesis: 1 for 500 at stumps on day one Working on problems connected to the Riemann hypothesis is a hard slog. In this talk I hope to illustrate how far mathematicians have come, how far there is to go, what definitely doesn't work, and what may work. 07/08/13 03:00 pm Wednesday Sophie Hautphenne (The University of Melbourne) A computational look at multitype branching processes Branching processes with finitely many types have been studied extensively; the extinction probability is known to be the minimal nonnegative solution of a finite system of equations, and a simple extinction criterion is given by the spectral radius of the mean progeny matrix.  The analysis of multitype branching processes becomes more complicated when the number of types is allowed to be infinite, or when the branching process undergoes catastrophes or evolves in a random environment. This talk addresses our contribution to the computation of the extinction probability vector, and to the characterization of extinction criteria, for such specific processes. 07/08/13 12:30 pm Wednesday Mark Flegg (University of Oxford) The application of mathematics to multiple scale reaction-diffusion processes in biology. Dynamic systems involving reaction and active and inactive transport of multipleindividuals occur in many fields of science, from physics and engineering to chemistry and biology. In biochemistry and systems biology, reaction-diffusion systems are common and are responsible for highly complex processes such as cellular "decision making". Classically, these types of systems have been treated by mathematicians using deterministic approaches such as the reaction-diffusion equation. However, in many cases in biology, these deterministic approaches are highly inappropriate and either lead to an inaccurate mathematical representation of the phenomena being modelled or a complete breakdown of the qualitative model predictions. In such situations it is appropriate to use a stochastic model representation. A significant international research effort in the last decade has focused on describing complex reaction-diffusion environments, such as a cell, in a mathematical framework that is computationally efficient and accurate. In this presentation, I will present a brief background of this field of research, leading up to the introduction of multiple scale techniques developed at the University of Oxford which are capable of integrating the two common types of stochastic modelling approaches and a deterministic representation into a single mathematical framework. I will discuss how these multiscale approaches are a crucial tool for modelling two particular subcellular processes which have been studied in further detail; Min protein pole-to-pole oscillations in E. Coli and quantifying calcium-induced calcium release puff frequencies from the endoplasmic reticulum. The application of mathematics to biology and other sciences is an expanding area of research, due to the vast increase in experimental data and improved capabilities of computer technology. As a potential future member of MAXIMA, a Monash academy designed for interdisciplinary mathematical collaboration, I would also like to briefly mention some other areas of applied mathematics in which I am involved. 07/08/13 11:00 am Wednesday Jennifer Flegg (University of Oxford) Applying mathematics to epidemiology and physiology: antimalarial drug resistance and nonhealing wounds This talk will cover the use of applied mathematical models in two very different biological applications: (1) epidemiology – spatiotemporal trends of resistance to drugs used to treat malaria and (2) physiology – investigating the clinical implications of a simple model of wound healing. (1) Malaria parasites have repeatedly evolved resistance to anti-malarial drugs, thwarting efforts to eliminate the disease and contributing largely to an increase in mortality. In this talk, I will explore a mathematical model developed to track the geospatial and temporal trends in the distribution of molecular marker mutations that are associated with resistance to the anti-malarial drug sulphadoxine-pyrimethamine. Prevalence data of molecular markers are used to inform a Bayesian geostatistical model. Predictive surfaces of the molecular marker mutation over sub-Saharan Africa from 1990-2010, which allow the space-time trends in the parasite resistance to be quantified as well as provide insight on the spread of resistance in a way that the data alone do not allow, will be presented.  (2) While the wound healing process is undoubtedly complex, I will discuss a deterministic mathematical model of wound healing, formulated as a system of partial differential equations. While the model equations can be solved numerically, emphasis will be placed on the use of asymptotic methods to establish conditions under which new blood vessel growth can be initiated. These conditions are given in terms of key model parameters including the rate of oxygen supply and its rate of consumption in the wound. Under the model framework, the use of clinical treatments that have the potential to initiate healing in nonhealing wounds, such as hyperbaric oxygen therapy, will be discussed. In both applications, mathematical models have been used to gain important biological insights. These insights will be discussed along with how the work can be extended. 