Mathematics Postgraduate Seminar Series - 2017
Seminars in 2017, Semester 1
Monday March 6
Dominic Tate (Sydney University)
Introduction to (G, X)-Structures
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In 1872 Felix Klein outlined the Erlangen Program with the aim of studying the intrinsic geometry of surfaces using group theory and projective geometry. This idea is formalised in the notion of (G, X)-structures wherein the geometric features of a given surface are locally modelled on that of a smooth manifold X and preserved by the action of a group G on X. I plan to give an introduction to the language and fundamental features of (G, X)-structures, including the Developing Map and Holonomy Representation. This will provide the framework for a discussion of the Teichmuller space and its parameterisation through Fenchel-Nielsen coordinates.
Monday March 13
Eric Hester (Sydney University)
Fresh Insights on Dead Water
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Dead water refers to a mysterious increase in resistance experienced by boats in density-stratified waters.
The problem has been documented since ancient times, and studied scientifically for over a century.
However, past investigations have been limited in several important ways.
For the first time, we study the phenomenon using state-of-the-art numerical simulations.
We reproduce the effect and demonstrate that it is most pronounced in strongly nonlinear regimes poorly modelled by current theory.
The most exciting development is a new trailing vortex found behind the boat experiencing the effect.
This robust structure is consistent with sailors accounts, but has been missed in previous scientific studies.
We expect these results to lead to actionable ways to mitigate dead water in the real world.
Monday March 20
Giulia Codenotti (Freie University, Berlin)
Triangulated spheres and Pachner moves
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In the world of discrete geometry-somewhere between combinatorics and geometry- Pachner moves (or bistellar flips) are an important tool to "build up" complex triangulations from more simple ones while preserving topological properties. I will give an introduction of basic objects in discrete geometry such as polytopes and simplicial complexes and show with examples how Pachner moves can help both to devise new questions and to answer others, spanning topics from combinatorics to topology.
Monday March 27
Jakub Tomczyk (Sydney University)
Gaussian Product Conjecture
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The aim is to present Gaussian Product Conjecture which proposes lower bound for moments of multivariate normal distribution. The conjecture is related to problems in functional analysis and combinatorics. I will present partial results and possible lines of attack.
Monday April 03
Alexander Majchrowski (Sydney University)
Mean curvature flow with surgeries and the level set flow
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An intuitive talk on mean curvature flow with surgeries, developed by Huisken and Sinestrari. And the techniques used to show that this flow converges to the well known weak solution of level-set flow studied by Evans and Spruck. This talk is based on the work of John Head.
Monday April 10
Inflammation and PDEs Structured in Cellular Quantities
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Macrophages are cells which rapidly populate inflamed tissue and clear pro-inflammatory material which enables the inflammatory response to resolve.
Funnily, these cells eat their dead and recycle accumulated substances.
When this cannibalistic cell population is expressed as a distribution (using PDEs) across accumulated substances, we observe that a substantial proportion of cells contain an extraordinary amount of material.
This is due to biomagnification where accumulated substances increase in concentration along a chain of cells which have consumed at least one dead cell.
Certain substances are cytotoxic in excess and induce necrotic (bad) cell death which sustains cell recruitment to the site of inflammation.
This creates a positive feedback loop which sustains both cell recruitment and necrosis and hence amplifies inflammation in time.
I will support and explore this theory using a non-local partial integro-differential equation whose steady state can be determined analytically.
I will conclude with the analysis of a non-local partial differential equation which suggests that cell division/proliferation can halt biomagnification and promote inflammation resolution.
Monday April 24
Alexander Kerschl (Sydney University)
Solving polynomial equations with radicals or why there are no general solutions for polynomial equations of degree 5 and higher
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The history of solving polynomial equations dates back to about 2000 BC for which we have written evidence that the old Babylonians already solved quadratic equations. Throughout the centuries people tried to formulize and solve these equations in general. Finally, in Italy during the 16th century scholars discovered the general solutions for cubic and quartic equations but the general quintic could not be solved. Nowadays we know that there is no solution using radicals for the general quintic and higher degree polynomial equations but historically it took until the early 19th century to give a proof for this fact. In 1799 Ruffini and Gauß were the first to formulate that there is no general solution for degree 5 and higher. Following them Cauchy, Wantzel, and especially Abel worked to help to finish Ruffini's first draft of a proof and led to the famous Abel-Ruffini Theorem in 1824. Independently and without knowing about Abel's proof a young Frenchman named Évariste Galois laid the groundwork of what is known today as Galois theory. Galois gave us a beautiful general approach to deal with solvability of polynomial equations of any kind and, moreover, his work led to solve two of the three classical problems of ancient mathematics. Unfortunately, he died way to young at the age of 20 after being severely injured in a duel. My talk will aim to lead throughout the centuries of the quest to solve polynomial equations and explain why there can't be a solution for the general quintic.
