Academic visitors to the Department of Applied Computing

During the latter part of the Autumn term Dr Stuart Hall, a research lecturer in Mathematics in the Department, hosted a visit from Dr Wafaa Batat. Wafaa is a researcher at the École Nationale Polytechnique d’Oran, Algeria who works in a field of mathematics known as differential geometry. She studies a type of geometry known as a Ricci Soliton which is a generalization of the type of shape that occurs in Einstein’s theory of general relativity.

During her visit they studied a similar geometry known as a quasi-Einstein metric and found a new way of describing some four-dimensional solutions of the quasi-Einstein equation found in 2004 by three physicists Hong Lü, Don Page and Christopher Pope. Each solution has two two-dimensional holes (a bit like the space inside a football) in it and using their new description, Wafaa and Stuart were also able to say how large the holes are relative to one another (technically this is known as determining the cohomology class of the metric).

What is very exciting about this work is how one could adapt the ideas to try to find solutions to the quasi-Einstein equations on four-dimensional shapes with three two-dimensional holes in them. It turns out that the equations in this case are much more complicated and it is so far impossible to even say if they can be solved, let alone trying to describe a solution. In 2014 Stuart and his collaborator Dr Thomas Murphy from California State University Fullerton wrote a computer program to search for solutions to Einstein equations on spaces with three holes and successfully recovered a solution that was shown abstractly to exist in 2007 by mathematicians Xuixiong Chen, Claude LeBrun and Brian Weber (this was a major breakthrough as it was the first new solution of Einstein’s equations found in 4 dimensions for about 20 years). Tommy visited the Department in January 2015 and some progress was made in adapting the algorithm from Einstein metrics to quasi-Einstein metrics. Stuart and Tommy hope to finish the new algorithm and set it running in the spring of this year.

Wafaa’s visit was funded by a grant from the London Mathematical Society (which Stuart is a proud member of) and Tommy’s visit was partly funded by a Dennison Research Grant from The University of Buckingham. The results of both visits appear in the preprint “Conformally Kähler geometry and quasi-Einstein metrics” written jointly by Stuart, Tommy, Wafaa and Ali Jizany. Ali is a DPhil student in the Department and is studying mathematics. He checked many of the equations used in the paper and provided some Matlab calculations of certain parameters that are needed to describe the solutions.