Mathematics

Project

Machine learning and, more broadly, artificial intelligence will continue to change everyone’s life profoundly. Mathematics, statistics, and computer science play an important role in advancing machine learning algorithms (e.g., to make algorithms more reliable) and theoretical research into machine learning can advance our understanding into why certain methods are successful or not.

In this project, you will study some theoretical foundations of machine learning algorithms and how techniques from probability theory, geometry, and graph theory can be leveraged to aid the design of machine learning algorithms.

The exact direction of the project will be decided at its start and depend on the interests and experience of the summer student.

Project

This project will involve using the scientific computing language Julia and implementing (previously developed) software to compute basins of attractions of various attractors such as equilibria and periodic orbits in systems which exhibit bi-stability. You will further examine how the boundaries of the basin change as parameters are varied.

Prerequisites: MATHS 260, and at least some programming experience.

Project

Continuous time recurrent neural networks (CTRNN) are systems of coupled ODEs inspired by the structure of neural networks. In recent work, we have shown that parameters in the CTRNN can be chosen so that the resulting phase space contains a network attractor. This project will use computational methods to investigate the dynamics near these network attractors as parameters are varied.

Prerequisites: MATHS 260, and at least some programming experience.

Project

Modelling traffic is of interest to estimate the density of vehicles, for instance in an urban area to determine the noise level, or on a suspended bridge to check the maximal authorised weight. Other features may also be modelled like the presence of the red traffic light (shock) or green traffic light (rarefaction, see figure).

In this project, we propose to study a simplified traffic model numerically. The main goal is to test different numerical schemes for the advection equation, then the Burgers equation, and exhibit the schemes' intrinsic properties. In particular, some schemes will induce artificial numerical viscosity, which can alter considerably the expected solution.

Requirements

Interested students should be familiar with differential equations (MATHS 260 or similar) and numerical methods (MATHS 270 or similar). Some knowledge on partial differential equations (MATHS 361, 362 or similar) is a plus.

Project

After an episode of heavy rain, a potential risk is the overflow of dams, resulting into a dam break. In such event, a large uncontrollable water flow will spread in the valley nearby causing devastating damages.

The goal of this project is to simulate a dam break in a simple way. We propose to use finite volumes schemes to solve shallow water equations numerically in one-space dimension. The shallow water equations are a system of partial differential equations that model the water height and the momentum of the water when the dam breaks. The solution also depends on the topology of the ground, and the numerical schemes will need to be adapted accordingly.

Requirements 

Interested students should be familiar with differential equations (MATHS 260 or similar) and numerical methods (MATHS 270 or similar). Some knowledge on partial differential equations (MATHS 361, 362 or similar) is a plus.

Project

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.

Requirements

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

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 timecourse 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?

This project will explore recent developments in this area.

Requirements

Interested students should have some familiarity with systems of differential equations (MATHS 260 or similar), linear algebra, and numerical methods (MATHS 162, 270 or similar).

Project

Proofs are a cornerstone of mathematics. Yet, mathematics education research shows time and again how challenging proving is for newcomers to this activity. Proving is especially challenging for students who transition from secondary to university mathematics.

The Scholarship Student will analyze written proofs created by first-year students. The aim of the analysis is to identify common issues, blind spots, and successes, to make conclusions about students’ mathematical ways of thinking behind the proofs.

My former Scholarship Students presented the results of their Summer Projects at educational conferences, and submitted papers to international research journals.

This project is intended for all students with a solid mathematical background and a genuine interest in education. Students who completed MATHS 202, MATHS 303, MATHS 399 and/or have some teaching experience are especially encouraged to apply.

Project

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

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.

Project

Many of the equations important in mathematics and its applications have strong links to geometry. This includes many equations important for physics. Geometry can often be used to solve the equations or, in other instances, to show that the equations cannot be solved!

The project will be to investigate these phenomena in very simple situations. It is suitable for a student with a good background in calculus and linear algebra, or a physics background.

Project

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

You will work in a research team with two other scholars on three different projects focussed on students’ voice in mathematics education. You will:

1. Engage with survey data related to student learner identity and capture an authentic student voice by creating a playscript
2. Analyse data on students’ memorable events from their mathematics studies and how these shaped their attitudes towards the subject
3. Review the use of creative writing methodologies for conducting mathematics education research

Applicants should be interested in mathematics education, and willing to learn about alternative research methodologies.

Project

In recent years, we have seen an increase in instructional maths videos shared online, both as part of standard university teaching, as well as on websites such as YouTube, Khan Academy, etc.

Acknowledging this trend is likely to continue to grow, this project aims at identifying strengths and weaknesses of mathematics teaching in online platforms.

The scholarship student will collect and analyse online instructional videos, addressing aspects such as interactivity, engagement, gesturing, and visualisation, with the aim of offering consequent implications for research and practice.

Applicants should be interested in mathematics education, and be curious about recent changes and trends in online and in-person teaching and learning.

Project

One way to approximate a function on a set K is by polynomial interpolation: construct a polynomial whose values agree with the function at a number of points (called nodes) distributed on K. If K is an interval [a,b], equally spaced nodes may give polynomials that are badly behaved near the endpoints, even if the function is very nice i.e. analytic (see: Runge phenomenon on Wikipedia). Replacing equally spaced nodes with Chebyshev nodes solves the problem.

In this project, discover why Chebyshev nodes work. Properties of good interpolation nodes are based on potential theory.

