- » Climate models for Snowball Earth
- » Solar radiation and tipping points in Earth’s history
- » Attribute-based encryption
- » Hotelling games onrectangulargrids
- » Analyzing animal movement data
- » Modelling EEG data in the transition to unconsciousnes
- » Hamiltonian cycles in vertex-transitive graphs
- » Modelling and analysis of clustered ventilation in the lung
- » Movement of immune cells and the role of HIV
- » Lorenz and computational chaos
- » The Cayley-Hamilton theorem andinvariants ofmatrices
- » Vanishing sums of roots of unity
- » Algebraic curves
- » Programs for gifted and talented school students in mathematics
- » Teachers unpack mathematical conventions
- » Lie ring structures of nilpotent subgroups of symplectic groups over integer rings
- » Nilpotent subgroups of classical groups over rings
- » Latin squares and NP-completeness
- » Finding equiangular lines and their (projective) symmetries
- » Useful redundancies - finite tight frames
- » Generalised inverse limits and spectacular subsets of the Hilbert cube
- » Okounkov bodies associated to complex algebraic varieties
- » Excluded volume effects for particle diffusion
- » Algebraic geometry and cryptography
- » Bifurcations in a system with three time scales
- » Snowball Earth events: the role of greenhouse gases in the radiative budget

## Climate models for Snowball Earth

### Project code: SCI157

There is evidence that Earth has gone through *Snowball p*eriods, i.e. times when it was completely covered in ice from equator to pole. A few differential equation models supporting this hypothesis have been developed. The goal of this project is to identify which aspects of the various models are consistent with Earth’s history. Is one model “better” than another? The student will do a numerical exploration of the dynamics and compare results to the climate record.

**Requirement: **

Maths 260 and MATLAB. Background in an undergraduate PDE course (361) is a plus.

## Supervisor |

## Solar radiation and tipping points in Earth’s history

### Project code: SCI158

Variations in Earth’s axial tilt, the *obliquity*, are known to impact the annual average solar energy received at a given latitude. For example, large obliquity results in warmer polar regions which sets the stage for melting ice sheets. The project involves estimating obliquity and modelling the associated incoming solar radiation at key times in Earth’s history such as onset and termination of glacial cycles. This will require working with data in MATLAB.

**Requirement:**

Maths 260 and MATLAB

## Supervisor |

## Attribute-based encryption

### Project code: SCI159

Public-Key encryption is a powerful mechanism for protecting the confidentiality of stored and transmitted information. Traditionally, encryption is viewed as a method for a user to share data with another targeted user. While this is useful for applications where the data provider knows specifically which user he wants to share information with, in many applications the provider will want to share data according to some policy, i.e., depending on the receiver’s credentials, e.g., a user can decode an e-mail if she is a “FRIEND” or “IMPORTANT”. Recently developed methods of attribute-based encryption heavily use linear secret-sharing schemes. It is this connection with linear secret sharing that we will make emphasis on.

**Prerequisites:**

MATHS 328

## Supervisor |

## Hotelling games onrectangulargrids

### Project code: SCI160

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:**

MATHS 250

## Supervisor |

## Analyzing animal movement data

### Project code: SCI161

The ongoing development of GPS devices small enough to be carried by animals has resulted in vast amount of animal tracking data being made available. The analysis of such data requires the development of new methods and techniques, many of which are mathematically very interesting. This project involves using and developing mathematical techniques for the analysis of previously collected animal tracking data.

Requires some knowledge of Matlab.

## Supervisor |

## Modelling EEG data in the transition to unconsciousnes

### Project code: SCI162

Recent studies have shown that there are interesting changes in resting-state EEG data as patients transition to unconsciousness under the effects of anaesthesia. We have recently developed a differential equation model of this system, using dynamical systems methods and a heteroclinic network. The project involves numerically investigating the dynamics of the differential equations and how they can be best fit to the EEG data.

Requires Maths 260.

## Supervisor |

## Hamiltonian cycles in vertex-transitive graphs

### Project code: SCI163

A vertex-transitive graph is, informally, a graph in which all the vertices are “identical” with respect to the structure of the graph. A Hamiltonian cycle in a graph is a cycle going through every vertex exactly once. There are only five known connected vertex-transitive graphs without Hamiltonian cycles and some mathematicians conjecture that there are no others. The general problem is probably hard, but various cases can be investigated, including graphs of small order, where a computer can be very helpful.

**Pre-requisite:**

MATHS 320 (Some knowledge of group theory is essential for this project; some experience with computing could be helpful.)

