# Mathematics Postgraduate Seminar Series

To be added to or removed from the mailing list, or for any other information, please contact Alex Casella, Adrianne Jenner, or Dominic Tate.

## Seminars in 2017, Semester 2

The seminars will be held at 5:00 pm on Mondays in Carslaw Room 535A.

Monday July 31

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Monday August 7

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Monday August 14

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Monday August 21

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Monday August 28

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Monday September 4

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Monday September 11

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Monday September 18

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Monday October 9

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Monday October 16

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Monday October 23

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Monday October 30

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## 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

Hugh Ford (Sydney University)

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 formulise 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

Gennady Notowidigdo (UNSW)

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.