Mathematics

Project description

Everyone has a story about learning mathematics. Many of these are happy stories. But some are not. For some people, mathematics is so tinged with distress and confusion that they opt out of the subject as soon as they can, avoiding it into adulthood, often at personal cost. This project explores how stories might be used to invite mathematics-averse adults to reconnect with the subject. It will involve searching and reviewing research literature, and analysing data. Experience in MATHS 302 is useful.

Project description

New Zealand’s declining achievement in mathematics at the school level is a severe problem that can negatively impact national development and prosperity. Substantial declines in achievement indicators have been recorded in the large-scale international studies: the Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA). Out of 64 countries assessed in TIMSS in 2019, New Zealand scored significantly lower than all of the OECD countries taking part, except for Chile and France, and significantly lower than the centre point. Over time, the trend is particularly concerning for high school students, with Year 9 average achievement being the lowest recorded since 1995. This is in sharp contrast with other countries, with 13 out of 33 improving their performance from 2015 to 2019, whereas New Zealand is one of the only four countries with decreased achievement. Furthermore, only 12% of New Zealand’s 15-year-olds scored at the top two levels in mathematics compared to Singapore’s 37% in the latest PISA cycle. Whereas at the bottom end of the performance distribution, 22% of New Zealand’s cohort are ‘low achievers’ contrasted to 2% of the students assessed in China.

This project will give you an opportunity to participate in research activities to shape new research projects to identify factors contributing to decline in national mathematics performance.

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

Project description

The underrepresentation of women in university mathematics departments is a complex issue stemming from both historical and contemporary barriers. Despite progress in gender equality, women are still significantly underrepresented as mathematicians in academia. Societal stereotypes, lack of role models, and institutional biases all contribute to this disparity. From a young age, girls often face discouragement from pursuing STEM fields, resulting in fewer women entering and remaining in mathematical careers. In universities, the scarcity of female faculty members creates a cycle where young women struggle to envision themselves in these roles, further reducing their numbers.

We urgently need comprehensive research to uncover the specific factors contributing to the underrepresentation of women in mathematics. This summer research project aims to investigate these issues in depth, exploring the systemic barriers women face at different stages of their academic careers.

Project description

The delta invariant is a quantity associated to a Fano variety X (which is a kind of geometric object). The delta invariant of X is greater than 1 if and only if X is K-stable. K-stability is a geometric property that a lot of mathematicians are currently interested in: https://en.wikipedia.org/wiki/K-stability_of_Fano_varieties
The aim of this project is to understand as much as possible the delta invariant and how it is computed. More precisely, we will consider a recent (2020) formula for the delta invariant by Rubinstein, Tian, & Zhang, and look at all the ingredients that go into it.

Background of Stage 3 Algebra & calculus would be most suitable: 320/328 + 340/333.

Project description

Due to the potential threat of quantum computers, the research community is re-evaluating the security of a number of protocols and security systems in widespread use. Post-quantum cryptography is the study of protocols that can be used with today’s computers, but that are secure against an attacker in the future who has a quantum computer. It is one of the hottest areas of research in mathematical cryptography.

The project will study post-quantum public key cryptosystems and their underlying mathematics. The main areas of mathematics that are used for post-quantum crypto are: lattices, error-correcting codes, multivariate polynomial equations, and isogenies of elliptic curves.

Depending on the student’s interest, the project can involve programming. The student will also have the opportunity to work with PhD students in post-quantum crypto.

Preferred skills: Abstract algebra, linear algebra, programming and knowledge of computer systems, algorithms and complexity.
Preferred pre-requisite courses: MATHS 320 or 328.

Project description

The transfinite diameter of a set is a positive real number that indicates how “spread out” its points are. It can be interpreted as a constant for normalising classes of polynomials on the set. It can also be interpreted as the minimum energy of a class of integrals. These relationships are explained by potential theory, which was worked out in the 20th century for one complex variable. In higher dimensions there are fewer results but they are a lot richer. Prerequisite: MATHS 253

Project description

The discipline of origami or paper folding has received a considerable amount of mathematical study, with applications in fields such as robotics, biotechnology, and industrial design. Despite this, even simplified models of folding are generally not well understood.