06/08/13 12:00 pm Tuesday Elsa Hansen (Harvard School of Public Health) Studying pathogen spread: mathematical modelling and statistical analysis In this two-part talk I will discuss (i) theoretical results on the optimal deployment of treatment during a disease outbreak with de novo resistance and (ii) the statistical analysis of count data to characterize parasite proliferation within a single host. Numerous modelling studies have shown that if treatment is administered to a population at a constant rate during an outbreak with de novo resistance, then the total outbreak size can be reduced by limiting this treatment rate. Here we use optimal control theory to consider how this result changes if the treatment rate is allowed to vary as a function of time. We show that, in contrast to previous results, it is in fact optimal to use the maximal possible treatment rate provided the onset of treatment is delayed an appropriate amount. A strength of this analysis is the use of a standard SIR type model which facilitates the derivation of an analytic expression characterizing the optimal delay. This work is done in collaboration with Troy Day from Queen’s University. The second part of this talk will examine different methods of using count data to characterize the malaria parasite’s invasion of red blood cells. Two common measures of invasion are (i) the parasite multiplication rate (PMR) and (ii) the selectivity index (SI). The PMR is often interpreted as the parasite’s average invasion ability and the SI is used to indicate how heterogeneous this invasion ability is. Using data collected from invasion assays we describe the relationship between PMR and SI and detail how this relationship can lead to incorrect conclusions about the degree of heterogeneity in the invasion process. We propose a modification of the SI to mitigate this issue. This work is done in collaboration with Caroline Buckee, Tiffany Desimone and Manoj Duraisingh from the Harvard School of Public Health. 01/08/13 03:00 pm Thursday Ian Sloan (University of New South Wales) Lifting the curse of dimensionality: numerical integration in very high dimensions Richard Bellman coined the phrase "the curse of dimensionality" to describe the extraordinarily rapid increase in the difficulty of most problems as the number of variables increases. One such problem is numerical multiple integration, where the cost of any integration formula of product type obviously rises exponentially with the number of variables. Nevertheless, problems with hundreds or even thousands of variables do arise, and are now being tackled successfully. In this talk I will tell the story of recent developments, in which within a decade the focus turned from existence theorems to concrete constructions that achieve the theoretically predicted results even for integrals in hundreds or thousands of dimensions and many thousands of points. The theory has been shaped by applications, ranging from option pricing to the flow of a liquid through a porous medium, the latter modelled by a partial differential equation with a random permeability field. 24/07/13 10:00 am Wednesday Anthony Piro (California Institute of Technology) Faint, fast, and few One of the fastest growing areas in astronomy right now is wide-field, high-cadence surveys. These projects allow the investigation and discovery of explosive, transient phenomena that are intrinsically dim (faint), occur on short timescales (fast), or are very rare (few). Soon ground-based laser interferometers (LIGO, Virgo) will reach sufficient sensitivity to study many of these same events in an entirely new way using gravitational waves. I will discuss theoretical work to complement these efforts by providing interpretations of discoveries, predicting new classes of transients, and guiding observing strategies. This has implications for many areas of astrophysics and fundamental physics, from helping identify the elusive progenitors of Type Ia supernovae, explosive events used to discover Dark Energy, to studying the structure of neutron stars, which contain some of the densest matter found anywhere in the Universe. 23/07/13 12:00 pm Tuesday Bernhard Müller (Max Planck Institute for Astrophysics) Core-collapse supernova simulations: past, present and future Core-collapse supernovae, the luminous outbursts marking the deaths of massive stars, have been a foremost problem in computational astrophysics for decades. After more than forty years, the mechanism responsible for the explosion is still not fully understood because the complex interplay of multi-dimensional hydrodynamics, neutrino transport, nuclear physics, and general relativity has proved difficult to capture in numerical simulations. However, as I shall demonstrate in this talk, the latest generation of simulations has considerably advanced our understanding of the successful ingredients for robust explosions. Moreover, we now have detailed predictions for neutrino and gravitational wave emission from the supernova core for a growing set of successful explosion model, which will be invaluable for a close look at the engine in the case of nearby galactic event. Predictions for nucleosynthesis yields that illuminate the role of specific progenitors in the chemical evolution of the galaxy are also becoming available. Finally, I will adumbrate some remaining challenges in supernova theory such as the connection between first-principle explosion models and classical photon-based astronomical observations. 