# Geometry and Mathematical Physics

There has long been a deep relationship between geometry and physics, from the descriptions of planetary motion by Newton and Kepler in the 17th century, through the differential geometry of Riemann and Poincaré which plays a crucial role in Einstein’s theory of relativity, to modern applications of algebraic geometry and integrable systems to mathematical physics, which are the main topics of research in our group.

Algebraic geometry studies geometric objects defined by the vanishing of polynomial equations. Whilst the subject has deep historical roots, modern algebraic geometry is largely built on pioneering work by Serre and Grothendieck in the mid-20^{th} century. Its power lies in its ability to unify many different areas of mathematics, including analysis, topology, number theory, and cryptography. This versatility has been utilised in many recent mathematical advances, including Wiles’ famous proof of Fermat’s Last Theorem. It has also made algebraic geometry an attractive tool for physicists developing the next generation of fundamental theories, and it plays an important role in modern quantum field theories and string theory.

The theory of integrable systems studies differential equations which are exactly solvable and possess regular behaviour. Algebraic geometry plays an important role in this theory too, through deep links discovered by Jacobi, Weierstrass, and Kovalevskaya. Integrable systems are important tools in physics, where they can be used to model many phenomena. These include the movement of a solid object about its centre of mass, the behaviour of waves moving through shallow water, and the propagation of light through an optical fibre, along with many more. Recently, this theory has also included the study of discrete and quantum integrable systems, which have deep links with spectral theory, representation theory, and the theory of special functions.

The Geometry and Mathematical Physics group hosted the Loughborough Institute of Advanced Studies thematic programme in Geometry in 2018/19. You can read about the theme here.

Dr Tarig Abdelgadir has recently organised conferences on algebraic geometry in Ghana and Tanzania

Dr Alexey Bolsinov leads the “Nijenhuis Geometry” project, investigating a new area of research that studies geometric structures defined by Nijenhuis operators. The vision for this project is presented in this paper. Loughborough will be hosting a number of workshops attached to this project, the first of which took place in February 2020. Details of the workshop can be found here.

Dr Alan Thompson is one of the coordinators of the UK Algebraic Geometry Network. The UK Algebraic Geometry Network was set up in 2023 with funding from the Isaac Newton Institute; it aims to foster collaboration between algebraic geometers working in the UK, by organising major annual conferences to bring academics together, schools and workshops for graduate students, and by providing seed funding to allow academics in the early stages of collaborative projects to travel to visit one another.

Dr Alan Thompson is also one of the organisers of the “COW” algebraic geometry network, which usually runs around 6 geometry events per year at locations across the country:

The Geometry and Mathematical Physics group has recently hosted conferences in Loughborough on:

- Bolsinov-60, November 2022
- International Conference on Integrable Systems and Applications, September 2020

Loughborough also organises the annual series of “Integrable Days” workshops:

Integrable Day 2022

Integrable Days 2021

Integrable Days 2020

**Ben Gormley:**Integrable Hamiltonian equations in three dimensions and their dispersive deformations (supervised by Professor Jenya Ferapontov and Dr Vladimir Novikov)**Jack Graham:**Geometric aspects of integrability (supervised by Professor Sasha Veselov and Dr Artie Prendergast-Smith)**James Jones:**Degenerations of K3 surfaces (supervised by Dr Alan Thompson)**Joseph Prebble:**K3 surfaces polarised by lattices of high rank (supervised by Dr Alan Thompson)- Sam Evans: q-Deformed Numbers (supervised by Dr Brian Winn and Prof Alexander Veselov)