## Supervisor |

## Modelling and analysis of clustered ventilation in the lung

### Project code: SCI164

Asthmatic patients typically exhibit a characteristic pattern of “clustered” ventilation during an asthma attack (as shown by magnetic resonance imaging (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.

## Supervisor |

### Pre-requisites

An A pass in Maths 260 and a good grade in Maths 250.

## Movement of immune cells and the role of HIV

### Project code: SCI165

The way that immune cells, such as T cells, move about within the body is crucial to the effectiveness of the immune system; without efficient movement the body is unable to fight disease or infection. Crucially the movement of these cells may be guided by a fibrous network, creating a scaffold for cell movement. HIV infection may damage this network,

leading to permanently impaired immune function, even when HIV is later effectively suppressed. This project will study the role of this network via modelling, and analysis of the resulting models. The tools involved are ordinary differential equations, and the geometric structure of the network itself.

**Recommended preparation**:

Maths 260 and some matlab experience. No prior biological knowledge is expected, just an interest and willingness to learn.

## Supervisor |

## Lorenz and computational chaos

### Project code: SCI166

Edward Lorenz, the famous meteorologist from MIT who discovered chaotic behaviour, has devoted a lot of research to validating his numerical results against theoretically known facts. In one of his papers, he wrote: “Anyone who has devoted much time to solving nonlinear differential equations numerically has almost surely encountered computational instability --- a rapid and unbounded amplification of the variables.” Decades on, we have a lot more theoretical understanding of complicated dynamics, but Lorenz' simple example systems still harbour many surprises. Can you explain the numerics?

**Prerequisites:**

Maths 162 - Modelling and Computations

Maths 260 - Differential Equations

## SupervisorHinke Osinga |

## The Cayley-Hamilton theorem andinvariants ofmatrices

### Project code: SCI167

Consider an n × n matrix A and its characteristic polynomial

## Supervisor |

The Cayley-Hamilton theorem asserts that

Of interest in this project will be the coefficients of the characteristic polynomials, i.e., the c_{k} = c_{k}(A). Observe that these depend polynomially on the entries of A. Furthermore, as A and GAG^{-1} (for an invertible matrix G) have the same characteristic polynomial, c_{k}(A) = c_{k}(GAG^{-1}); we call c_{ k}invariants of n×n matrices. The goal of this project is to show all GL_{n}-invariants come from the c_{k} and investigate invariants under other group actions.

**Prerequisites:**

MATHS 255 and at least one of the MATHS 32? courses. A good grasp of linear algebra and a bit of computer programming (preferably Mathematica) is desired.

## Vanishing sums of roots of unity

### Project code: SCI168

Consider a regular n-gon in the plane. What is the number of interior intersection points made by its diagonals? To answer this simple problem we need to study roots of unity and their vanishing sums. For instance, if α is a primitive 3-rd root of unity, and β is a primitive 5-th root of unity, then

## Supervisor |

Of particular interest are small sums of this form, which have been classified up to length 9. In this project we shall explore minimal vanishing sums of roots of unity and how they are related to the original geometric problem.

**Prerequisites: **

MATHS 255 and at least one of the MATHS 32? courses. Familiarity with complex numbers and a bit of computer programming (preferably Mathematica) is desired.

## Algebraic curves

### Project code: SCI169

A plane algebraic curve is the set of points in the Euclidean plane whose coordinates are zeros of some polynomial in two variables. Their properties (e.g. tangents, singularities, smoothness, asymptotes, rationality, genus, desingularization, etc.) are the objects of study in algebraic geometry and topic of a new course offered to stage 3 and graduate students in 2017. The project will be to investigate potential topics and activities for such a course, and prepare course materials.

**Prerequisites:**

MATHS 255 and at least one of the MATHS 32? courses. Familiarity with LaTeX is desired.

## Supervisor |

## Programs for gifted and talented school students in mathematics

### Project code: SCI170

In NZ gifted and talented students are not categorized as priority learners, and instead they are considered as a subgroup 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 highachievement. In this way, schools are obligated and encouraged to cater for gifted and talented students on their own.

The goal of this project is to map the programs that schools in the Auckland area provide for students who are gifted and talented in mathematics. The person working on this project will analyse school websites, contact Heads of the Department and teachers, visit schools and observe lessons. The objectives of the project are to explore the criteria that schools use for defining giftedness and talent, programs and activities that they suggest, and profiles of participating students.

**Prerequisites:**

This project is suggested 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.

## Supervisor |

## Teachers unpack mathematical conventions

### Project code: SCI171

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 analyse the data that was collected from Canadian teachers of mathematics. The teachers chose mathematical conventions and unpacked them to form a virtual dialogue between teacher and student characters. The objectives of the project are to categorise teachers’ conventions, validate their explanations and to explore what did teachers learn from the activity.