Monday May 01
Brent Giggins (Sydney University)
How to Predict the Weather - An introduction to Chaos, Data Assimilation and Ensemble Forecasting
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In 1961, Edward Lorenz discovered that the atmosphere has a finite limit of predictability, even if we have a perfect model of the atmosphere and the initial conditions are known almost perfectly. This was a catalyst for the fields of numerical weather prediction and chaos theory, which is the study of dynamical systems that exhibit sensitive dependence to small perturbations in the initial conditions - often referred to in popular culture as the "butterfly effect". In this talk, we will examine what it means for a dynamical system to be "chaotic" and look at ways to characterise chaotic behaviour both globally and locally. We will look at this through the context of weather and climate forecasting - the main example of chaotic behaviour in natural systems - and summarise the basic components needed for numerical weather prediction. In particular, we will examine the topics of Data Assimilation and Ensemble Forecasting in generating optimal initial conditions for a weather or climate model and consider the practical problems that arise. Finally, we will look multi-scale dynamical systems and illustrate the challenges of forecasting over multiple time and length scales.
Monday May 08
Philip Bos (Sydney University)
Modular Forms and Number Theory - an insight into the Ramanujan conjectures
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The vector space of modular forms will be explained as complex-valued functions on the upper half plane with periodic-like properties. They are in some sense the hyperbolic geometrical equivalent of periodic functions of Euclidean space. We will explain that sense.
As periodic functions on the one hand, we can develop a Fourier series expansion for modular forms.
As a vector space on the other, they permit linear operators and in the 1930 the German mathematician Hecke, developed the so-called Hecke operators. With such a development, we discover that the vector space of Hecke operators form a unitary commutative algebra called the Hecke Algebra.
When we apply these two results together, we can solve significantly difficult analytic number theory questions. The gifted Ramanujan had great insights into such relationships but could not prove all his conjectures. We will give an example of the Ramanujan tau function and outline the method of Hecke that shows the proof as a "natural" consequence of the above ideas.
This is a far as we will go in our talk, though Hecke's student Petersson went further showing Hecke operators are Hermitian with respect to the Petersson inner product, allowing us to derive bases for the vector spaces of modular forms. Continuing in this direction and far further allowed Andrew Wiles to solve Fermat's Last Theorem.
Monday May 15
Tetrahedron centres over a general metrical framework
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In this talk, the three main centres of tetrahedra (centroid, circumcentre and Monge point) will be obtained over an arbitrary symmetric bilinear form. I will also talk about the Euler line (unifying the three main centres of a general tetrahedron) and the twelve-point sphere, as well as discuss the existence of orthocentres in a general tetrahedron.
Monday May 22
Sarah Romanes (Sydney University)
Thinking like a Bayesian - an Introduction to Bayesian Inference
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Almost all of the statistical inference methods learnt at the University of Sydney concern what is referred to as frequentist inference. A major alternative to frequentist inference is Bayesian inference, named after Reverend Thomas Bayes (1701 -1761). Bayesian inference has many advantages over frequentist inference, including (but not limited to) allowing for better accounting of uncertainty, and producing results that are both highly interpretative and intuitive.
However, Bayesian inference is not without its drawbacks. Intractable integrals that appear in Bayesian statistics must be evaluated numerically, and can be quite complex. The computational complexity of Bayesian statistics has been a major obstacle for its application in previous years, however with modern computational power Bayesian approaches to statistical problems are much more feasible and implementable by researchers. In this presentation, I will introduce the basic concepts of Bayesian inference - (including topics such as the posterior, prior choice, and numerical approximations to Bayesian inferences) in a light-hearted presentation accessible to all levels of statistical background.
Monday May 29
Joel Gibson (Sydney University)
A different approach to representations of the symmetric group
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The representation theory of the symmetric group is closely connected to the combinatorics of Young tableaux, however the usual way in which this is realised can seem unnatural. The aim of this talk is to present a more recent approach to the subject by Okounkov and Vershik, which pays close attention to the structure of the nested group algebras \(\mathbb{C}S_{n} \subseteq \mathbb{C}S_{n + 1} \) and the restriction of \(S_{n+1}\)-representations to \(S_n\). By constructing an extremely nice basis of any irreducible representation of \(S_n\), along with a corresponding diagonal subalgebra of \(\mathbb{C}S_n\), we will be able to "do weight theory" to a representation, from which the connection with Young tableaux will emerge naturally.
Monday June 05
Matthew Cassel (Sydney University)
A Multiple Scale Approach to Sunspots
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Very strong magnetic fields exist throughout the universe. Sunspots are one such example. These localised regions of reduced temperature on the solar surface were first observed in the 8th century BC. Spots arise due to complex interactions between temperature, convection and magnetism. Whilst the dynamics have been examined since the 1930s, this interplay is still poorly understood. Examining the quantitative behaviour of the governing equations presents a significant challenge. We discuss a multiple scales approach to deriving a new set of governing equations and show that we can reproduce dynamics consistent with the theory, in turn highlighting the importance of non-linear corrections to the system.
