A one dimensional analysis of singularities of the d-dimensional stochastic Burgers equation.

This thesis presents a one dimensional analysis of the singularities of the d- dimensional stochastic Burgers equation using the 'reduced action function'. In particular, we investigate the geometry of the caustic, the Maxwell set and the Hamilton-Jacobi level surfaces, and describe some turbulent phenomena. Chapter 1 begins by introducing the stochastic Burgers equation and its related Stratonovich heat equation. Some earlier geometric results of Davies, Truman and Zhao are presented together with the derivation of the reduced action function. In Chapter 2 we present a complete analysis of the caustic in terms of the derivatives of the reduced action function, which leads to a new method for identifying the singular (cool) parts of the caustic. Chapter 3 investigates the spontaneous formation of swallowtails on the caustic and Hamilton-Jacobi level surfaces. Using a circle of ideas due to Arnol'd, Cayley and Klein, we find necessary conditions for these swallowtail perestroikas and relate these conditions to the reduced action function. In Chapter 4 we find an explicit formula for the Maxwell set by considering the double points of the level surfaces in the two dimensional polynomial case. We extend this to higher dimensions using a double discriminant of the reduced action function and then consider the geometric properties of the Maxwell set in terms of the pre-Maxwell set. We conclude in Chapter 5 by using our earlier work to model turbulence in the Burgers fluid. We show that the number of cusps on the level surfaces can change infinitely rapidly causing 'real turbulence' and also that the number of swallowtails on the caustic can change infinitely rapidly causing 'complex turbulence'. These processes are both inherently stochastic in nature. We determine their intermittence in terms of the recurrent behaviour of two processes derived from the reduced action

The new normal: What does maths and stats support and teaching look like post pandemic?

The workshop “The new normal: What does maths and stats support and teaching look like post pandemic?” took place on the 1st of December 2022 and discussed the changes to mathematics and statistics support since the lifting of restrictions after the pandemic. The event consisted of five short talks where presenters explored the changes that had occurred to mathematics and statistics support at their institutions and concluded with a discussion on how to effectively combine online and face-to-face support and how to increase engagement in all forms of support. This report summarises the talks and discussion, concluding with some thoughts on the changes still required and how we can support each other

Engagement and online mathematics enrichment for secondary students

Online may not be the ideal format for a mathematics enrichment event, but in some circumstances, it may be the only option available. This article considers a mathematics enrichment programme consisting of a series of masterclasses which were held live online for secondary students in the UK during the Covid-19 pandemic. The series of masterclasses were part of the Royal Institution of Great Britain’s Mathematics Masterclass Programme which runs annually across the UK. In this study, we investigate how and to what extent students were able to engage with this series of online masterclasses. Learner engagement is researched through in-session observations, student work, attendance data, participant feedback and interviews. While the online masterclass series lost some of its traditional in person features, such as hands-on live social interaction and a university environment, it appeared that the participants perceived the online sessions as interactive enabling them to both enjoy the sessions and enjoy learning mathematics in the sessions. The evidence found suggests that the participating students could engage behaviourally, emotionally and cognitively in online mathematics enrichment. However, constructing mathematical knowledge in online sessions can be difficult for some students and social interaction may need to rely on existing social groups among school friends

Further Mathematics, student choice and transition to university: part 2—non-mathematics STEM degrees

This is the second paper reporting the results of a study investigating student choices of optional post-16 advanced-level (A-level) Mathematics and Further Mathematics qualifications in the UK and their impact on the transition from school to university mathematics. Here, the opinions of non-mathematics Science, Technology, Engineering and Mathematics (STEM) undergraduate students (all of whom had previously studied A-level Mathematics) were accessed via a survey and individual interviews. The study found that Further Mathematics qualifications are perceived as advantageous for non-mathematics STEM degrees by students once they are at university but not when making A-level choices. While the students often perceived mathematics positively, this appears to influence the choice of A-level Mathematics but not Further Mathematics. The lack of support from teachers and parents, the lack of perceived utility of Further Mathematics qualifications and a perception that Further Mathematics is only useful for studying a mathematics degree could all be factors affecting the uptake of Further Mathematics. The identified perceived impact of Further Mathematics on the university transition is linked to studying more pure mathematics which may give students a better understanding of how to apply mathematics in the context of their degree. Some comparisons between the findings in Parts 1 and 2 of the study are included which suggests that the Further Mathematics qualification is better serving students intending to study a non-mathematics STEM degree rather than mathematics itself

Further Mathematics, student choice and transition to university: part 1 - Mathematics degrees

The transition from studying mathematics at school to university is known to be challenging for students. Given the desire to increase participation in science, technology, engineering and mathematics subjects at degree level, it is important to ensure that the school mathematics curriculum is providing suitable preparation for the challenges ahead, and yet remains both accessible and popular. This two-part study investigates student choices of studying the post-16 A-level Mathematics and Further Mathematics qualifications in the UK and their impact on the transition from school to university mathematics. Student opinions were accessed via a survey of undergraduate students and also individual interviews. This first part of the study considers the responses of mathematics undergraduate students and finds that both those who studied Further Mathematics and those who did not perceive studying Further Mathematics as advantageous for their degree courses. However, the advantages identified mostly relate to the familiarity with topics, while students still feel unprepared for studying more abstract and proof-based mathematics. The study found that some factors which may be beneficial for transition currently lie outside the mainstream school mathematics syllabus and include studying through blended learning provided by the Further Mathematics Support Programme, practicing more advanced extension exam papers and attending university outreach events. The choice of Further Mathematics is found to be influenced by the attitudes of the students, their teachers and their parents, to both mathematics as a subject and to Further Mathematics as a qualification as well as student perceptions of Further Mathematics and their plans in terms of degree and university choice

Choosing Further Mathematics

'Education in the UK is failing to provide the increases in the numbers of school-leavers with science and mathematics qualifications required by industry, business and the research community to assure the UK's future economic competitiveness' (The Royal Society, 2008: 17). Furthermore, the proportion of students in Wales following mathematics courses post 16 is lower than in England (GSR, 2014). In particular, although the situation has improved, fewer students in Wales choose to study further mathematics (FM). This paper explores the reasons for student choices in mathematics and FM in order to make recommendations about how to increase participation. Phase one of the study used a questionnaire to access the opinions of students studying mathematically based courses in sixth forms and colleges to explore the reasons behind their choices and the factors influencing their progression or otherwise in mathematics. In phase two, small focus groups of students in selected schools and colleges were interviewed to enrich the questionnaire data and provide further insight into their decisions. The study identified a lack of information from peers, siblings, parents and teachers about FM as a factor restricting choice. Current models of delivery contribute to the false perception that FM is harder than mathematics and only suitable for the most talented mathematicians. We suggest: developing teachers' knowledge and skills so that whenever possible students can be offered FM as a fully timetabled subject; promoting FM to parents; and establishing student champions to encourage participation

The Divine Clockwork: Bohr's correspondence principle and Nelson's stochastic mechanics for the atomic elliptic state

We consider the Bohr correspondence limit of the Schrodinger wave function for an atomic elliptic state. We analyse this limit in the context of Nelson's stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler's laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Kelperian orbit for eccentricities greater than 1/\sqrt{2} which do not occur classically.Comment: 42 pages, 18 figures, with typos corrected, updated abstract and updated section 6.