*Alycia Winter was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.*

I have always been passionate about mathematics. This is due to how you can find it anywhere. You use it when you shop or when you cook. It is also hidden in less obvious things like music or a shell of a snail. One of my favourite uses for maths is magic.

Magic is an ancient past time, which has been mesmerising people around the world for centuries. Magic comes in all shapes and forms, all of which are based on illusion and sleight of hand, meant to deceive an audience into believing the impossible. Behind all these tricks there are simple explanations, which allow the magician to captivate others. Mathematics is the basis behind a range of tricks, both the simple and the more involved.

One type of trick that relies of maths is card tricks. The Perfect Shuffle is commonly utilized by magicians to perform a range of magic tricks which relies on the mathematical concept of arrangement to have its unique cyclic property. The Reed-Solomon code magic trick which relies on field theory. Card algebra is a small trick that relies on basic algebra to “guess” an audience members card.

Another card trick, or more so an idea which is utilized in tricks is the Gilbreath Principle which is a card theorem. This card trick uses a deck of cards arranged in alternating colours. The deck is then cut in two so that the bottom card of each deck is a different colour. These halves are then riffle shuffled together. Pairs of cards are taken from the top of this deck and it is seen that every pair consists of a black and red card.

This trick can be shown to work with inductive proof. Assume that the first card to fall, in the riffle shuffle, is a black card. If the next card to fall is from the same half then it will be red. Similarly if it is from the other deck, the card will also be red. Thus there will be a red black pair. This leaves the cards in the same initial condition, with opposite colours sitting on the base of each half. This allows the same logic to able to be applied to each dropping of cards.

A similar result can be gained with the four suits in a deck, as the Gilbreath Principle generalises. It could even be extended to a group of five cards or even a double deck using a similar method to the original.

My interest in optimisation first started when I was still in high school. The idea of being able to maximise or minimise anything seemed almost fanatical to me. Later I applied for the joint degree in Mathematical Science & Education at Federation University with the hopes of achieving two goals. The first goal was to deepen my knowledge in mathematics, the second is to successfully attain a career that will let me travel and share my knowledge of mathematics with the world. My first goal was achieved.

After coming to the belief that my knowledge in mathematics had reached the pinnacle, I was attracted to the research scholarship provided by AMSI. This scholarship was an eye opener, I discovered that I knew very little of mathematics. The papers written by my professor seemed almost incomprehensible to me; it almost felt as if I was staring at a foreign language. So I decided to start all over again. I revised everything I knew of calculus and then dedicated some of my time studying functional analysis and convex analysis. I had learnt of the many different spaces such as the *Banach*, *normed *and *metric *spaces etc. After my personal study was complete I felt I was ready to tackle the alien language of mathematics again.

Soon after I started learning and comprehending what a subdifferential was, how to calculate these subdifferentials and the final task, creating the illustrations, the presentation for AMSI was due. It was quite nerve wracking, but it was a wonderful and enlightening experience. I met many other students and professors and made many friends. I was shocked at the level of mathematics other students displayed and was deeply impressed; my urge to conquer mathematics grew.

I am glad I took on this research project, although it was very frustrating at times. It was a beneficial learning experiences that I will never forget.

In primary school I remember taking nets of different shapes likes cubes and pyramids, and gluing their sides together to get the three-dimensional shape. But it was only in the project I undertook this summer when I really thought about how the shape behaved at its edges and vertices. For example, the sum of the angles around the apex of a square pyramid, like the ones at Giza, isn’t 360 degrees like on a flat piece of paper, but 240 degrees (the 60 degree angles of the equilateral triangle multiplied by 4). If you tried to “flatten” the shape around this vertex, your shape becomes distorted. This is an example of a shape with positive curvature. On the other hand, if the sum of the angles were 360 degrees, then flattening the shape at the vertex would not distort the shape at all.

By analogy, if the sum of the angles around some vertex happened to be more than 360 degrees, then your shape would have negative curvature around that point; try visualising that in three dimensions!