Prerequisite: MATHS 250

Project

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

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 Schr\’’odinger (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 Draup- ner 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, Ob- servation 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

Active fluids occur in different systems, from animal herds [1], schools of fish [2], flock of birds [3], insect swarms [4] and bacterial colonies [5-7]. We want to consider the over-damped dynamics of a collection of such particles when they are also subject to close range interactions with their neighbours that align them. A mean-field description of such a system involves describing the evolutions of the self-propulsion speed, and a measure of the nonlinear interaction [8-10].

Two dimensional computations of the model will allow us to explore the parameter space and identify the possible structures that can arise [11].

An introduction to numerical methods and some programming is desirable for this project.

References

[1] J. K. Parrish and W. M. Hamner (eds), Animal Groups in Three Dimensions, Cambridge University Press (1997).
[2] S. Hubbard, P. Babak, S. Sigurdsson and K. Magnusson, Ecol. Model. 174, 359 (2004).
[3] Physics Today 60, 28 (2007); C. Feare, The Starlings, Oxford University Press (1984).
[4] E. Rauch, M. Millonas and D. Chialvo, Phys. Lett. A 207, 185 (1995).
[5] C. Dombrowski, L. Cisneros, S. Chatkeaw, R. E. Goldstein and J. O. Kessler, Phys. Rev. Lett. 93, 098103 (2004); A. Sokolov, I. S. Aranson, J. O. Kessler and R. E. Goldstein, Phys. Rev. Lett. 98, 158102 (2007); E. BenJacob, I. Cohen and H. Levine, Advances in Physics 49, 395 (2000); B. M. Haines, I. S. Aranson, L. Berlyand and D.A. Karpeev, Phys. Biol. 5, 046003 (2008).
[6] J. H. Kuner and D. Kaiser, J. Bacteriol. 151, 458 (1982); M. S. Alber, M. A. Kiskowski and Y. Liang, Phys. Rev. Lett. 93, 068102 (2004).
[7] X. -L. Wu and A. Libchaber, Phys. Rev. Lett. 84, 3017 (2000); X. -L. Wu and A. Libchaber, Phys. Rev. Lett. 86, 557 (2001).
[8] D. Mizuno, C. Tardin, C. F. Schmidt and F. C. MacKintosh, Science 315, 370 (2007).
[9] V. Schaller, C. Weber, C. Semmrich, E. Frey, and A. R. Bausch, Nature 467, 73 (2010).
[10] F. J. N´ed´elec, T. Surrey, A. C. Maggs and S. Leibler, Nature 389, 305 (1997).
[11] A. Gopinath, M. F. Hagan, M. C. Marchetti and A. Bhaskaran, Phys. Rev. E 85, 061903, 2012.

Project

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

Pursuit–evasion is 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 in discrete time steps, that is, players make one move after the other rather than moving continuously.

Depending on your background or interests, in this project you will either study how the geometry and topology of the environment influences possible strategies, or the effect of random moves on the game.

The project is suitable for students who have enjoyed an introduction to combinatorics (e.g. COMPSCI 225), background in geometry, topology, or probability theory will be an asset.

Project

New Zealand’s declining achievement in mathematics at the school level is a severe problem that can negatively impact national development and prosperity (Mullis et al., 2020; OECD, 2019). The country’s underachievement in mathematics has been identified as an issue for some time. For example, the 1994/1995 Third International Mathematics and Science Study reported that the standard of mathematics learning in New Zealand was below the averages of fifty other countries in number (place value, fractions, and computation), measurement, and algebra. In response to this, the government committed $75 million to introduce the Numeracy Development Project (NDP) from 2000 (Young-Loveridge, 2010). The NDP was part of a complete review of New Zealand’s curriculum, which began in the 1990s and emerged in the completed form with the 2007 National Curriculum (Ministry of Education, 2007).

However, New Zealand’s average mathematics achievement has continued to decline even more. This is evidenced by substantial declines in achievement indicators 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, which accounts for all participating countries. 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 (Morrow et al., 2021). This pressing issue needs to be researched.

This project will give you an opportunity to participate in research activities to shape new research projects such as literature reviews, data extraction and/or analysis.

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.

References

Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. Retrieved from Boston College, TIMSS & PIRLS International Study Center website: http://timssandpirls.bc.edu/timss2015/international-results/

OECD (2019). PISA 2018 Results (Volume I): What Students Know and Can Do, PISA, OECD Publishing, Paris, https://doi.org/10.1787/5f07c754-en

Young-Loveridge, J. (2010). A decade of reform in mathematics education: Results for 2009 and earlier years. Findings from the New Zealand Numeracy Development Projects 2009, 15-35.

Morrow, N., Rata, E., & Evans, T. (2022). The New Zealand mathematics curriculum: A critical commentary. STEM Education, 2(1), 59-72. https://doi.org/10.3934/steme.2022004

Project

The aim of the project is to identify and develop effective ways to provide professional development to mathematics teachers at the 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 and will allow you to participate in research activities such as undertaking a literature review, data collection and/or analysis.

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

Blended learning, the integration of face-to-face and online teaching and learning, is being widely adopted as the ‘new normal’ in course delivery across higher education. In mathematics courses, this new mode of instruction is commonly seen at all levels, yet the extent to which it is effective raises important questions about its pedagogical merit.

The recent unprecedented worldwide shift to online teaching as the emergency response to the Covid-19 pandemic opens up new frontiers in investigating novel educational delivery at scale.

Participating in this project will give you an opportunity to be part of such research investigation. You may be engaged in conducting a literature review, data collection and/or analysis.

This project is suitable for all students with a solid mathematical background and an interest in educational issues. Students who completed MATHS 302 or have some tutoring experience are encouraged to apply.

Project

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

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 student's background.

Project

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 he complex numbers, where the multiplication is not commutative (in general).