In this project, you will study the theory of polyomino folding. A polyomino is a collection of unit squares in the plane, glued together edge-to-edge. You will investigate the question whether a given polyomino can be folded into a unit cube (every face of the cube is covered by at least one, but possibly more unit squares of the polyomino); the picture below shows three polyominos all of which can be folded into a cube – I encourage you to try it. Depending on your background and interests, this project may include algorithmic considerations, sufficient criteria for foldability, or topological obstacles to finding an actual folding from partial folding information.

This project is suitable for any student with an interest in combinatorics. Some programming skills or background in topology would be ideal.

Project description

Pursuit–evasion games are a family of two player games in mathematics and computer science in which one player attempts to track down or capture another player in an environment. There are countless variants of these games, depending on the environment, the precise rules of how players move, the information available to each player, and various other parameters.

In this project, you will study pursuit evasion games that take place on a graph, and in particular the effect of random moves on the game.

The project is suitable for students who have enjoyed an introduction to combinatorics (e.g. COMPSCI225), background in probability theory, and particularly stochastic processes will be an asset.

Project description

Voting is a common method of collective decision making, which enables the participating voters to identify the best candidate for the society on the basis of their individual rankings of the candidates. The Gibbard-Satterthwaite theorem implies that
regardless of the voting rule there exist occasionally voters, called manipulators, who can change the election outcome in their favour by voting strategically. When several such manipulators exist, voting becomes a game played by these voters. The project will be devoted to the study of such games for various voting rules which, in particular, will
include voting in a jury setting.

Prerequisites: MATHS 250.

References:

U. Grandi, D. Hughes, F. Rossi, and A. Slinko. Gibbard-Satterthwaite games for k-approval voting rules. Mathematical Social Sciences, 99:24-35, 2019.

Fabrizi, S., Lippert, S., Pan, A. et al. A theory of unanimous jury voting with an ambiguous likelihood. Theory and Decision, 93:399-425, 2022.

Project description

Kleinian groups are discrete groups of orientation-preserving isometries of hyperbolic isometries. They sit at the intersection of group theory, complex analysis, geometry and topology, you will learn the basics and can explore properties of these groups either theoretically or aided by computer.

Project description

In recent years there have been some interesting results in pure mathematics which have been assisted by AI by groups in Oxford and Sydney, in particular in geometry and topology and in representation theory. As this is a very new line of enquirey you will learn the basics alongside us, try to replicate some of the results obtained and explore the potential for new results.

Project description

These spaces, called CAT(0) spaces, are a metric generalisation of Riemannian manifolds where triangles are thin. They play a central role in geometric group theory and also encompass singular spaces such as building which are a geometric description of algebraic groups. In recent years Stephan Stadler has obtained ground-breaking results on higher-rank CAT(0) spaces. You will explore these results.

Project description

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). Recently, it has been shown that there are six equiangular lines in H^2 (there cannot be more), where H is the quaternions.

The aim of this project is to study quaternionic equiangular lines, specifically to find examples in more than two dimensions. In particular, finding a set of ten to fifteen equiangular lines in H^3 would be a very interesting new result. This project requires linear algebra over the quaternions (an extension of the complex numbers, for which multiplication is not commutative). It is likely that such lines can be constructed as orbits of groups of matrices over the quaternions (e.g., reflection groups). To do this, a symbolic algebra package such as maple or magma will be used.

Pre-requisites: Linear algebra and MATHS 320.

Project description

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.

Project description

This is an elementary project to see whether the Cauchy-Schwarz inequality, which holds for R^n and C^n holds for quaternionic space H^n, with a natural analogue of the Euclidean inner product, and then to apply it do some questions in spherical design theory where it came up. The quaternions H are an extension of the complex numbers, where the multiplication is not commutative (in general).

Project description

The Horn problem is a classical problem of Linear Algebra which is about characterising the eigenvalues of a sum of two Hermitian matrices in terms of the eigenvalues of the summands.

The answer is given in terms of a complete set of inequalities and it was solved by Klyachko and Knutson-Tao in the late 1990s. The proof combines clever ideas in geometry and combinatorics.

The aim of this project is to learn about this result and explore some new directions, for instance:

1. What is the analogous statement for a product of unitary matrices, i.e. the multiplicative Horn problem?

2. Can one give a probabilistic description of the Horn problem?

Project description

Tipping points in dynamical systems are rapid transitions between two distinct states. In a physical sense, one might think of ecosystem collapse, or rapid changes in climate systems. In a mathematical sense, tipping points can be driven by noise, or bifurcations. Depending on the type of tipping point, there are different warning signals which will occur prior to the tipping point, which may be useful for managing these systems. (Most interest to date has been in applications to climate or ecology.)

