Significant research is being carried out in the Department in these areas of Mathematics:
- Graph theory and combinatorics
- Mathematical and computational relativity
- Mathematics of evolutionary biology
- Numerical analysis and uncertainty quantification
- Point patterns and processes
- Quantitative Genetics
- Semiclassical and harmonic analysis
- Stochastic integro-differential equations and their applications
Research group: Stochastic integro-differential equations and their applications
Fractional derivatives are almost as old as their integer-order counterparts, but recent successful applications have reinvigorated the field. The recently discovered link between fractional derivatives and stochastic processes with heavy tails provides a fresh perspective. Stochastic solutions to some of these fractional PDEs are Levy motions, generalising the Brownian motion solution to the classical diffusion equation. The extension of the theory to include stochastic integro-differential equations allows the investigation of the effect of added noise/stochasticity to the system. Heavy tailing occurs whenever there is a power-law probability of catastrophic events, and therefore the approach has promising applications in hydrology, ecology (invasion of species), epidemiology, chemical engineering (for example, build-up on electrodes), physics (for example, rays going through the atmosphere), economics (stock-market), meteorology (rainfall patterns, flood events), etc., wherever these power-laws are observed.
Borid Baeumer and Mihály Kovács are working on theoretical aspects of the problem such as well-posedness, uniqueness and convergence of numerical approximations as well as collaborating with scientists on applying the results. These collaborations are invaluable as they often lead to new and relevant theoretical questions as well.