School of Mathematics

Student Research Article - Ross Grassie

PhD Student Ross Grassie has written the following article as part of our series of Student Research Articles!

My research:

Since Einstein first introduced the world to his ideas on relativity in 1905, symmetry has emerged as one of the key ingredients with which we can seek to understand our universe. Earlier theoretical physicists, such as Newton and Maxwell, placed the discovery of dynamical laws as the primary goal of the physicist. These laws would dictate how objects moved about space and time, and, with this knowledge, we could describe the natural world. Einstein's critical insight was that each of these laws arises as a direct consequence of symmetry. In particular, the equations describing these laws are invariant under the symmetries of space and time.

Einstein came to this revelation when creating special relativity, and, in this theory, it is the Poincaré group that is best suited to describe space and time. The resounding success of both special relativity and general relativity, which would appear ten years later, meant that the Poincaré group took centre stage for much of the twentieth century. However, where did this leave Newtonian physics and its Galilean group of symmetries? Newton's ideas played a crucial role in developing our understanding of the universe for nearly two centuries before Einstein was born. Surely these theories must still be vital for informing us on how the natural world works; indeed, there's an abundance of physical phenomena classical ideas are better suited to describe. Thus, we arrive at what may feel like an odd conclusion. We need multiple theories to provide the sharpest picture of nature. However uncomfortable for a field that is famously interested in finding one omnipotent theory of everything, we may proceed to ask: can we find all the theories we need for a complete and detailed picture of nature?

As a starting point for this investigation, we can remove all objects from our universe and focus exclusively on finding all the possible models of an empty space-time. Given that a specific symmetry group describes each model, we can aim to use these groups as the organising principle in a classification of space-times. Using this approach, my supervisor José Figueroa-O'Farrill and his collaborator Stefan Prohazka identified 16 different groups we could use to describe space-time.

Having found all of these models, we can now see what happens when we want to introduce objects into our universe. The first question we may ask ourselves in this evolution of our investigation is, can we continue to use symmetry as our guiding principle? Interestingly, the answer to this question is yes.

After being placed front and centre in the 1900s and 1910s due to the success of general relativity, it was only natural to explore symmetry's role in describing sub-atomic particles when particle physics took off during the birth of quantum mechanics in the 1920s. These investigations led to researchers again using groups, but this time they were describing the interactions between the building blocks of matter. Therefore, with symmetry again playing a crucial role, this time in the physics of objects, it stands to reason that it may be kept as our guide.

We can now ask, which particle symmetry do we want to look at first? For a theoretician, one choice may instantly come to mind, supersymmetry. Since its proposal in the 1960s, supersymmetry has received an incredible amount of interest, and for several good reasons. This symmetry promises to solve numerous open problems in the physics literature and is a foundational concept underlying one of the most famous ideas in physics, string theory. Additionally, and crucially for my research, supersymmetry can be elegantly woven with the symmetries of space-time to create a particular group called a supergroup. Analogously to the way groups are associated with models for space-time, supergroups are related to models for superspace; a mathematical enhancement of space-time that ties supersymmetry and space-time symmetry together.

With this enhancement of the symmetries of space-time, we can aim to use these supergroups as the organising principle in a classification of superspace models. Using this approach, my supervisor and I identified 27 different supergroups we could use to model superspace. Notice that the addition of supersymmetry caused a jump in the number of models from 16 to 27. Therefore, perhaps unsurprisingly, as we add more to our theories, the number of models will only increase. Although this may cause some panic (there are a lot of things in the universe we need to add to our theories), it should also cause some excitement. Going out, finding different and exotic particle symmetries to incorporate, and discovering all the possible models of physics is a fascinating field of research.

But, before we go adding any more to our theories, our result means we currently have 27 different models; 27 playgrounds to put objects into and find out how they move and interact; 27 different worlds waiting to be explored.

