Research in the Mathematical Sciences Institute

Some recent highlights

The Monge mass transfer problem: The Monge problem of optimal mass transfer has been one of the most famous problems in mathematics during the last two centuries. Originally proposed by Monge in the 1780s in the prosaic form of finding the best way to move a pile of soil with the least amount of work. A partial solution by Kantorovic in the early 20th century led to the development of linear programming. In 2001, Neil Trudinger and Xu-Jia Wang (Applied and Nonlinear Analysis) found a simple approach to the problem and published the first correct proof.

Banach spaces: Very few Banach spaces are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on the space is fully understood. Indeed, since the late 60's the only such spaces were, up to isomorphism, Hilbert spaces, and the classical sequence spaces. The latter all have three element lattices. With colleagues in Denmark and England, Rick Loy (Analysis and Geometry) has added a new member to this family, giving a space, built up from finite-dimensional Hilbert spaces, whose lattice is a four element chain.

Category theory: Amnon Neeman (Algebra and Topology) published a 449-page research monograph in the prestigious Annals of Mathematics Studies series. The book proves many new theorems about triangulated categories. Neeman also found a counterexample to a 40-year-old result, which had been extensively applied in the literature.

Kato square-root problem: Alan McIntosh (Analysis and Geometry) was recognised widely in the national press and international arena for his part in the solution of a 40-year-old maths problem - the Kato square root problem. This was achieved in collaboration with mathematicians from France and the USA.

Research programs

The principal research activities of the MSI are divided into eight programs with the following main interests:

Algebra and Topology: algebraic geometry, algebraic K-theory and homotopy theory, as well as the more traditional areas of finite and discrete groups, and representations of Lie algebras

Analysis and Geometry: several complex variables, Banach algebras, spectral theory of operators, harmonic analysis on Lie groups, manifolds and Lipschitz surfaces, microlocal analysis on manifolds with corners, non-commutative geometry, and applications to pde's and Maxwell's equations

Applied and Nonlinear Analysis: nonlinear pde's, variational problems, minimal surfaces and affine maximal hypersurfaces, monotonicity formulae, interior second derivative and interior curvature bounds, harmonic maps, heat flow, and the theoretical aspects of numerical analysis

Astronomy and Astrophysics: modelling of accretion disks, modelling of stars and stellar atmospheres, and fluid mechanical problems.

Computational Mathematics: from theoretical analysis of numerical algorithms to practical implementation of software on parallel supercomputers; applications include data mining, optimization and solving pde's

Mathematical Physics: exactly solved models in statistical mechanics, related combinatorics, spin ladders, chiral Potts model, theoretical morphology and stromatolites, algebraic geometry and quantum field theory

Statistical Science: applied statistics, bioinformatics, statistical genetics, biometrics, medical statistics, epidemiology, survival analysis, bootstrap methods, curve estimation, spatial statistics, data mining and robust statistical inference

Stochastic Analysis: applied probability with interests in general properties of a range of models as well as in particular models; applications include mathematical finance, telecommunication systems and epidemics

In addition, the MSI hosts two virtual centres:

Centre for Bioinformation Science (CBiS)
Centre for Financial Mathematics

Research reports

Members of the MSI are encouraged to submit papers as part of the CMA Research Reports series, which are then made freely available for download.