In my project this summer, I looked at the corresponding case one dimension up. I looked at tetrahedra, which are like the pyramids but where the base is also a triangle, and examined how gluing them in different ways influenced its curvature properties. Here, your nets are three-dimensional tetrahedra glued together at faces. This gluing produces a shape in four-dimensional space, just as gluing together squares in a specific way at the edges produces a cube in three-dimensional space. The result is that you end up with shapes which do not necessarily fit in three-dimensional space. For example, imagine gluing one of those triangles of a Giza pyramid, to a triangle next to it, of the same pyramid. This would be like “folding in” one of the faces into the one next to it, which would considerably distort the shape of the pyramid. Then if you go around the edge shared by those two faces, you would only need to travel the *dihedral angle* between the two faces, which is considerably less than 360 degrees, and the shape would be positively curved around this edge. The curve produced by going around an edge in this way is called an *edge link*, which is comprised of pieces of circles.

This process can also be done with vertices, producing pieces of spheres glued together in what is called the *vertex link*. If you can show that there is no length-minimising loop, called a *geodesic loop*, with length less than 360 degrees in either the vertex links or the edge links, then you know that the original shape is nonpositively curved. Nonpositive curvature is an extremely important property spanning areas in physics (general relativity) and chemistry (knot theory of molecules), as well as mathematics.

What can and can’t a computer do This question is central to the area of Mathematics now referred to as computability theory. In the early 1900’s, there was interest in not only finding solutions to specific questions in mathematics, but also in finding algorithms which can solve these problems for us. The first step towards achieving this is to first answer the question “what is an algorithm”?

Godel, Church, and Turing all came up with different looking, but logically equivalent definitions of exploring what kinds of problems can be solved by following an algorithm, and what kinds of problems can’t. Perhaps surprisingly, there are some general problems for which there exists no algorithmic solution. For example, it has been proven that there exists no algorithm which takes as input a mathematical statement, and returns “true” if the statement is true, and “false” if the statement is false. Note: the phrase “Mathematical statement” here means a statement of 1^{st} order logic.

For this project, we focused on Church’s approach, which is that of the lambda calculus. Essentially, lambda calculus is a formal language used to describe functions. According to Church, an algorithm is a list of instructions to follow, where the tasks in the list can be described and executed in this language.

There has been a huge rush of developments in computer science over the past fifty years, and it is natural for one to wonder what the effects this explosion has had in the world of Mathematics. Alongside this muse, one might also consider the fact that it is frequently mentioned how hand in hand Pure Mathematics goes with Computer Science. This is an example of a somewhat strange phenomenon where the strength of such a connection seems so obvious, that one can’t help but to feel almost embarrassed when the inevitable realisation that explicit relationships are actually far from common knowledge comes to dawn. Alas, such relationships do exist, and they extend beyond the monogamy of Mathematics and Computer Science, indeed certain areas of Logic also become helpful along the way. We wish to explore such connections, whilst coming to grasp with the foundations of these topics outside of Mathematics, as this content is rarely introduced at undergraduate level, at least as far as the Mathematics department is concerned.

In the third year of my mathematics degree, our lecturer introduced the class to an area of statistics called Bayesian statistics. Up until that point, I had a little idea about this branch of statistics as we were mostly taught about classical statistics throughout our undergraduate degree.

I was quickly drawn to the topic as our lecturer went on explaining some of the cool applications of Bayesian statistics. I learnt that in the Second World War, Bayesian statistics was used to break the enigma code and find German U-boats; in the Cold War, Bayesian statistics helped find a missing H-bomb and the U.S. and Soviet submarines; Bayesian methods were also applied to find a missing Air France plane in 2011.

At its heart, Bayesian statistics is about how to use new information to update our initial belief about something to get an improved belief. To me, this makes intuitive sense as in our own lives, we all learn from experience. From time to time, our opinions about people or events are either confirmed or adjusted whenever we have new information.

I did some further reading after that lecture and was amazed as the list of impressive applications of Bayesian statistics kept growing. Many of those applications intertwined with several significant historical events. On top of that, Bayesian statistics tells a fascinating story in its own history. It was discovered, dead, buried and rediscovered at various time points since the 18th century. Today, Bayesian statistical methods are widely used and highly praised by many practitioners.

I was obviously very attracted to the field of Bayesian statistics so I could not have been more excited when I received the opportunity to do a summer research in this area. My research looked at two application software packages that apply Bayesian statistical methods into analyzing data. With the help of my supervisor, Dr Christopher Drovandi, I looked at the algorithms underlying these tools and compared their performances in different situations. This research has brought me a step closer to what I really want to do, which is making great research ideas more easily accessible to the wider audience in the real world.