23/07/13 10:00 am Tuesday Robert Izzard (University of Bonn) The origin of the elements and the critical role of binary stars In a few millennia of recorded history, mankind has made great strides in its understanding of the Universe. We now know that the Big Bang created almost all visible matter as just two elements, hydrogen and helium, which were later turned into carbon, oxygen, iron and the 90+ other chemical elements by nucleosynthesis in stars. Many of these stars are gravitationally bound as close binary systems which may evolve quite differently to single stars leading to exotic phenomena such as thermonuclear novae and gamma-ray bursts. Binaries are also crucial to galactic chemical evolution, particularly through type la supernovae which make most of the iron in the Universe. Furthermore, ancient binary stars preserve the chemistry of their long-dead companion and offer unique insight into nucleosynthesis when our galaxy was young. By comparing statistical models of binary star populations with observations, we can pin down the uncertainties in stellar astrophysics models. For example, chemically peculiar stars, such as carbon-enhanced metal-poor and barium stars, are difficult to reproduce both in number and binary orbital properties with conventional stellar models. Much of my recent work has involved statistically testing new theories of mass transfer and orbital interactions to explain these and other unusual binary stars. Statistical methods are also predictive and I will report on new calculations of a binary-star source of the most contemptuous element in the Universe: lithium. Canonical Galactic chemical evolution models of lithium make too little by a factor of three. Mass ejection from binary stars offers an alternative, hitherto ignored, source of lithium which helps to solve this problem but creates a few more along the way. 11/07/13 03:00 pm Thursday Luciano Rezzolla (Albert Einstein Institute) Modelling the most catastrophic astrophysical events The detection of gravitational waves is eagerly expected as one of the most important scientific discoveries of the next decade. A worldwide effort is now working actively to pursue this goal both at an experimental level, by building ever sensitive detectors, and at a theoretical level, by improving the modelling of the numerous sources of gravitational waves. Much of this theoretical work is made through the solution of the Einstein equations in those nonlinear regimes where no analytic solutions are possible or known.  I will review how this is done in practice and highlight the considerable progress made recently in the description of the dynamics of binary systems of black holes and neutron stars. I will also discuss how the study of these systems provides information well beyond that contained in the gravitational waveforms and opens very exciting windows on the relativistic astrophysics of GRBs and of the cosmological evolution of massive black holes. 05/06/13 12:00 pm Wednesday Julie Clutterbuck (Australian National University) Stability for noncompact solutions to mean curvature flow I will describe some recent results on the stability of some solutions to mean curvature flow. This is joint work with Oliver Schnurer and Felix Schulze. 05/06/13 10:00 am Wednesday Roozbeh Hazrat (University of Western Sydney) Graph algebras From a graph (e.g., cities and flights between them) one can generate an algebra which captures the movements along the graph. This talk is about one type of such correspondences, i.e., Leavitt path algebras. Despite being introduced only 8 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, a theory originated and nourished in Australian universities which has become an area of intensive research globally. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered. In this talk, we introduce Leavitt path algebras and then try to understand the behaviour and to classify them by means of (graded) K-theory. We will ask nice questions! 04/06/13 12:30 pm Tuesday Peter McNamara (Stanford University) Categorifying quantum groups The general idea of categorification has been pursued with increased vigour in recent years. We will discuss what it means to categorify a quantum group, and present some results on the combinatorics and representation theory which appears in the resulting structures. 18/04/13 03:00 pm Thursday Alexander Guterman (Moscow State University) Tropical linear algebra and its applications Tropical algebra (sometimes called max algebra) is the set of real numbers with additional symbol, -∞, with unusual way to define the operations, namely, the sum of two elements is their maximum, and the product is their sum. Under these operations tropical algebra is an algebraic structure called a semiring. Note that there is no subtraction in this semiring, however addition and multiplication are commutative, associative, and satisfy usual distributivity laws. Tropical algebra naturally appears in modern scheduling theory and optimization. Tropical arithmetic allows to reduce difficult non-linear problems to the linear problems but over tropical algebra. Therefore, to investigate these problems it is necessary to develop linear algebra in the tropical case. The main purpose of our talk is to discuss tropical linear algebra and its different applications. We plan to consider the modern progress in the theory including our recent results. 