A bit about me:

First Year

During the first academic year of my PhD, I took several SMSTC courses on algebra, geometry, and topology. SMSTC courses aim to give each PhD student a broad knowledge of mathematics, extending beyond their research focus to create wellrounded mathematicians. Given my background in computer science and theoretical physics, and, therefore, not having undertaken any pure mathematics before, these courses were a fantastic opportunity to add to my mathematical toolkit. In particular, the algebra and geometry courses were incredibly useful: they gave me an excellent platform from which to contribute first to Jose and Stefan's paper on the geometry of particular homogeneous spaces, and, subsequently, to complete my superspace project with Jose. Because of the extensive range of topics discussed during these courses, I now comfortably sit in on many seminars discussing algebra and algebraic topology.

In the summer of 2018, I completed a range of calculations for the paper. More specifically, for the homogeneous spacetimes classified in, I determined numerous geometric properties, including the action of the kinematical symmetries, their vielbein and soldering form, the space of affine connections, calculating their torsion and curvature, and the symmetry algebra of the invariant structures. As Stefan was in Brussels at the time, we used Github to keep exchange drafts of the paper and Skype for weekly meetings.

Second Year

The second year of my PhD was my first opportunity to propose a project of my own, which lead to the paper. During this project, in addition to the tools presented in the paper itself, I developed my mathematical knowledge further with Lie (super) algebra cohomology and a more indepth understanding on Clifford algebras. I also learnt how to schedule my time to achieve several goals at once. I created a personal Slack channel to set targets, manage my time, and store ideas and notes from talks. It was also during this time that I began making a conscious effort to write more presentations. Within the school of mathematics, I gave talks at the Mathematical Physics group meeting, the BMS study group within the mathematical physics department, and the postgraduate colloquium. The first talk was based on and discussed the Hopf algebra structure underlying the renormalisation in quantum field theory; the second concentrated on the introduction of superrotations into the BMS algebra; the final talk was a general audience level discussion on the homogeneous spacetimes presented in. Outside the department, I gave presentations on my research at the Young Theorists' Forum in Durham and the Strings, Cosmology, and Gravity Student Conference in Munich. During the summer of 2019, I also attended the String Math conference in Uppsala.

Having spent the majority of my first year completing courses, I strove to engage more actively in the research community within the School of Mathematics; therefore, I became a postgraduate representative (PG rep). As a PG rep, I sat in on several school meetings on topics such as postgraduate admissions and the running of SMSTC courses, while also representing PhD students at school forums. Other aspects of this role included the arranging of social events for PhD students, the organising of the school Christmas party, and running the annual school trip to Firbush, an outdoor activity centre in the Trossachs.

Third Year

This last year, I focussed more on building projects. Applying the techniques from to the Carroll superspace in, I aimed to analyse the supersymmetric BMS group. However, realising that this investigation required the description of a superCarroll space in one dimension lower than the definition I had, I set up the classification problem for this dimension. While completing this setup, discussions with my supervisor and a visiting researcher lead to a separate project, which I am currently finishing off. This paper is my first solo paper and classifies the N = 1 and N = 2 superextensions of the Bargmann and NewtonHooke algebras in 3 + 1dimensions. These are the symmetry algebras which describe nonrelativistic massive particles in flat and (A)dS space, respectively.

In addition to these three projects, I continued to attend workshops, seminars and lecture series on nonLorentzian geometry, which is the area of mathematical physics into which my research falls. I have also continued to progress my knowledge of both mathematics and physics through participation in seminars on homological mirror symmetry, Coulomb branch varieties, Lie algebroids and Poisson geometry, and exceptional field theory, and, subsequently, giving a presentation to the Mathematical Physics group on Coulomb branch varieties.

Throughout my PhD, I have tutored undergraduate and postgraduate mathematics courses, from the first and second year "Proofs and Problem Solving" and "Engineering Mathematics", to the final year "Quantum Information" and "Geometry of General Relativity". Wanting to improve my tutoring skills, I began the work to receive category 1 of the Edinburgh teaching award (EdTA); however, due to personal circumstances and the subsequent outbreak of Covid19, I was unable to complete the award this year. This award requires the writing of two brief statements detailing my knowledge of teaching methods and three reflective blog posts on my teaching practice, evidencing my commitment to professional values. It is accredited by the UK Professional Standards Framework (UKPSF) and is equivalent to an Associate Fellowship of the Higher Education Academy (HEA).