Seminars in 2017, Semester 2
The seminars will be held at 5:00 pm on Mondays in Carslaw Room 535A.
Monday August 7
Becky Armstrong (Sydney University)
Group actions, groupoids, and their C*-algebras
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C*-algebras were first introduced in order to model physical observables in quantum mechanics, but are now studied more abstractly in pure mathematics. Much of the current research of C*-algebraists involves constructing interesting classes of C*-algebras from various mathematical objects---such as groups, groupoids, and directed graphs---and studying their properties. Groupoid C*-algebras were introduced by Renault in 1980, and provide a unifying model for C*-algebras associated to groups, group actions, and graphs. In this talk, I will define topological groupoids and examine Renault's construction of groupoid C*-algebras. I will discuss several examples of groupoids, including group actions and graph groupoids, and will conclude with a brief description of my PhD research.
Monday August 14
Patrick Eades (Sydney University)
An Introduction to Geometric Optimisation Algorithms and Uncertainty
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The vast quantity of low quality data being generated around the world is one of the most important stories in computer science today. Traditional methods typically rely on the input data being correct, and the user hopes the solution is not too sensitive to changes in the input. In this talk I will introduce some classical problems in computational geometry and demonstrate some algorithmic solutions. Afterwards I will generalise the problems to handle uncertain input and show how this necessitates an entirely new approach to finding solutions. I will conclude with some comments about the current state of my own research.
This talk should serve as a light introduction to many topics and so no particular background is assumed.
Monday August 21
Nathan Duignan (Sydney University)
Regularisation for Singular Points of Planar Vector Fields
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Regularisation for vector fields is concerned with the continuation of solutions through singular points. Many of the core ideas and definitions of regularisation were developed to understand the collision singularities of the n-body problem. Although binary collisions are understood via the Levi-Civita regularisation, there are conjectures about the finite differentiability of nearby orbits of simultaneous binary collisions. This presentation will take a relaxed look at a modern, geometrical understanding of regularisation for singular points of planar vector fields. The primary aim is to introduce the key concepts underpinning the conjecture and provide a mechanism that causes orbits near regularisable singular points to become finitely differentiable.
Monday August 28
Adrianne Jenner (Sydney University)
How does pressure within a tumour affect the outcome of cancer treatment?
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Pressure within tumour's has been identified as one of the major culprits impeding cancer treatments. Tumour pressure is known to result in heterogeneous intratumoural distribution of cancer treatments and increase the metastatic potential of tumours. We use a sytem of ODEs to initially optimise in vitro data. We introduce heterogeneous population characteristics into our model through the use of gamma distributions. From this we then build an off-lattice agent based model for the interaction between tumour cells and a cancer treatment. Investigations are undertaken into the optimisation potential of pressure reducing treatments combined with alternate treatment application profiles to discern how to enhance homogeneous treatment diffusion without allowing the formation of metastasis.
Monday September 4
Pantea Pooladvand (Sydney University)
Modelling Diffusion of Anti-Cancer Viruses in Solid Tumours
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One of the biggest barriers in treating solid tumours is the inability of therapeutic vectors to
propagate throughout the tumour mass due to the high density of the tumour and tumour
stroma. The dense nature of many solid tumours can be attributed to an over-expression of
a collection of molecules known as the extracellular matrix (ECM). This thick and compact
structure acts as a physical barrier for many treatments by shielding the malignant cells and
reducing drug penetration and efficacy. One method, used to tackle the lack of diffusion
of therapeutic vectors, is by treating the tumour with an oncolytic adenovirus as they
are incredibly small and should be able to diffuse more efficiently throughout the tumour
mass. However, biologist find that treatment with oncolytic viruses still lacks diffusion and
penetration in solid tumours. In this presentation we aim to model the interaction between
the ECM and the virus population. We ask the question: how does the ECM affect the
diffusion of the virus?
Monday September 11
Bernard Ikhimwin (Sydney University)
Computational model of an initial lymphatic network
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The lymphatic vascular system consists of networks of vessels which play a key role in immune surveillance by transporting lymph and protein from the tissue space back to the circulatory system. In contrast to the cardiovascular system which has a central pump, the lymphatic vascular system has no central pump, hence the transport of fluid against gravity relies on local extrinsic and intrinsic pumping. The initial lymphatics lack smooth muscle cells, hence they cannot perform intrinsic pumping, instead lymph is moved by means of extrinsic pumping, taking advantage of the relative motion of adjacent tissues. We used a lumped-parameter model to describe the transport of lymph through an initial lymphatic network of the mesentery by extrinsic pumping. In this talk we shall look at three different scenarios which are (a) Impermeable initial lymphatic network, (b) permeable initial lymphatic network and (c) tapering-permeable initial lymphatic network. I will discuss the implication of these results.
