- » Hotelling Games on Rectangular Grids
- » Secret Sharing Schemes
- » Finding equiangular lines and their (projective) symmetries
- » Useful redundancies - finite tight frames
- » Earth Impactors
- » Group actions and Quasi-isometries
- » Octonions
- » Number theory and cryptography
- » Climate models for Snowball Earth
- » Asymmetric vortex patterns
- » Commutator subgroups of some symplectic nilpotent subgroups.
- » Understanding coupled calcium oscillations
- » Hamiltonicity of Cayley graphs
- » The Writing Mathematician
- » The mathematical experience of struggle
- » Polynomial convexity
- » Orthogonal polynomials
- » Giftedness and talent in mathematics
- » Unpacking mathematical conventions
- » Determinant Tic-Tac-Toe
- » Fingerprints of Wild Chaos
- » Computing tori with Chebfun
- » Balancing El Niños and La Niñas.
- » Grobner bases for matrix equations
- » Modelling and analysis of clustered ventilation in the lung
- » Classes of subgroups of the general linear group
- » Presentations for finite simple groups
- » The Allee effect in a model of predator-prey interaction

## Hotelling Games on Rectangular Grids

We will consider a Hotelling game where a finite number of retailers choose a location on a grid, given that their potential consumers are distributed uniformly on this grid (Imagine a city which streets form a rectangular grid.) Like in the classical Hotelling’s Main Street model retailers do not compete on price but only on location, given that each consumer shops at the closest store with greater probability but sometimes with smaller probability shops at more distant shops too. We will look for conditions under which a pure Nash equilibrium of this game exists.

Prerequisites: A in MATHS 250

## Secret Sharing Schemes

In an era of Fancy Bear and Cozy Bear, security of sensitive information is a major concern for governments and corporations. One of the answers to this challenge are secret sharing schemes introduced by Shamir (1979) which store secret information not on a single server but on a variety of them so that only from an authorised coalition of servers the secret can be learnt. Originally, secret sharing schemes were introduced for secure storage of cryptographic keys, missile launch codes and numbered bank accounts, but gradually have found numerous other applications, e.g., in attribute-based encryption, e-voting, and secure multiparty computation.

The set of all authorised coalitions of a secret sharing scheme is known as the access structure to the secret. An important challenge of the theory of secret sharing is to characterise those access structures that can carry an ideal (most informationally efficient and secure) secret sharing scheme. We will be looking at some aspects of this problem.

Prerequisites: A in MATHS 250

## 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).

The aim of this project is to construct and analyse complex spherical t-designs for t>2 (spherical 2-designs are equiangular lines).

These are "equally spaced" configurations of points on the complex sphere.

Ideally, one would obtain an explicit construction of an unknown t-design as the orbit of one or more vectors.

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

## Earth Impactors

Every once and a while, Earth is hit by an asteroid. In this project the student will use state-of-the-art software to perform computer simulations of different groups of asteroids. The aim of these simulations will be to estimate the probability that asteroids from each group will hit Earth.

## Group actions and Quasi-isometries

The project concerns the study of group actions on metric and topological spaces. You will learn how to construct a group presentation for an arbitrary group Γ acting by homeomorphisms on a simply connected space *X*. If *X *is a simply connected length space and Γ is acting properly and cocompactly by isometries, then this construction gives a finite presentation for Γ.

To gain greater understanding in the relationship between a length space and any group which acts properly and cocompactly by isometries on it, we will consider the group itself as a metric object, and we will introduce an equivalence relation called quasi-isometry which equates space that look the same on the large scale.

The goal is that at the end of the project you fully understand what you have just read, you will reach this understanding by a mixture of studying the literature, testing out ideas on particular examples and solving exercises. According to your interests more emphasis can be put on particular topics along the way.

Ideally you would have followed MATHS320 and MATHS332, but this is not strictly necessary.

## Octonions

You may recall being puzzled when you were told that -1 had a square root after all. In the meantime complex numbers are probably second nature to you. The question arises if there are more such intriguing algebraic structures.

Legend goes that Hamilton discovered the quaternions in a flash of genius and carved them into a Dublin bridge in 1843.