Not only did I learn so much more about an area I’m interested in, I gained so much joy through my summer research and became more aspiring to pursue further in this awesome area called Bayesian statistics!

**Reference:**

McGrayne, S.B. 2011, The theory that would not die: how Bayes’ rule cracked the enigma code, hunted down Russian submarines, & emerged triumphant from two centuries of controversy, Yale University Press, New Haven [Conn.].

Ever since I can remember, I’ve always been fascinated with the idea that mathematics is behind everything. Whether it was seeing the golden spiral everywhere, noticing nifty patterns and sequences popping up in places where you don’t expect them, or thinking about the sheer number of outcomes from seemingly small objects, like the permutations of a Rubik’s cube.

Growing up thinking about these things led me to a path of studying applied mathematics, with goals of one day devising my own mathematical links between properties that seem to correlate; an example being my honours thesis this year, which will involve establishing a link between the internal heat of a clump of maggots within a decomposing corpse and the mass of the clump. The field work that will be involved in this project is one part of mathematics that is exciting to me, that I will actually get to apply mathematics to real world results as opposed to just theorising about it.

One of my biggest influences for wanting to study applied mathematics is the website xkcd.com. The website initially started as a webcomic and the author Randall Munroe is a physicist who used to work for NASA. A few years ago he started a section on his website called ‘what if?’ in which he answered hypothetical questions posed by his fans which include topics like ‘What if there was a robot apocalypse? How long would humanity last?’ and ‘What would happen if a hair dryer with continuous power was turned on and put in an airtight 1x1x1 meter box?.’ All of these questions were answered mathematically, with careful research and considerations.

Now if he is able to answer these hypothetical situations with mathematics and models, what is to stop me from being able to model nearly anything in real life? The hypotheticals are amusing, though I’d much prefer to create and tweak models that are useful to humanity as a whole for example, my honours thesis will hopefully be a stepping stone to being able to better estimate the time at which a person has died.

I am thankful to AMSI and the University of Wollongong for giving me this opportunity to be introduced to the wonderful world of research this summer. The methods I have learned will prove invaluable to me as my career in mathematics continues.

This summer I looked at investigating a particular second order differential equation called a Hill equation. The study of differential equations is the study of change. There is a “rule”, the differential equation, which governs how a quantity will change. Our job is simply to let time run and observe how this quantity evolves.

The Hill equation is a simple looking thing: an operator consisting of the second derivative summed with a periodic function acts on a variable to produce that variable multiplied by a constant. The exploration of this equation however has revealed many intricate patterns and curious behaviours.

In general for any random periodic function we can’t write out an explicit solution to the differential equation. To understand the future behaviour of these variables therefore it is not enough to simply calculate their function and evaluate it at a later time. Indeed what is required is that we tease out information about the solution.

Aiding us in our endeavours is Floquet theory which simplifies the study of these periodic differential equations. It allows us to consider only the evolution of the system through a period; intuitively this makes sense, given that the “rule” which governs the change of the variable repeats itself we need only look at one repetition to begin to understand the variable.

We are interested in solutions to the Hill equation which do not go to infinity as time goes by. Applying the idea of Floquet theory we are interested in situations where once the variable goes through repetition its size does not change: any change either negative or positive, however small will result in the variable reaching infinity in either positive or negative time. For certain values of the constant within the differential equation, these solutions appear. We call the constant and eigenvalue in this case.

Thus far the study of Hill equations has been limited to real periodic functions. This is very neat since the operator is “self-adjoint” which means that the eigenvalues of this system are only real. Thus the subject of Hill equation is relatively simple: real periodic functions, real eigenvalues.

This project looks to extend the theory of Hill equation to complex periodic functions. Now the eigenvalues are no longer necessarily real and where they appear in the complex plane is hard to pin down. While it is just as hard to exactly find where these eigenvalues appear as it is to explicitly solve these equations, we’d like to narrow down and establish some patterns for where these solutions appear.

The answers to these questions have implications beyond just this differential equation. There is a connection between the eigenvalues of these equations and the behaviour of fluids. This simple second order differential equation is a rabbit-hole to a world of curious behaviour and intricate patterns. It’s been a pleasure trying to unpack this puzzle.