Monday September 18
Edward Selig (Sydney University)
A Generalisation of the Stochastic Conditional Duration Model
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Durations, in the context of financial econometrics, are defined as the time intervals between consecutive financial events. Ever since the emergence of high-frequency financial data, financial duration modelling has played a key role in analysing the time intervals between trades and understanding the behaviour of market argents operating in global market exchanges like the ASX. The random nature of durations led Engle and Russell (1998) to develop the Autoregressive Conditional (ACD) model which was then extended by Bauwens and Veredas (2004) into the Stochastic Conditional Duration (SCD) model. In this presentation, we present a generalised version of the SCD model called the Generalised Stochastic Conditional Duration (GSCD) model to allow for more flexibility of the latent variable which drives the dynamics of the durations. We look at the model’s properties, devise procedures to estimate its parameters and fit the model onto a dataset containing transaction durations of the IBM stock.
Yossi Bokor (Sydney University)
Resolving Singularities
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I will discuss methods for
resolving singularities of curves in both $\mathbb{C}^2$ and $\mathbb{CP}^2$. Beginning
with $\sigma$-processes, we work towards using Cremona Transformations to ensure that we
can construct a non-singular plane curve in $\mathbb{CP}^2$.
Monday October 9
John Wormell (Sydney University)
A user's guide to chaotic systems
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Chaotic behaviour is common in real-world dynamical systems, but chaotic systems are notoriously "impossible to predict". But nevertheless, what useful things can you say about the (long-term) behaviour of a chaotic system? The answer is largely in and around ergodic theory.
We introduce some statistical properties generally possessed by chaotic systems, sketch how people try to prove these properties exist, and discuss how they can be used to form a useful dynamical picture of a given system.
Monday October 16
Ishraq Uddin (Sydney University)
Cell crowding effects and tissue growth models: an incomplete survey
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This talk presents a brief survey of PDE models for the growth of tissues and cell populations influenced by cell crowding effects. We first discuss how such models may be obtained from agent-based and stochastic models of cell behaviour within limited space by taking an appropriate continuum limit. We also discuss multiphase models, which provide a purely continuous framework for modelling cell behaviour and also open doors to free boundary problems. We motivate our models with reference to specific biological systems including bacterial chemotaxis, neural crest cell colonisation, and atherosclerotic plaque formation. Along the way, we also discuss how the nonlinearities appearing in the diffusion and advection terms for specific models may be interpreted biologically, as well as the difficulties in defining where the boundary is for a cell population or tissue.
Monday October 23
Finite State Automata for Coxeter Groups
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Coxeter groups are abstract reflection groups with a straightforward group presentation consisting of generators and relations. Whilst the presentation is economical, it gives notoriously little information about the nature of its elements. For example, given an arbitrary string of generators, is it possible to know whether the expression is “reduced"? In 1993, members of the University of Sydney's School of Mathematics and Statistics, Brigitte Brink and A. Prof. Bob Howlett produced seminal work in the area of Coxeter groups which precisely answered the above question. They proved: “For every Coxeter group, there exists a finite state automaton which recognises the language of reduced words". Informally, in this context, a finite state automaton is a “machine" capable of “reading" words in the generators and giving an answer of “Yes" if and only if the group element represented by the word cannot be expressed with a shorter string of generators. The aim of this talk is to give a brief introduction to Coxeter groups and describe how we can build a finite state automaton to study Coxeter groups by following the work of Brink and Howlett.
Monday October 30
Giulian Wiggins (Sydney University)
Representation categories and reductive Lie algebras
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We introduce some algebraic toys: Algebras with a partition of unity (APoU) and their representations. Given such an algebra \(A\), we construct a category, CA, in which the category of representations of \(A\) is equivalent to the category of linear functors from CA to the category of vector spaces over the ground field.
As an application we take a reductive Lie algebra \(g\), and construct an APoU, \(U*g\) (Lusztigs idempotent form of \(g\)), whose representations are all the integral representations of \(g\). Then applying the above theory, we are able to take a \( (g,A) \)-bimodule \(P\) ( \(A\) is any algebra) satisfying certain conditions, and derive a presentation for the full subcategory of \(A\) whose objects are direct sums of the weight spaces of \(P\). If \(P\) contains a copy of every irreducible \(A\)-module then the Karoubi completion of this category is the whole category of representations of \(A\). As an example, we give a presentation of the category of permutation modules of \(S_n\) and discuss how a presentation of the category of representations of \(S_n\) may be obtained from this.
This talk is accessible to anyone with a basic knowledge of representations of finite dimensional algebras, and of the definitions of category and functor.