# Mathematics

## Higher-dimensional expanders

Expander graphs are highly connected finite sparse graphs which are of importance in number theory, group theory, geometry and combinatorics, with significant applications in computer science as basic building blocks for network constructions, error correcting codes and algorithms. In recent years there has been strong interest in defining and constructing higher dimensional expander graphs, one such construction arises as a finite quotient of a Bruhat-Tits building in analogy to the spectrally optimal Ramunujan expander graph of Margulis and Lubotzky, Phillips and Sarnak.

## Combinatorics of simple games

### Supervisor

Prof. Arkadii Slinko

### Discipline

Mathematics

### Project code: SCI131

We will be looking at compositions of simple games and their properties. In particular, we will investigate when the composition of two weighted simple games is weighted. 326 is desirable but not compulsory.

## How to turn a pea into the sun by playing ping-pong.

In 1924 Banach and Tarski show that given a solid ball in 3 dimensions there exists a decomposition of the ball into a finite number of disjoint subsets, which can be reassembled in a different way to yield two identical copies of the original ball. A key component in the proof is the ping-pong lemma, a technique first used by Felix Klein in the 19th century and used by Jacques Tits in the 1970s to prove the celebrated Tits alternative.

## Condorcet domains

Condorcet domains are sets of linear orders such that if a society restricts themselves to choosing from these they will never run in the paradox of intransitivity of the majority voting. Condorcet domains appear to be interesting combinatorial objects. In particular, they are closely related to median graphs.

## Equilibria in a circular market

A number of firms compete by choosing a location in a circular market, e.g., ice cream shops around a lake. We look to understand the equilibria of such a game.

## Dynamical systems analysis of a Braitenburg vehicle

A Braitenburg vehicle is a simple robot equipped with sensors that are directly connected to motors or actuators. Although no signal processing takes place, depending on the connections between sensors and actuators, suprisingly complex behaviour can be observed. This project will conduct a dynamical systems analysis of such a system as parameters are varied. Prerequisites: Maths 260.

## Heteroclinic networks in partial differential equations

A heteroclinic network is a special type of solution to a dynamical system that consists of a set of states connected by trajectories. This project would involve investigating the existence of heteroclinic networks in a system of coupled partial differential equations. This work has potential applications to understanding turbulence in fluid flow through a pipe. Prerequisites: Maths 260, Maths 361.

## Non-routine Problem Solving: Investigating the Impact on Creativity and Engagement of STEM Tertiary Students

This project aims to investigate the creative thinking skills and engagement of STEM (science, technology, engineering, mathematics) students as a result of solving non-routine problems during their learning. The participants comprise five groups of students from four diverse tertiary institutions who are studying different STEM subjects. Their learning will be enhanced by the addition of non-routine problem solving activities in semester 2, 2018. Learners’ creativity, engagement and intuition will be analysed to evaluate the effect of this innovative practice. We anticipate that wide implementation of this learning enhancement would improve employability of STEM students since innovative and creative thinking is a workplace requirement.

The main research question of this project is:

Does the use of non-routine problems enhance participants’ engagement and learning? Specifically, is the integration of non-routine problems and Puzzle-Based Learning associated with changes in participants’ engagement (emotional, cognitive, and behavioural) in lecture and/or their ability to inhibit intuitive thinking and exhibit creative thinking?

This project will give you an opportunity to be part of a major research project that was recently selected and funded by the Teaching and Learning Research Initiative (TLRI) – a government fund established to enhance the links between educational research and teaching practices to improve outcomes for learners.

This project is suitable for all students with a solid mathematical background and an interest in educational issues. Students who completed MATHS202, MATHS302, and/or have some teaching/tutoring experience are encouraged to apply.

## Professional Development of Mathematics Teachers

This project builds on a number of key ideas that emerged from the Discussion Group ‘Videos in Teacher Professional Development’ at the 13th International Congress on Mathematical Education (ICME13), July 2016, Hamburg, linking experiences from a number of professional development projects internationally. The aim of the project is to identify and develop effective ways to provide professional development to mathematics teachers at university and high-school level in New Zealand.

This project will give you an opportunity to contribute as a student-partner providing valuable insights from your perspective.

This project is suitable for all students with a solid mathematical background and an interest in educational issues. Students who completed MATHS202, MATHS302, and/or have some teaching/tutoring experience are encouraged to apply.

## How We Make Sense of Mathematics

Sense-making – the process through which we work to understand novel, unexpected, or confusing ideas – has become a critically important topic in the study of mathematics learning. In this project, we review and integrate existing theory and research in mathematics education and use insights gained from cognitive science and philosophy to unpack the sense-making process in mathematics.

The scholarship student will gain access to data on students’ various sense-making strategies that are to be explored and investigated from novel perspectives articulated in mathematics education, cognitive science, and the philosophy of mind.

Particularly students interested in taking an inter-disciplinary approach to mathematics learning are encouraged to apply.

