2023 Summer Research in Maths (SRIM) Projects

The aim of this project is to build computational tools in the Julia programming language for spike train analysis. The starting point will be to implement some core methods for reading in electrophysiological data sets, and performing "burst analysis" whereby groups of spikes occur in quick succession to generate a burst. After this, I would like the student to implement pairwise methods for detecting correlated activity in pairs of spike trains. Time permitting, the student would be able to develop new methods for detecting 'functional connectivity' between all neurons in a recording of 60--4000 neurons recorded simultaneously. The student will also need to develop good data visualisation methods for this project, as they are critical for the efficient communication of large volumes of data. Finally, I would like reproducible test data sets to be created, so that others can evaluate our methods, and compare with their own.

I have chosen the Julia programming language for this project. Most of our group's work to date has been in the R programming language. Part of the aim of the project would be to evaluate the utility of Julia. Based on earlier work from two summer students, I believe that Julia has a lot of promise https://sje30.github.io/catam-julia/ particularly in making code that is easy to write and fast. Efficiency (in both speed and memory usage) is an issue here as we have large volumes of data. In our current R implementations, we often have to rewrite key portions in the C programming language, which makes it harder to develop, debug and maintain.

A feasible output from this program would be to make the preliminary version of the toolkit freely available on the internet, visible through the Julia package manager. This would hopefully then seed further work in this area in collaboration with partners.

Some key papers from our earlier work:

Gelfman S, Wang Q, Lu Y-F, Hall D, Bostick CD, Dhindsa R, Halvorsen M, McSweeney KM, Cotterill E, Edinburgh T, Beaumont MA, Frankel WN, Petrovski S, Allen AS, Boland MJ, Goldstein DB, Eglen SJ (2018) meaRtools: An R package for the analysis of neuronal networks recorded on microelectrode arrays. PLoS Comput Biol 14:e1006506 Available at: http://dx.doi.org/10.1371/journal.pcbi.1006506.

Cotterill E, Charlesworth P, Thomas CW, Paulsen O, Eglen SJ (2016) A comparison of computational methods for detecting bursts in neuronal spike trains and their application to human stem cell-derived neuronal networks. J Neurophysiol 116:306-321 Available at: http://dx.doi.org/10.1152/jn.00093.2016.

DPMMS, Stats Lab

Zhu, Z., Wang, T. and Samworth, R. J. (2022) High-dimensional principal component analysis with heterogeneous missingness. J. Roy. Statist. Soc., Ser. B, 84, 2000--2031.

Follain, B., Wang, T. and Samworth, R. J. (2022) High-dimensional changepoint estimation with heterogeneous missingness. J. Roy. Statist. Soc., Ser. B, 84, 1023--1055.

Loh, P.-L. and Wainwright, M. J. (2012) High-dimensional regression with noisy and missing data: provable guarantees with nonconvexity. Ann. Statist., 40, 1637--1664.

While a great deal is known about the speed of convergence of classical L_2 orthogonal series to analytic functions in compact intervals, there is definite absence of theory once we remove any phrase ("classical", "L_2", "analytic" or "compact") from this sentence. Indeed, all we have are highly incomplete computational and asymptotic results that indicate that something strange is going on.

The project will have three components:
1. Learning new and non-classical means to construct L_2 and H_2^p orthogonal series on real intervals;
2. Computational experiments to indicate the speed of convergence and hint upon what exactly we should be proving;
3. An attempt to determine the rate of convergence in some situations, e.g. by using a variation on the theme of the Bernstein ellipse.

From a regional outbreak in Wuhan, Covid-19 became a global pandemic. In the early stages of the pandemic, travellers from Wuhan who subsequently tested positive for Covid-19 were likely to have contracted the disease during their stay in Wuhan. Using data (available from the paper referenced) on the dates travellers entered and left Wuhan, and the dates at which they reported symptoms, the goal of the project would be to estimate the distribution of the incubation period, that is the time from transmission of the virus to the onset of symptoms. The referenced paper makes some parametric assumptions about the form of this distribution, but the aim here would be to construct a method that avoids making such assumptions, and develop theoretical guarantees on its performance.

This project would likely be suitable for a 3rd year undergraduate with some prior knowledge of theoretical statistics as for example taught in the Cambridge Part II course Principles of Statistics.

High-dimensional data, where the number of variables is large compared to the number of observations, are becoming increasingly common across a range of scientific disciplines and industry. Classical statistical methods often perform poorly for such data, or do not work at all. There is therefore a need to develop new methods that are able to cope well with the high-dimensionality of the data.