**Prerequisites: **

This project is suggested 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.

## Supervisor |

## Lie ring structures of nilpotent subgroups of symplectic groups over integer rings

### Project code: SCI172

Study the lie ring structure of small ranks for the symplectic groups over the integer ring.

(Pre-requisites: Maths 320)

## Supervisor |

## Nilpotent subgroups of classical groups over rings

### Project code: SCI173

Study subgroups of some classical groups over some commutative rings.

(Pre-requisites: Maths 320)

## Supervisor |

## Latin squares and NP-completeness

### Project code: SCI174

## Supervisor |

Take an n by n grid, in which each cell is either blank or contains a symbol from the set {1, 2, ... , n}. We say that this grid is a partial Latin square if no row or column contains more than one copy of any of these symbols.

We say that a partial Latin square is completable if there is some way to fill in its blank cells with values from {1, 2, ... , n} without breaking our “no repeats” property in any row or column. (If this reminds you of Sudoku, you've got the right idea!)

Surprisingly, finding out whether a given square is completable is a difficult task. Students involved in this project will study Latin squares and learn about algorithmic complexity, a mathematically rigorous way to quantify what “difficult” means for certain kinds of problems. From there, students will work on open research problems on determining which kinds of Latin squares are hard to complete, and which are easier.

**Required:**

Maths 255, being comfortable with proofs.

**Recommended:**

Maths 225, Maths 326.

## Finding equiangular lines and their (projective) symmetries

### Project code: SCI175

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).

## Supervisor |

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

### Project code: SCI176

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.

## Supervisor |

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

**Pre-requisites:**

Maths 255

## Generalised inverse limits and spectacular subsets of the Hilbert cube

### Project code: SCI177

Certain subsets of the Hilbert cube can be found by working backwards through an infinite sequence of functions from [0,1] to [0,1]. The functions can be very simple but the subset obtained may have striking and unexpected properties. If we allow the function to be set-valued, the subset is called a generalised inverse limit (GIL). They are fun to obtain and there are many open questions to work on regarding their properties. You will have the opportunity to work with a group of students and graduates who are also working on GILs.

## Supervisor |

## Okounkov bodies associated to complex algebraic varieties

### Project code: SCI178

A complex algebraic variety is a geometric object that includes some well-known curves and surfaces. The Okounkov body of the variety is a real compact convex body that gives a simplified picture of its behaviour. Recently, Okounkov bodies have become useful in potential theory. These objects are allegedly difficult to compute. The aim of the project is to implement an algorithm that constructs an Okounkov body from the equations that define the variety.

(This project has no specific prerequisites.)

## Supervisor |

## Excluded volume effects for particle diffusion

### Project code: SCI179

The heat equation is a partial differential equation that models the conduction of heat in an object. This equation also models the concentration of small particles moving randomly; particle diffusion. The heat equation is valid if particles don’t interact. This project investigates what happens if particle size is large enough to cause particles to interact significantly. This project will suit students interested in partial differential equations and random processes.

## Supervisor |

## Algebraic geometry and cryptography

### Project code: SCI180

Modern public key cryptosystems are based on computational problems in number theory and algebraic geometry. The project will be to study some of these computational problems, the mathematics behind them, and the algorithms that are used to solve them. One of the central

Themes in this subject is elliptic curves, but currently there is major research interest in systems of polynomial equations corresponding to algebraic sets of higher dimension. This project is most appropriate for a student who has taken MATHS320 and/or MATHS328, and is interested in computer experimentation.

## SupervisorSteven Galbraith |

## Bifurcations in a system with three time scales

### Project code: SCI181

When trying to understand mathematical models of physical phenomena, we often look for qualitative changes in behaviour (i.e., bifurcations) that occur when one or more system parameters are varied. These bifurcations can be particularly complicated in systems where some variables evolve much faster than others, as is common in models of biological systems. This project will investigate bifurcations associated with complex periodic solutions in a simple model of cell dynamics with three time scales.

**Prerequisites:**

An A pass in Maths 260 and a good grade in Maths 250.

## Supervisor |

## Snowball Earth events: the role of greenhouse gases in the radiative budget

### Project code: SCI236

Typically, climate models for Snowball Earth assume a simplified form of outgoing longwave radiation that ignores the effect of greenhouse gases entirely. This project involves incorporating these effects into one such model and exploring their implications for the tipping point corresponding to Earth’s jump from a relatively warm state to a frigid ice-covered regime.

**Recommended preparation: **

Maths 260 would be useful background, but is not required.

## Supervisor |