## Research groups

Our research covers the spectrum of fundamental and applied mathematics.

## Algebra

Algebraic techniques are of central importance in modern mathematics. As such the algebra group sits naturally among a number of major research topics in the department, with connections to geometric topology via group theory, homotopy theory and number theory through representation theory, and algebraic geometry through geometric representation theory.

## Combinatorics

Research interests: extremal combinatorics, graph theory, and combinatorial number theory.

## Functional analysis

Research interests: operator theory, including unbounded operators, and abstract differential equations.

## Geometry

Research includes algebraic geometry, Riemannian geometry, homological mirror symmetry and symplectic geometry. There are links with representation theory in the algebra group.

## History of mathematics

History of mathematics is a multidisciplinary subject with close ties to the History Faculty. Research interests cover mathematics and its social context from the early modern period right up to the twentieth century.

## Logic

Research interests: analytic topology, geometric stability theory, and the model theory of p-adic fields and diophantine geometry.

## Mathematical & computational finance

Due to the multidisciplinary nature of the mathematical finance research, MCFG has strong links and close collaborations with the stochastic analysis group, OCIAM, and the numerical analysis group. The recent upsurge of interest in big data further strengthens such links.

## Mathematical physics

Research interests: gauge and gravity theories (quantum field theories), string theory, twistor theory, Calabi-Yau manifolds, quantum computation and cryptography.

## Number theory

Members of the number theory group work in analytic and combinatorial number theory, arithmetical algebraic geometry, and computational number theory, with numerous and deep connections to current issues in algebra, combinatorics, geometry, topology, logic, and mathematical physics.

## Numerical analysis

The numerical analysis group develops and analyses algorithms for mathematical problems related to partial differential equations, linear algebra, optimization and other areas. The is a strong involvement in applications, with particularly close connections with OCIAM, the Wolfson Centre for Mathematical Biology, and the Centre for Nonlinear PDE.

## Oxford Centre for Industrial and Applied Mathematics

Research interests: energy, industry, geoscience, networks, finance, methodologies.

Who's who in the Oxford Centre for Industrial and Applied Mathematics

## Oxford Centre for Nonlinear Partial Differential Equations

Research focuses on the fundamental analysis of nonlinear PDE, and numerical algorithms for their solution. Current areas of interest include the calculus of variations, nonlinear hyperbolic systems, inverse problems, homogenization, infinite-dimensional dynamical systems, geometric analysis and PDE arising in solid and fluid mechanics, materials science, liquid crystals, biology and relativity.

Who's who in the Oxford Centre for Nonlinear Partial Differential Equations

## Stochastic analysis

Research interests: rough path theory, Schramm-Loewner evolution, mathematical population genetics, financial mathematics, self-interacting random processes.

## Topology

The members of the topology group have very wide ranging interests in algebraic, geometric and differential topology. Both high- and low-dimensional manifold theory (including knot theory) are represented. Particular research foci are topological quantum field theory and geometric group theory.

## Wolfson Centre for Mathematical Biology

Development and application of mathematical and computational methodologies for the understanding of key problems in the biological and medical sciences.

## Case studies

Examples case studies articulating the impact of applied mathematics include

- Influencing HIV/AIDS policy in India through mathematical modelling
- Helping the ‘Greeks’ to run faster
- Mathematics in the design and manufacture of novel glass products
- HYDRA: Rolls-Royce's standard aerodynamic design tool
- How networks shape the spread of disease and gossip
- How predictable is technological progress?
- Understanding droplets and surfaces
- Mathematical theories of consciousness
- The Universal Structure of Language