To me, the decision to do postgraduate study in biostatistics was a long, unnoticed process. It began in the second semester of my first year. I had returned to University, hell-bent on study genetics and make an impact in cancer research, when in my first lecture towards this goal, my lecturer stood in front of the whole class and professed, “If you’re here because you don’t want to do math, you’re not going to do well”. He then went on to elaborate that genetics research had advanced to a point where basic statistics knowledge was a given and commonly, complex analysis was required in order to produce significant studies. Begrudgingly, I took my lecturers advice and made a late enrolment into a first year statistics subject. The subject was focused on how statistics can be related to problems in life sciences, and to my surprise, I loved every minute of it! For the first time in my education, I was given tangible evidence math was used to solve real-life problems that interested me – not just how to determine how tall a tree is if it casts a shadow x meters long.

Following my new found interest, I began enrolling in more statistics subjects at second-year level and eventually I had to make the decision, did I want to major in genetics or statistics? By the end of my second-year, the decision was a no-brainer, Statistics!

Seeing how statisticians can turn a seemingly confusing mass of numbers into something profound and meaningful really hit it home for me. They are todays modern magicians, speaking a language unheard of and incomprehensible to the average ear. They maintain the ability to make sense of the ever increasing data being produced by todays society and translate it, so that it can be used for good, whether that be redirecting resources, determining the cause of disease, implementing useful policies or evaluating the efficacy of new drugs. It truly is a side of math that be used in any industry and is an invaluable skill set to hold and I’m excited to see where my new path takes me.

Maths isn’t something I ever set out to study at university, it’s more something that I’ve fallen into. Growing up I liked how pure it was as a subject, how you could start with a couple of really simple facts and then derive everything you need to know from there. I especially enjoyed applying maths to solving real world problems, as in physics and chemistry.

When it came time to choose a university degree I was planning on going with engineering. In my country Victorian hometown this was the main career path for anyone good at the maths/physics/chemistry trio. It seemed like a safe option and everyone around me was telling me it was good idea.

But I wasn’t sold on it, and I felt like I needed to explore my options some more. So I did what any good Millennial would do, and I trawled the internet for advice. I spent hours searching through forums, looking what people with similar interests to me were studying. Eventually I came to a post talking about ‘computer science’. This was this first time I’d even heard that this course existed, but it sounded like the sort of thing I’d enjoy. The post mentioned how it was pretty maths intensive, so doing it together with a few extra maths units would be beneficial.

So I took the advice of an anonymous stranger on the internet and I enrolled in a Mathematics/Computer Science double degree. At the time I didn’t really even understand what computer science was, but I somehow managed to convince my mum that it was a good idea. I figured that if I hated it, I could always transfer into engineering after a semester.

As it turns out it was probably one of the best decisions I’ve ever made. Maths teaches you about really powerful concepts and computer science gives you the ability to create something real from them. I’d really recommend the computer science/maths combo to anybody interested in the applied side of maths.

My Research Project was on the Survival Data Analysis for Improving Patient Outcomes. For a patient to undergo elective joint replacement surgery (JRS) clinicians must decide that coexisting chronic conditions are stable and impose minimal surgical risks. However the increased risk of patient outcomes after JRS due to comorbidity and multi-morbidity is not always addressed (multi-morbidity is the presence of additional diseases or disorders co-occurring with a primary disease or disorder). We now know that patients presenting with JRS are likely to have multi-morbidities. So the management of patient care for patients with multi-morbidities needs to be improved.

The aim of the study is to develop a statistical model, to predict post-operative patient pain and function using individual (pre-operative) patient multi-morbidity data. The ultimate purpose of the statistical analysis is to provide clinicians with a diagnostic tool for assessing the potential benefits of knee and hip replacement surgery for patients with multi-morbidities.

Through doing this research I was able to learn about the nature of receiving data from medical services. Not only is the paper work lengthy but so is the clean up. I realised not only how hard it is to collate and model the data in a way that produces meaningful results but also how important this is. With minimal understanding of the data and of the methods of analysis that you are using the results can easily prove misleading. Often I would find myself with an unintuitive result that was more a reflection of the limitations of my data set than any form of breakthrough. On the other hand understanding the data set and survival analysis allowed me to make well-supported conclusions even if they are built on gapped data. What I really learnt here is the importance of people. The importance of human minds to understand where our results is just as crucial as the results itself.

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