This project will focus either on tipping points in spatial systems, or application of tipping points to physiological systems (asthma in particular), depending on the interests of the student. Interested students should have at least some background in differential equations and bifurcations (MATHS 260 or similar) and numerical methods (MATHS 162, 270 or similar).

Project description

Suppose that, from some observable dynamical system, we have only a set of observations of this system at discrete times. The problem of equation discovery is: can we reconstruct, from these observations alone, the dynamical system which governs the underlying behaviour of this system? Importantly, we want not just to predict the future time course beyond the observations (i.e. in a black-box fashion) but to gain understanding of the underlying system; in the case of a system of differential equations, can we reconstruct the underlying equations themselves given only the observations?

Project description

The formation of rogue waves is of interest, from North sea waves [1-3], waves in tanks [4-7], to waves in nonlinear optics[8-11]. Most common models used to investigate rogue bursts have used the nonlinear Schrödinger (NLS) equation and its variants. However, such integrable settings and analytical solutions are rare in higher dimensions. So we propose to use the model of a dissipative system: which describes interaction between standing waves in domains of moderate aspect ratio. When spatial reflection symmetry is broken, the left and right running waves can interact strongly producing a spatially and temporally localised extremely large amplitude event, i.e., a rogue burst [12]. This project will involve writing a time-stepper to advance the initial value problem in time. Some knowledge of numerical methods and/or dynamical systems will be an advantage but is not crucial.

References:
1. N. Mori and P. C. Liu, Analysis of freak wave measurements in the Sea of Japan, Ocean Eng. 29, 1399 (2002). 

2. S. Haver, A possible freak wave event measured at the Draupner jacket January 1 1995, Rogue Waves 2004: Proceedings of a Workshop, Brest, France (unpublished). 

3. D. A. G. Walker, P. H. Taylor, and R. E. Taylor, The shape of large surface waves on the open sea and the Draupner New Year wave, Appl. Ocean Res. 26, 73 (2004). 

4. A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Rogue Wave Observation in a Water Wave Tank, Phys. Rev. Lett. 106, 204502 (2011).

5. A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves, Phys. Rev. X 2, 011015 (2012).

6. M. L. McAllister, S. Draycott, T. A. A. Adcock, P. H. Taylor, and T. S. Van Den Bremer, Laboratory recreation of the Draupner wave and the role of breaking in crossing seas, J.Fluid Mech. 860, 767 (2019).

7. G. Xu, A. Chabchoub, D. E. Pelinovsky, and B. Kibler, Observation of modulation instability and rogue breathers on stationary periodic waves, Phys. Rev. Research 2, 033528 (2020).

8. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Optical rogue waves, Nature (London) 450, 1054 (2007).

9. J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Instabilities, breathers and rogue waves in optics, Nat. Photonics 8, 755 (2014).

10. B. Frisquet, B. Kibler, P. Morin, F. Baronio, M. Conforti, G. Millot, and S. Wabnitz, Optical dark rogue wave, Sci. Rep. 6, 20785 (2016).

11. A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation, Phys. Rev. Lett. 119, 033901 (2017).

12. P. Subramanian, E. Knobloch and P. G. Kevrekidis, Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice, Phys. Rev. E 106, 014212 (2022).

Project description

Wild chaos is a new form of unpredictable behaviour that can occur in higher-dimensional dynamical systems. Only very few examples are known, but a previous summer project student discovered a measure of wild chaos that can identify such behaviour in experiments. Your task will be to test the approach on data that is thought to come from a wild chaotic attractor in a three-dimensional discrete dynamical system, defined by a map. If it works, what happens as parameters change and wild chaos is lost?

Prerequisites: MATHS 260 and good Matlab or Python coding skills.

Project description

Chebfun is an extension to Matlab that allows for arithmetic operations on functions, performed to machine precision, using Chebyshev polynomials. Recent updates include periodic function approximations. Your task will be to develop a new application of Chebfun for the computation of an invariant closed curve, or torus, in a two-dimensional discrete dynamical system, defined by a map. In two dimensions, the torus will be either attracting or repelling. Are you up for the challenge of doing it for a three-dimensional system when the torus can be of saddle type?

Prerequisites: MATHS 162, MATHS 250, and good Matlab coding skills.