4 June 2021

Q+A: Dr Robert Kurinczuk on the Langlands programme

This week we're delighted to welcome Dr Robert Kurinczuk to the University of Sheffield as he begins an EPSRC Early Career Fellowship on 'Integral Structures in the Langlands Programme'. Here he talks about his career so far and plans for the future.

Robert Kurinczuk

Can you tell us a bit about your career so far? What sort of topics have you been working on as a researcher?

I completed my PhD in 2012 under the supervision of Professor Shaun Stevens at the University of East Anglia, where I studied infinite dimensional symmetries in a branch of algebra called representation theory. Following on from this, I have held two postdoctoral fellowships, first at the University of Bristol then at Imperial College London, where I have continued my research and branched out into new areas.  The infinite dimensional symmetries I studied in my PhD, are amazingly connected to finite dimensional symmetries which are fundamental in number theory, and I have increasingly looked towards strengthening these connections and what my work on the infinite dimensional side corresponds to in the arithmetic world of finite dimensional symmetries. 

What made you want to join the School of Mathematics and Statistics in Sheffield for your fellowship?

Sheffield has excellent Number Theory and Algebraic Geometry research groups whose research interests connect to themes in my own research, making the university an excellent environment to pursue my work. The school also has great undergraduate and postgraduate programmes, and during my fellowship I hope to engage with many different people on many different topics in mathematics.

Your fellowship will focus on the Langlands programme. What is the Langlands programme, and what do you find interesting about it?

The Langlands programme is a vast web of predictions connecting arithmetic and analysis. These connections are not only beautiful from a theoretical point of view, they are proving powerful as a tool. Famously, Andrew Wiles proved Fermat’s Last Theorem (FLT) by establishing a very small example of these predictions which allowed him to translate FLT from the world of arithmetic to the world of analysis where it could be tackled with new tools. Recently, mathematical physicists have been working on connecting the predictions in the Langlands programme to bridges between physical theories (quantum field theories, string theories), with the idea that one can simplify calculations in one physical theory by moving to another. I have always believed in some sort of universality in mathematics and in science, without lines drawn by disciplines (algebra, analysis,…), and that the Langlands programme witnesses this is one of the reasons I am interested in it. 

Langlands programme

What are you looking forward to about the fellowship, and what do you hope to accomplish?

I am looking forward to working in a team, with a postdoctoral research associate and PhD student, and forming international collaborations to deepen and understand further the amazing connections in the Langlands programme. The shared excitement of collaborative research is something which I have always enjoyed. 

The Langlands programme has classically dealt with symmetries built on top of the complex numbers and the connections that it predicts have loosely been phrased as correspondences (taking an object in one world and associating to it an object in another).  Recent trends have shown that this reveals just the tip of the iceberg, and I would like to contribute new and novel results which see beneath the surface.  As I have five years, I also hope to uncover some surprises along the way –  this is part of the excitement of research!

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