21/03/13 03:30 pm Thursday Michael Coons (University of Newcastle) An arithmetic excursion via Stoneham numbers In this talk we will aim to give an explicit example of a "random" number (at least as much as "random" and "explicit" make sense together). We will come at this example via an excursion through easily computable numbers arising from finite automata and some of their generalisations. 07/03/13 03:00 pm Thursday Hyam Rubinstein (The University of Melbourne) The solution of the Poincaré conjecture The Poincaré conjecture was one of the most celebrated questions in mathematics. It was amongst the seven millennium problems of the Clay Institute, for which a prize of \$1 million was offered. The Poincaré conjecture asked whether a 3-dimensional space with 'no holes' is equivalent to the 3-dimensional sphere. In 2003 Grigori Perelman posted three papers on the internet ArXiv outlining a marvellous solution to the Poincaré conjecture, as part of the completion of Thurston’s geometrisation program for all 3-dimensional spaces. Perelman introduced powerful new techniques into Richard Hamilton’s Ricci flow, which 'improves' the shape of a space. Starting with any shape of a space with no holes, Perelman was able to flow the space until it became round and therefore verified it was a sphere. A brief history of the Poincaré conjecture and Thurston’s revolutionary ideas will be given. Hamilton’s Ricci flow will be illustrated. Famously, Perelman turned down both the Clay prize and a Fields medal for his work. 17/12/12 03:30 pm Monday Pascal Buenzli (The University of Western Australia) Understanding complex system behaviours from underlying structures - Mathematical modelling in fluctuation-induced phenomena and in bone remodelling Many physical and biological systems possess structures or behaviours across multiple scales. Mathematical modelling enables us to link these scales and explain the emergence of these structures and behaviours from underlying mechanisms. I will present two distinct examples based on the material balance equation in which this approach is implemented: (1) fluctuation-induced forces in a nonequilibrium fluid and (2) the renewal of bone tissue in our skeleton. (1) Macroscopic objects immersed in a fluctuating medium constrain the medium's fluctuating modes. This may induce forces between the macroscopic objects, as in the "Casimir effect'. I will present the case of asymmetric objects immersed in a nonequilibrium fluid whose density fluctuations obey a stochastic reaction-diffusion equation. (2) Throughout life, our bones are continually renewed to repair micro-cracks. This renewal process is operated by groups of bone-resorbing cells and bone-forming cells that are organised into standalone functional units called 'basic multicellular units' (BMUs). A system of reaction-diffusion-advection equations expressing the complex network of cell-cell communication mediated by several signalling molecules explains the emergence of these organised units. 17/12/12 02:00 pm Monday Chris Brook (Universidad Autonoma de Madrid, Spain) Using simulated galaxies as probes of the cold dark matter paradigm We solve the non-linear equations that govern structure formation within a Universe dominated by cold dark matter with a cosmological constant, and model the behaviour of baryons using smoothed particle hydrodynamics along with physically motivated models for star formation and energy feedback. We show how feedback energy from massive stars, including supernova explosions, is absolutely crucial in resolving the properties of structures predicted by the cosmological paradigm with the properties of structures observed in real galaxies. 17/12/12 11:00 am Monday Bronwyn Hajek (University of South Australia) Problems in viscous extensional flow and electrokinetics In this talk, I will give an overview of two research projects. Honey dripping from an upturned spoon is an everyday example of a viscous extensional flow. The study of these types of flows is motivated by a wide range of applications such as ink-jet printing, the drawing of glass fibres and rheological measurement by fibre extension. I will discuss the effect of the underlying physical mechanisms such as gravity, inertia and surface tension on the evolution of a viscous drop. In particular, we use an Eulerian formulation which overcomes the problem of excessive stretching of grid elements which is often seen with Lagrangian models. Electrokinetic techniques are used to gather specific information about concentrated dispersions such as electronic inks, mineral processing slurries, pharmaceutical products and biological fluids (eg blood). But, like most experimental techniques, intermediate quantities are measured, and consequently the method relies explicitly on theoretical modelling to extract the quantities of experimental interest. I will describe a self-consistent cell-model for the electrokinetic behaviour of concentrated suspensions. This model takes account of the underlying physical processes such as the transport of ions in solution, the electric potential and the hydrodynamic flow, and their effect on the behaviour of the suspension. 