While these may seem obscure at first, as e.g. multiplication is no longer commutative, they turn out to be very useful to describe rotations in 3-dimensional space.

Going even further the octonions, which are at the heart of this project, are of vital importance for theoretical physics and turn out to be intimately linked to Lie groups and Lie algebras, very central objects in mathematics connecting algebra and analysis.

You will study these fascinating objects by a blend of literature study and solving exercises, emphasis can be put on a subtopic of your choice as we proceed.

Ideally you would have followed MATHS320, but this is not strictly necessary.

## Number theory and cryptography

Modern public key cryptosystems are based on computational problems in number theory and algebra. The project will be to study some of these computational problems, the mathematics behind them, and the algorithms that are used to solve them. This project is most appropriate for a student who has taken MATHS320 and/or MATHS328, and is interested in computer experimentation. However there are a range of research projects within this theme that can be customised for the individual student and their background and ambitions.

## Climate models for Snowball Earth

There is evidence that Earth has gone through *Snowball *periods, i.e. times when it was completely covered in ice from equator to pole. How could the planet have entered such a frigid state? Current research points to a complex interplay between Greenhouse gases and the rise of oxygen in an atmosphere that was previously dominated by other gases. In this project, the student will explore how different interplays could have caused a transition to Snowball Earth using ordinary differential equations models.

Prerequisites: Maths 260, Proficiency in MATLAB.

## Asymmetric vortex patterns

Equations for two or more interacting vortices in a geophysical fluid are known to produce a vast number of symmetric patterns. Recently, completely asymmetric patterns have been discovered. This project is an exploration into the symmetry/asymmetry of vortex patterns and how these patterns depend on the underlying physics of the fluid (e.g. shallow water versus magma). The project will involve using methods from pure maths, specifically computational algebraic geometry, to answer questions in applied maths, specifically fluid dynamics and differential equations.

Prerequisites: No prerequisites required, but Maths 260 along with interest in both pure and applied maths is a plus.

## Commutator subgroups of some symplectic nilpotent subgroups.

The study of central series of a nilpotent group is fundamental in the theory of nilpotent groups. The upper central series of symplectic nilpotent subgroups are know, and the commutator subgroup is the first term of the lower central series of the symplectic nilpotent subgroups, which can be calculated using matrix multiplication.

Prerequisite: Maths 320.

## Understanding coupled calcium oscillations

Parotid cells secrete saliva, and this is controlled by oscillations in the intracellular calcium concentration. Although the cells are coupled by gap junctions, and thus the oscillations are weakly coupled, we still have very little understanding of how this coupling controls the properties of the resultant oscillations.

In this summer research project the student will investigate the mathematical theory of coupled oscillations, learn how to construct bifurcation diagrams, and how to use these bifurcation diagrams to increase our understanding of coupled calcium oscillations behave in actual cells. The outputs from the modelling will be compared to experimental data from laboratories in Japan and the USA.

A suitable student will have excellent grades in Math 260 (and preferably in Math 270) as well as interest in biological applications of mathematics.

## Hamiltonicity of Cayley graphs

A Cayley graph is a graph that can be obtained from a group, on which the graph acts naturally. (https://en.wikipedia.org/wiki/Cayley_graph)

It is conjectured that all connected Cayley graphs of order at least 3 admit a Hamiltonian cycle. (https://en.wikipedia.org/wiki/Hamiltonian_path)

A summary of some of the known results can be found here: https://arxiv.org/pdf/1009.5795v3.pdf

The project would involve understanding some of techniques that have been used so far and trying to see if these methods, perhaps together with some computer calculations, can be used to deal with some of the small open cases. (Order 72, for example.)

Prerequisites:

Having taken MATHS 320 (Algebraic Structures) or some equivalent would be useful, but not strictly necessary.

## The Writing Mathematician

To what extent are mathematics and writing related? How do these two competencies develop? Are there some productive similarities between mathematics and writing? This project is for mathematicians who are interested in writing – both for a general or academic audience. A background in mathematics is a must, and preference will be given to students who have some writing experience at university.