## What do mathematics teachers notice and how do they talk about it?

This project explores what mathematics teachers pay attention to in complex learning-teaching environments, how they interpret what they attend to, and to what extent this informs their decision-making in the classroom. The Summer Research Scholar will learn to critically review literature in a research project and conduct fine-grained analyses of teachers' responses. The ideal candidate has some background understanding in mathematics education (or learning science), is literature, and hopes to do a doctorate in mathematics education.

## Can undergraduates convert $100 to a foreign currency?

Why do we need to study math? This is one of the common questions that students raise all over the world. One of the common answers that teachers give, touches upon the usefulness of mathematics in everyday-life situations.

The scholarship student will check the claimed useful in the case of Stage-I students. Specifically, after 13 years of mathematics studies, more than 100 students were asked to put their arithmetical knowledge in use for converting $100 to a foreign currency. The scholarship student will evaluate students’ success, analyse the methods that they used, and identify common misconceptions.

This project is intended for all students with a solid mathematical background and a genuine interest in educational issues. A background in statistics is an advantage. Students who completed MATHS202, MATHS303, and/or have some teaching experience are encouraged to apply.

## The mistakes that students make as a window into their mathematical thinking

In the daily teaching-and-learning reality, we treat mistakes as something to be avoided. If you think about assessment, for example, it often acts as an institutionalized punishment for those who struggle with the subject. From the perspective of mathematics education research, however, these struggles provide a window into silent mechanisms of students’ thinking. Hence, the enhanced interest in the errors that students make.

The scholarship student will gain access to data where Stage-I students were encouraged to articulate some of their mathematical difficulties and mistakes. The scholarship student will categorize these into misconceptions and analyse possible ways to overcome them.

This project is intended for all students with a solid mathematical background and a genuine interest in educational issues. Students who completed MATHS202, MATHS303, and/or have some teaching experience are encouraged to apply.

## Understanding neuron bursting patterns from a mathematical perspective

Neurons may exhibit different types of bursting patterns. These patterns are characterized by intervals of fast spiking in their action potential (voltage) separated by rest periods. Of particular interest are two bursting patterns that appear to be very similar in the observed experimental data but are different in terms of the mathematical mechanism by which they are generated; these are so-called square-wave and pseudo-plateau bursting. (See the figure below. Can you spot the difference between the two bursting patterns?)

In this project, you will learn about bifurcation theory and the use of numerical techniques for dynamical systems. These tools will help you understand the mathematical mechanisms for generating the two shown patterns of bursting and investigate the transition between the two through a hybrid-type bursting.

Prerequisites: Grade A in Maths 260. No biological knowledge is required.

## Quotients of finite groups by isomorphic normal subgroups

Given a finite group and two isomorphic normal subgroups, the respective quotients are not necessarily isomorphic. On the other hand, they have some properties in common, for example, their order. What other properties must they share? This is what we would like to investigate. (While not obvious, this question has some connections with the concept of local action in arc-transitive digraphs.)

MATHS320 is a prerequisite. Willingness to learn how to use a computer algebra system would also be helpful.

## Coupling the northern and southern hemispheres in conceptual climate models

The classical Budyko energy balance model describes the evolution of Earth’s latitudinal temperature

distribution via a partial differential equation. This equation depends

on the incoming solar radiation, outgoing long wave radiation (which in turn

depends on greenhouse gases), and latitudinal heat transport. A typical

simplification of the model is to assume that Earth is symmetric across the

equator, with ice caps covering each pole.

Moreover, the solar forcing incorporated into the model is typically

taken from Northern hemisphere measurements. The goal of this project is

to forego these simplifications and couple two energy balance models, one for

each hemisphere, taking into account the different parameter values and

forcings for the north and south. This

project will involve numerical simulations of the equations in MATLAB, and so

some familiarity with MATLAB is expected. Coursework in ordinary

differential equations is a prerequisite (Maths 260 or equivalent).

## Three-vortex collapse in geophysical flows

The so-called Euler point-vortex equations describe the motion of N vortices interacting in a planar, inviscid and incompressible fluid. If N=3, certain conditions can lead to a collision of the vortices. It has been long known that for these equations, such collisions can only happen in a self-similar fashion, i.e. the triangle formed by the three vortices maintains its shape as the vortices approach collapse. Recently, it has been suggested that non-self-similar collapse can occur in a separate set of physically relevant equations. The goal of this project is to explore three-vortex collapse in other sets of point vortex equations. Background in differential equations (Maths 260) and computational software such as MATLAB is required for this project.

## Derived subgroups of nilpotent subgroups of symplectic groups over the integers.

Nilpotent groups play a basic role in the study of group theory.

Many important examples are obtained from nilpotent subgroups of a classical group.

The structure of derived series is fundamental in the understanding

of nilpotent of a nilpotent group. The project is to study the structure of the

first term of the derived series of nilpotent subgroups of a symplectic group over the integers.

