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.

Image created by Artie Prendergast-Smith (Loughborough) using surfer.

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-20th 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.

Dr Hamid Abban was the theme lead for the Loughborough Institute of Advanced Studies semester in Geometry, 2018/19. You can read about the theme here.

Dr Hamid Abban is the Principal Investigator for the Loughborough mini-Centre for Doctoral Training in Geometry in Robotics, Algorithms, and Design (GRAnD). Read more about the programme 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 organisers of the “COW” UK 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:

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

Integrable Days 2020
Integrable Days 2019
Integrable Days 2018

  • Sobhi Berjawi: Second-order PDEs in three (four) dimensions carrying Einstein-Weyl (self-dual) characteristic conformal structure (supervised by Professor Jenya Ferapontov and Dr Vladimir Novikov)

  • Erroxe Etxabarri Alberdi: Extremal metrics on singular Fano 3-folds (supervised by Dr Hamid Abban)
  • 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 Yang-Baxter dynamics (supervised by Professor Sasha Veselov and Dr Artie Prendergast-Smith)
  • Tim Grange: Birational geometry, in particular positivity of divisors on blowups of products of projective spaces and characterisation of log Fano varieties and Mori dream spaces in this setting (supervised by Dr Alexey Bolsinov and Dr Artie Prendergast-Smith)
  • James Jones: Degenerations of K3 surfaces (supervised by Dr Alan Thompson)
  • Ronan Kerr: Moment polytopes for spherical integrable systems (supervised by Dr Alexey Bolsinov and Professor Sasha Veselov)
  • Laura Mallinson: Classification of complete intersection flips in dimensions 3 and 4 (supervised by Dr Hamid Abban)
  • Casper Oelen: Symmetry in infinite dimensions and automorphic Lie algebras.
  • Joseph Prebble: K3 surfaces polarised by lattices of high rank (supervised by Dr Alan Thompson)
  • Luis Santana Sánchez: Algebraic geometry and commutative algebra, in particular the study of certain Mori dream spaces via the theory of Newton-Okounkov bodies and their connection with open conjectures on the postulation of fat point schemes (supervised by Dr Alexey Bolsinov and Dr Artie Prendergast-Smith)