## The mathematical experience of struggle

Clarity is highly valued when it comes to teaching and learning mathematics, and understandably so! Yet the experiences of struggle and even confusion are also fundamental (but perhaps less valued) parts of the mathematical experience. Recent research encourages us to engage students in mathematical struggle, but how can we do that without turning students off mathematics? This project will explore and critique some ways of creating opportunities for productive mathematical struggles in classrooms, and discuss some benefits and challenges of a positive view on struggle.

## Polynomial convexity

What is a convex set? One way to characterize convexity uses linear functions. K is convex if it has the following property: if the value of any linear function at a point z is smaller than the maximum value of the linear function on K, then z is in K. We can use this to develop various notions of convexity using different classes of functions. For example, replacing `linear functions' with `polynomials' gives the notion of polynomial convexity. Such sets have interesting geometric and analytic properties.

## Orthogonal polynomials

A system of orthogonal polynomials is a (suitably normalized) infinite collection of polynomials that is an orthogonal basis for the vector space of all polynomials with respect to some inner product, usually given by an integral. For integrals over the unit circle in the complex plane or a real interval, these polynomials have been well-studied (orthogonal polynomials on the unit circle/real line: OPUC/OPRL). In particular, 1) these polynomials obey recurrence relations; 2) the sequence of orthogonal polynomials has interesting asymptotic behaviour (as the degrees increase). In this project we will look at these polynomials, and also look more generally at orthogonal polynomials on polynomial lemniscates (of which OPUC are an example).

## Giftedness and talent in mathematics

In NZ gifted and talented students are not categorized as priority learners, and instead they are considered as a sub-group of students with special needs. Students with special needs are strongly associated with disabilities, and thus there is no actual funding that the government allocates for promoting excellence and high-achievement at school level. In this way, schools are obligated and encouraged to cater for gifted and talented students on their own.

The student working on the project will delve into the complexity of giftedness and talent in mathematics with a special focus on identification, intellectual needs and catering. The practical parts of the project may include designing tasks and workshops, interviewing school and university students, their teachers and lecturers. Opportunities may be available to explore teaching and learning that occurs in a preparation camp to International Math Olympiad.

Prerequisites:

The projects are proposed to students who are interested in the mathematics teaching and learning processes. Students with high-achievements in Maths 202 or 302 will be given preference.

## Unpacking mathematical conventions

Why does a^{0} equal 1? Why isn’t 1 defined as a prime number? Why are absolute values of real numbers, moduli of complex numbers and determinants of matrices denoted by the same symbol of ‘**⬛**’? These questions are concerned with mathematical conventions, i.e. choices of the mathematics community regarding definitions of concepts and symbols. Usually a convention is presented to students by a teacher, without delving into a deep discussion about it. However, in some cases unpacking possible reasons for establishing a convention can lead to insightful mathematical ideas and new understanding.

The person working on the project will engage in identifying mathematical conventions that worth unpacking. In addition, s/he will analyse data collected from Canadian teachers of mathematics who unpacked conventions in a form a virtual dialogue between Teacher- and student-characters. The project will involve categorization of teachers’ conventions, validation of their explanations and exploration of what teachers learned from this activity, if at all.

Prerequisites:

The projects are proposed to students who are interested in the mathematics teaching and learning processes. Students with high-achievements in Maths 202 or 302 will be given preference.

## Determinant Tic-Tac-Toe

In **determinant tic-tac-toe**, two players (Player 1 and Player 0) take turns writing 1's and 0's on a 3x3 board. When the board is filled, the players calculate the determinant of the resulting 3x3 matrix; if it is 0, player 0 wins, and if it is nonzero player 1 wins.

Surprisingly, though, we know very little about **generalizations **of determinant tic-tac-toe. For instance, you could give one player a "handicap" by letting them fill in a few cells at the start; you could play on larger boards (like a 4x4, or 5x5, or in general a nxn board); you could let players play values other than 0 or 1; you could even add a third player! For many of these generalizations, winning strategies are unknown, despite the fact that the underlying mathematical objects (determinants of 0-1 matrices) are heavily studied and remarkably useful objects.