Great strides have been made in this direction over the last couple of decades, and we now have well-established methods for providing point estimates of high-dimensional parameters in a variety of common statistical models. Whilst this represents great progress, much remains to be done. One of the key goals of statistical methodology is uncertainty quantification, and there is a need to develop high-dimensional analogues of classical statistical tools for forming confidence intervals and performing hypothesis tests, for example. One approach that has gained great traction in recent years is that of the debiased Lasso (see references). While in theory this can provide confidence intervals for parameters even in challenging high-dimensional settings, there remain substantial challenges in using it in practice in terms of both its statistical and computational performance. This project could look at ways of addressing these issues, with the aim of developing a software package for the statistical computing language R implementing the methodology. This project would be particularly suitable for those with prior computing experience in R and / or C++.

The goal of this project is to develop provably convergent and efficient algorithms for computing spectral properties of operators (e.g., PDEs, integral operators etc.) on surfaces. Surface-bound phenomena arise in a wide variety of applications, including electromagnetics, plasma physics, biological pattern formation, bulk–surface diffusion processes, biomechanics, and fluid dynamics. However, computing spectral properties in the ensuing infinite-dimensional Hilbert spaces is challenging! This project will combine several recent breakthroughs in this area.

To deal with eigenvalue problems (discrete spectra), the student will combine methods based on complex contour integration and residuals to compute error bounds. These methods will make use of high-order methods that solve PDEs on surfaces. As well as combining and generalising previous ideas, there is considerable scope here for novelty. For example, by combining with eigenvalue asymptotics, this may lead to the first method that can accurately compute high frequency eigenmodes on generic surfaces. Another avenue could be solving time-dependent PDEs via contour methods. A second part of the project will focus on so-called continuous spectra. These provide a generalisation of diagonalisation of linear operators in infinite dimensions. Again the tool here is solving shifted linear systems through surface PDE software. Finally, if time, the project will look at the foundations of computation for surface spectral problems – i.e., proving rigorous theorems on what can and cannot be done.

This is a potentially high impact project that touches upon a range of ideas in analysis and computing. A successful outcome will be transforming the results of this project into a paper for a peer-reviewed journal. There is room to complete this final outcome after the completion date, and such an outcome would also provide a competitive edge for the student if they apply to further studies (i.e., PhD).

This project will involve evolving configurations of cosmic strings using an adaptive mesh refinement code. Cosmic strings are topological defects that may have formed due to a phase transition in the early Universe, and are a potential source of dark matter axions and gravitational waves.

There is scope for discussion about the type of project that would interest the student. Some possible ideas could be:
a) investigating simple string/anti-string configurations and their decay channels
b) investigating cosmic string `cusps' (potential GW sources) and/or networks
c) learning about numerical relativity and its applications/challenges in cosmic string simulations
d) a more computing-focussed angle involving learning about CPU vs GPU architectures and optimising code for GPUs

My publications in this area: https://arxiv.org/abs/2211.10184, https://arxiv.org/abs/2201.03458, https://arxiv.org/abs/2112.10567, https://arxiv.org/abs/1910.01718

Code website: https://www.grchombo.org/

Background reading: the papers referenced in my first author papers above, and Vilenkin and Shellard, Cosmic Strings and Other Topological Defects

Topological quantum computation (TQC) remains one of the most promising roads towards practical quantum computers. In a topological quantum computer, computations are performed via the braiding of anyons, exotic quasi-particles that arise in 2 dimensional models exhibiting topological order. The topological nature of the braiding process makes it particularly robust against local perturbations, and this resistance to noise is what makes TQC such a prominent contender for a practical implementations in which noise is inevitable. Furthermore, it can easily be shown that even simple models, such as those with Fibonacci anyons, are universal for quantum computation.

Despite the promise of TQC, the exotic nature of these anyonic excitations makes it challenging to engineer systems that exhibit them in the lab. A different model, known as permutational quantum computing (PQC), is based on the same principles but instead works with particles that have a symmetric braiding. This means that the topological nature of these braiding operations is irrelevant, and the swapping of two particles is just a permutation. The advantage is that this model can be realized using e.g. ordinary spin-1/2 particles, which are much less exotic than anyons. In this setting, it connects to an intriguing model of quantum computation whose origins can be traced back to Roger Penrose: Permutational Quantum Computing (PQC).

Inspired by these developments, this project aims to reinvestigate the classical simulation of TQC. Recent work on generalized symmetries recasts exotic anyonic excitations in a more conventional language, and brings these anyons much closer to the types of excitations used in PQC. It is expected that many of the tools introduced to classically simulate PQC can be generalized to TQC. Given that TQC is universal however, the classical algorithms proposed for PQC must break down at some point, and one of the goals of this project is to pinpoint exactly where and why this happens.