Efficient numerical methods for 3D underwater acoustic wave propagation

Adaptive sparse grids automatically search for the best resolution in each dimension and build a model of the data. This project will study techniques, implementations and merits of such adaptive schemes for use in predictive modelling using sparse grids.

Our projects cover computational methods used in biology, chemistry, physics, finance and machine learning. They prepare applied and pure mathematicians for collaborative work with scientists and engineers and for doing a PhD in this area.

Depending on student interest and background I can supervise a variety of topics related to algebraic topology.

Analysis (harmonic, functional, and/or stochastic) of partial differential equations.

Ancient solutions to geometric flows

Applications of geometric flows in geometry

Arithmetic algebraic geometry

How many solutions does the equation x2+y2 = z2 have if x, y, z are taken from Z/pZ? How many square-free polynomials of degree n are there with coefficients in Z/pZ? Questions of this kind have deep and seemingly unexpected connections with the arithmetic and geometry of algebraic varieties.

The discovery of strong 17 per cent circular polarisation in the IR in the Orion OMC-1 star forming region has provided a new impetus to the view that the homochirality of natural amino-acids seen in biological systems on Earth may have been triggered by asymmetric photolysis of biogenic amino acids in space.

Clustering, ie finding groups of similar objects, is a central theme in data mining. While the k-means algorithm is one of the most popular at the moment, strong contenders are based on the estimation of density

Combinatorial and Geometric Group Theory

About 50 percent of stars in our Galaxy occur in binary systems. The orbits of such systems evolve with time due to a variety of mechanisms involving mass and angular momentum loss.

Homotopy groups are extremely difficult to compute, and even the homotopy groups of spheres are only understood through some small range of degrees.

Can one write a computer program to decide whether two given rings are isomorphic? Or whether a given group is the trivial group? Or whether a given CW complex is a sphere? These are questions on the inteface of logic, theoretical computer science, and mathematics with fascinating results and open problems.