EPSRC Reference: 
EP/Y008812/1 
Title: 
Complex quantum topology 
Principal Investigator: 
Jordan, Professor D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematics 
Organisation: 
University of Edinburgh 
Scheme: 
EPSRC Fellowship 
Starts: 
01 April 2024 
Ends: 
31 March 2029 
Value (£): 
961,167

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Our research is concerned with the mathematical notion of symmetries of space and time, especially in the most physically relevant dimensions three (space) and four (space and time). Two of the most important tools for understanding space and time are the notions of invariants: computations we can do which allow us to tell distinct spacetimes apart; and symmetries: transformations we can apply to space and time, which preserve its essential structure. These notions are important for our mathematical understanding of space and time, but at a more fundamental level they encode the basic physical laws of the universe: invariants can be understood as an abstraction of experiments: things we can measure; meanwhile the elementary particles which make up our physical universe are understood in modern terms as "representations" of universal symmetries. These two facts together mean that when we understand the mathematics of space and time, we are also understanding the full spectrum of possible physical states of our universe, as well as what quantities we can meaningfully measure.
The most easily understood mathematical symmetries are the finite symmetries. We see an order two symmetry when we see ourselves refleced in a mirror, and we see a symmetry of order 24 when we look at a perfect cube. The role of such finite symmetries as they enter into our physical universe is completely understood mathematically. Much more difficult symmetries to understand are infinite, continuous symmetries  what are called complex and noncompact in this proposal. A great example of such symmetries is the symmetries of a perfect sphere, such as the surface of the earth: we can rotate a sphere along a continuum of possible axes, and along each axis we can rotate along a continuum of angles. If we now wish to study quantum mechanical systems taking place on the surface of a perfect sphere, we must understand the "representation theory"  the possible ways of translating these rotational symmetries into measurable numerical quantities.
While the notion of invariance and measurement bring to mind physical problems, one of the most remarkable and revolutionary developments in the past century has been that such a priori physical challenges provide deep and essential insight into mathematics. This is a paradigm shift from the historical interaction of mathematics and physics, wherein physical problems were solved by mathematical formalism. Today, we find that mathematicians are often unable to answer the challenges, to realise the dreams, of physicists, and that in striving to heed the call, we in fact discover deep and surprising new mathematics, structures that are interesting, important, and indeed profound, even separate from their interest to physicists.
The essential problem this research seek to address is this: mathematicians have mastered how to spread finite symmetries through space and time, in a way that is fruitful both for the mathematical study of three and fourdimensional space and time, and also gives essential insight to mathematical physicists. Mathematicians have also mastered how to analyse the representation theory of continuum symmetries, as they occur in isolation. What has not yet been mastered, and what this research concerns, is how to construct invariants of space and time which respect and incorporate continuous symmetries in variation throughout space and time and not in isolation. This is realised by both mathematicians and physicists alike as an important conceptual challenge, and the call to answer this challenge has motivated some of the most important mathematics over the preceding three centuries. We will build on this progress to solve a number of challenging problems at the forefront of this important area of reasearch.

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