In this project, we will consider several variations of the determinant tic-tac-toe game and attempt to find winning strategies for as many as possible.

Background required: Linear algebra skills (e.g. you should be comfortable with the linear algebra in 250/253), the ability to write a proof.

## 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 Matlab coding skills.

## Computing tori with Chebfun

“Chebfun” is an extension to Matlab that allows for arithmetic operations on functions rather than discretisations, performed to machine precision. The Chebfun project is led by (Sir Lloyd) Nick Trefethen (FRS) from Oxford University, who will be visiting Auckland in December this year. This project will give you the opportunity to learn Chebfun with the help from Nick Trefethen himself, and to develop a new application of this tool to the computation of invariant manifolds based on the so-called graph transform.

Prerequisites: Maths 260 and good Matlab coding skills.

## Balancing El Niños and La Niñas.

An El Niño or La Niña is characterised by an abnormal warming or cooling of the Pacific Ocean, respectively. These climatic events are known to have far-reaching global impacts, such as flooding in South America, disease outbreaks in India and droughts in New Zealand. Yet, there are still many aspects of the El Niño/La Niña climate system that are not well understood; for example, why do El Niños always seem to be stronger than La Niñas? In this project, the student will consider a conceptual climate model and investigate how accurately such a simple model can reproduce the relative strengths between El Niños and La Niñas.

Prerequisites: Maths 260, experience in MATLAB, C++ or Python.

## Grobner bases for matrix equations

In algebraic geometry a Gr¨obner basis is a particular kind of generating set of an ideal in a polynomial ring *K*[*x*1*, . . . , xn*]. A Gr¨obner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension or the number of common zeros. Gro¨bner bases are of the main practical tools for solving systems of polynomial equations.

Gro¨bner basis computations are a multivariate, non-linear generalization of both Euclid’s algorithm for computing polynomial greatest common divisors, and Gaussian elimination for systems of linear equations.

In this project we shall study modern algorithms for producing Gr¨obner bases and investigate whether they can be useful in solving *matrix equations*.

## Modelling and analysis of clustered ventilation in the lung

Asthmatic patients typically exhibit a characteristic pattern of ‘clustered' ventilation during an asthma attack (as shown by MRI).

These clusters originate via an interesting dynamic phenomenon in which airways may be either open or closed, and which groups closed airways with closed airways, and open airways with open airways, leading to clusters. This project will study a model of clustered ventilation defects using tools from ordinary differential

equations and linear algebra. Recommended preparation: Maths 260 and some Matlab experience. No prior biological knowledge is expected, just an interest and willingness to learn.

## Classes of subgroups of the general linear group

A project of long-standing interest is the classification of certain classes of subgroups of the general linear group. We are particularly interested in certain classes of subgroups of GL(p, C) where p is a prime and C is the field of complex numbers. Various authors describe the subgroups, sometimes describing generating matrices. This project will seek to understand some of the machinery used in such classifications, and set up explicit lists of the subgroups which can be used in computational algebra packages. 320 is essential background, prefer 720; some interest in computation will be useful.

## Presentations for finite simple groups

The finite simple groups are the building blocks or composition factors for other groups. If we know a finite presentation for each composition factor, then we can construct one for the group. This project will study certain classes of presentations for the composition factors, seeking to verify their correctness. We hope to make the presentations available explicitly in a form which can be used in computational algebra packages. 320 is essential background, prefer 720; some interest in computation will be useful.

## The Allee effect in a model of predator-prey interaction

In population dynamics, the Allee effect refers to a process that reduces the growth rate for small population densities. The interactions between two species of predators and prey influenced by the Allee effect can be modelled by iterating a planar noninvertible map. The dynamics are organised by invariant sets of the map, that is, by fixed points, their stable and unstable manifolds (which can act as extinction thresholds for the different species) and curves that bound regions with different numbers of preimages of the map. These sets interact with each other when the parameters of the system are changed. The goal of this summer project is to compute these invariant sets and to study their interactions when the Allee effect is switched on.

Basic knowledge of Matlab is required.