10/12/12 02:00 pm Monday Valentina Wheeler (Potsdam Universitat, Germany) Mean curvature flow with free boundary Proposed in 1956 by Mullins, the mean curvature flow was initially used to model the formation of grain boundaries in annealing metals and later on found to have many more applications in fire fronts, image processing and moving interfaces. The flow is the steepest descent gradient flow of the area functional and it is designed to minimise the area of a surface by evolving it in the direction of the unit normal with speed proportional with its mean curvature. We are interested in presenting a particular case of mean curvature flow of graphs with free boundary in a half space, which is the next natural step in the setting of free boundaries from the seminal work on entire graphs of Ecker and Huisken. We start our talk by defining the flow, followed by examples and a discussion about some results from the literature relevant to our work. We show that the mean curvature flow of graphs evolving in a half space with boundary on a fixed hyperplane exists for all time and asymptotically approaches a selfsimilar solution of mean curvature flow with boundary on the same fixed hyperplane. If time permits we will give brief hints towards the proofs. 10/12/12 11:30 am Monday Enrico Carlini (Politecnico di Torino, Italy) Waring problem: from natural numbers to polynomials and tensors Lagrange proved that any natural number is the sum of four squares. Inspired by this result Edward Waring asked to compute the number g(d), defined as the minimal number of summands needed to write any natural number as the sum of d-th powers. Waring problem can be stated in different settings, such as polynomial and tensor algebras. Very little is known in these situations, especially when dealing with a given specific form or tensor. However, special cases are of great relevance for applications: consider, for example, Strassens'celebrated result on matrix multiplication. In this talk we will analyse in some detail the homogeneous polynomial version of Waring problem, showing known results and challenging research questions. 10/12/12 10:00 am Monday Glen Wheeler (Otto-von-Guericke Universitat Magdeburg, Germany) Unexpected phenomena: Energy gap, spherocytosis, and the Helfrich model One beautiful quality of mathematical models is that they allow us to understand a priori unexpected and surprising phenomena. In this talk I shall describe some recent progress toward better understanding of four such phenomena, focusing on the blood disease spherocytosis and the Helfrich model. People with spherocytosis have spherically shaped red blood cells, and suffer from a range of complications. The condition is eventually fatal, with the spleen recognising spherical cells as damaged and destroying them systematically. The only available 'treatment' is a splenectomy, which is far from desirable. One fundamental and unsolved problem is to understand how red blood cells could be spherical in the first place. Using methods from geometric analysis, we (joint with Dr. James McCoy) have discovered an *energy gap phenomenon for the Helfrich functional*, which gives a partial mathematical explanation of the appearance of the spherical shape of red blood cells. Interpreting this result in terms of the underlying Helfrich model gives a physical explanation, and allows us to conclude when blood cells may become spherical, and when they may not. I finish the talk by outlining an approach to tackling the larger picture, and how this might fit together for an eventual treatment for the disease. 07/12/12 11:00 am Friday Daniel Mathews (The University of Melbourne and Contextual Systems, Melbourne) Contact topology and holomorphic invariants via elementary combinatorics In recent years a great amount of progress has been achieved in understanding the structures of symplectic and contact geometry. Powerful invariants of 3-manifolds and their contact structures have been defined, such as Heegaard Floer homology and contact homology, based on theories of generalised Cauchy-Riemann equations, holomorphic curves, and moduli spaces. These invariants also possess some of the properties of topological quantum field theories. In simple cases, these structures reduce to intriguing elementary combinatorial and algebraic results. I will discuss these elementary results, and give a brief introduction to some of the ideas of contact topology and holomorphic curves. 06/12/12 03:00 pm Thursday Andrei Okounkov (Columbia University, USA) The index and the vertex I will report on a joint work with Nikita Nekrasov. After a review of Dirac operators and their indices, I will explain how and why one computes the indices of Dirac operators acting on instanton moduli spaces for algebraic surfaces and, more recently, algebraic 3-folds.