Prerequisite: Maths 320

## External symmetries of regular hypermaps

A regular hypermap is a discrete structure with lots of symmetry. It’s a bit like a map on a surface (made up of vertices, edges and faces), and interesting examples can be constructed from groups. This project involves finding out about regular hypermaps, and constructing examples with ‘extra’ symmetries that permute the roles of the hyper-vertices, hyper-edges and hyper-faces. Knowledge of group theory (e.g. from Maths 320 or 328) is essential, and use of the MAGMA computer system would be helpful.

## Regular actions of alternating and symmetric groups

A regular permutation group is a transitive group in which only the identity element has fixed points. This project involves finding out some specific things about regular groups of degree n! that are isomorphic to the alternating group An or symmetric group Sn, to help answer an open question posed recently by Pierre-Emmanuel Caprace. Knowledge of group theory (e.g. from Maths 320 or 328) is essential, and use of the MAGMA computer system would be helpful.

## Computing tori with Chebfun

"Chebfun” is an extension to Matlab that allows for arithmetic operations on functions, performed to machine precision. Rather than working with a discretisation of the domain of the function, Chebfun will generate a polynomial approximation of the function and perform the arithmetic in a truly functional sense. This project will give you the opportunity to learn Chebfun and to develop a new application of this tool to the computation of closed curves or tori that are fixed `points’ of a two-dimensional iterative system.

Prerequisites: Maths 162, Maths 250 and good Matlab coding skills

## Fingerprints of wild chaos

Wild chaos is a new form of unpredictable behaviour that can occur in higher-dimensional dynamical systems. Only very few examples are known and it is as yet entirely unclear how one could measure and identify such behaviour in experiments. This project considers data that is known to come from a wild chaotic attractor; your task will be to characterise it so that it can be distinguished from other kinds of behaviour, in particular, from ordinary chaotic behaviour.

Prerequisites: Maths 260 and good coding skills, e.g., in Matlab.

## K-theory and Index Theory

K-theory is an invariant that was introduced by Sir Michael Atiyah and Friedrich Hirzebruch in 1959 and bears all the hallmarks of a great mathematical structure.

It draws upon and relates several areas of mathematics, and allows for various district realisations. It is the natural habitat for the Atiyah-Singer index theory, one of the greatest mathematical achievements of the 20th century.

Over the past two decades, K-theory has also found a prominent place in physics, particularly in the classification of topological insulators in condensed matter physics and in the study of chiral anomalies in particle physics and D-brane charges in string theory.

The aim of this project is to learn about K-theory and explore some of its applications

## Loop Groups

Loop groups are an important class of groups that are ubiquitous in mathematics.

They have a beautiful structure theory and have deep connections to number theory (modular forms) and theoretical physics (gauge theory and string theory).

This project is focused on learning about the properties of loop groups and some of their applications.

## Unknotting knots

How do you know if two knots are the same? One way to approach this problem is to associate a number or a polynomial to knots. A famous example is the Jones polynomial.

If these so called “knot invariants” are different, then the knots have to be distinct.

The aim of this project is to learn how to describe knots mathematically and how to compute their invariants.

## Finding equiangular lines and their (projective) symmetries

There are three equiangular lines in R^2 (there cannot be more), and there are four equiangular lines in C^2 (there cannot be more).

There is a conjecture that there are d^2 equiangular lines in d-dimensional complex space (these lines have been constructed numerically to high precision as an orbit of the Heisenberg group).

Recently all sets of up to twelve real equiangular lines have been constructed by finding the classes of switching equivalent graphs on twelve or less vertices. The aim of this project is to analyse all of the "interesting" sets of equiangular lines that occur in this way.

The idea is to recognise them as the orbit of a single line under a projective unitary action of a finite group. A recently develop Fourier transform for projective unitary actions of finite groups offers a neat way to do this.

Pre-requisites: Linear algebra and Maths 320.

## Useful redundancies - finite tight frames

In the game of ``battleships'' the position of a ship is given by two coordinates. These Cartesian coordinates appeal to mathematicians because they use the minimum amount of information. It is possible to give three coordinates for a ship with the property that if one is lost, or changed, then the position is still known precisely.

Such representations are called finite tight frames, and are increasingly used in applications precisely because of this useful redundancy.

The project will investigate an aspect of finite tight frames appropriate for the students background.

Pre-requisites: Maths 255

## Coupled oscillations in mathematical biology

Oscillations in the concentration of certain types of chemical are known to be an important mechanism for cellular signalling. These types of oscillations often result from the interaction of biological processes occurring on different timescales, and this gives the oscillations some special features. This project will use numerical and theoretical methods to investigate the complex dynamics that occurs when two or more oscillators with multiple timescales are coupled.

Prerequisites for this project are an A pass or better in Maths 260 and a good grade in Maths 250.