Research in Mathematics and Statistics



Seminars: This week This semester History


The school has the following research groups (hide details):

Algebra

The algebra group has worked extensively on tight closure and related questions in commutative algebra. Another major theme is the study of local cohomology; this connects with the work of the Topology group via the work of Greenlees on the local cohomology of equivariant cohomology rings. On the noncommutative side, the group has studied classes of rings such as (generalised) Weyl algebras, rings of difference operators, quantum algebras, and Frobenius skew polynomial rings.


Algebraic Geometry

The group members have a variety of interests including combinatorial algebraic geometry, moduli spaces, derived categories, enumerative invariants, mirror symmetry and cluster varieties.


Analysis

The analysis group is a school-wide collection of mathematicians who share a common interest in the development and use of analytic techniques. We encompass real, complex, functional, harmonic, numerical and stochastic analysis; operator algebras and analytic K-theory; analysis on groups, graphs, manifolds and other structures; ordinary, partial and stochastic differential equations; chaos, fractals and dynamical systems; applications of analytic methods to concrete problems in e.g. number theory, topology, probability theory, fluids and physics.


Category Theory

The group works on triangulated categories, Quillen model categories, various types of higher categories and other aspects of category theory, often with a view to applications in homotopy theory. The group has also worked extensively on operads.


Differential Geometry

Differential Geometry at Sheffield is concerned with new structures developed in response to recent work in mathematical physics and fundamental problems in differential geometry.

Kirill Mackenzie is primarily concerned with the multiple Lie theory which he initiated, an extension of the Lie theory of Lie groups and Lie algebras to double and multiple Lie groupoids and Lie algebroids. This work relies very much on the use of Poisson structures and in turn Poisson group(oid)s and Poisson actions give rise to double structures, the integrability of which is a major problem. Multiple Lie theory has given rise to the idea of multiple duality: the ordinary duality of vector spaces and vector bundles is involutive and may be said to have group Z2; double vector bundles have duality group the symmetric group of order 6, and 3-fold and 4-fold vector bundles have duality groups of order 96 and 3,840 respectively. An idea of double and multiple Lie theory can be obtained from Mackenzie's 2011 Crelle article (see below) and the shorter 1998 announcment, "Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids" (Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 74-87) and from the 2011 paper of Th. Voronov which uses supergeometric methods. "Q-manifolds and Mackenzie Theory" Comm. Math. Phys. 315, 279-310, 2012.

Simon Willerton has worked on the role of hyper-Kähler manifolds and gerbe-connections in topological quantum field theory and is interested in how curvature relates to `magnitude', a metric space analogue of the Euler characteristic.

Ieke Moerdijk works, among many other interests, on Lie groupoids and Lie algebroids, especially étale groupoids and orbifolds and their relations with foliation theory. See in particular his 2003 book with Mrcun.

The gif above is a rotating hypercube (or tesseract) from http://en.wikipedia.org/wiki/Tesseract The outline of a 4-fold vector bundle is a hypercube.


Environmental Dynamics

Research of this group is well known in the areas of Synthetic Aperture Radar, HF Radar Remote sensing, Turbulent diffusion and Meteorology. Some of its research is facilitated by NERC Centre of excellence for Terrestrial Carbon Dynamics and the group members enjoy a wide range of research collaborations.


Fluid Dynamics

This group's research covers a wide spectrum of topics in fluid dynamics. It covers vortex dynamics and turbulence, including basic problems of the Navier-Stokes equations, differential geometric characterisation of Lagrangian stability and possible singularity formation in Euler flows. The group also has interests in engineering fluid dynamics, particularly in microfluidic rheology and in interfacial flows, including the Marangoni effect. Acoustic waves, e.g. digitised speech patterns, are also studied.


Mathematical Biology

The Mathematical Biology Group uses a wide range of mathematical and statistical approaches to study questions in biology and medicine. Members of the group collaborate widely both within the University of Sheffield and further afield, through joint supervision of PhD students and postdoctoral researchers, and through membership of interdisciplinary research centres. Weekly group meetings bring together members of the group based in SoMaS with students and postdocs associated with the group but based in other departments.

See the Mathematical Biology home page for further details.

Current research interests include
  • Animal movement and biological invasions
  • Cancer growth and progression
  • Dynamics of genetic regulatory networks
  • Dynamics of infectious diseases
  • Genetic epidemiology and statistical genetics
  • Mechanisms of evolution and development
  • Pattern formation in plant and animal development
  • Statistical ecology
  • Tissue morphogenesis


Number Theory

The interests of the Number Theory group are centred on modular forms, automorphic representations and their congruences, Galois representations and their modularity, elliptic curves, arithmetic algebraic geometry, and the Bloch-Kato conjecture on values of L-functions. See the staff pages below for further details.

Please click here for the upcoming talks in the number theory seminar.

Please click here for the upcoming talks in our learning seminar.

Prospective postgraduates should note that a strong student with a good all-round knowledge of undergraduate pure mathematics would still need to do large amounts of preparatory reading to begin with. Their work is likely to have as much to do with algebraic geometry, algebraic topology, analytic functions and representation theory as with undergraduate courses on elementary number theory.


Particle Astrophysics and Gravitation

The group's interests are in Cosmology, Gravitation and Black Holes. It seeks to understand how the universe expands, why its expansion is accelerating, the classical and quantum behaviour of black holes, and the fundamental theory of space and time.


Plasma Dynamics Group



Probability

Sheffield has a proud tradition of research and teaching in both probability and statistics, dating back to the early 1950s under Geoffrey Jowett and Hilda Davies. In 1965 Professor Joe Gani was appointed as the first professor and head of the new Department of Probability and Statistics which separated from the Mathematics Departments. He established an MSc course and PhD programme which have now developed into three MSc courses and a large PhD group covering a wide range of areas including many joint projects with other university departments. The research group in probability has 5 academic staff (including 2 professors), and 3 postgraduate students. Together with statistics, the group has a seminar series with external invited speakers, and regular informal research meetings, led by members of the group.


Linked with the group is The Applied Probability Trust, which publishes two major international journals (Journal of Applied Probability and Advances in Applied Probability, both founded by Joe Gani) and which sponsors an annual lecture in Sheffield given by a leading international figure. This APT lecture takes place within the contact of a Sheffield Probability day.


Research in probability includes: branching processes; random walk; large deviations; fractals and random graphs; Levy processes; probability on groups; stochastic analysis; stochastic differential and partial differential equations and inference for stochastic processes.


Solar Physics and Space Plasma Research Centre

The work of this Centre is at the forefront of addressing theoretical and observational issues in solar and solar system physics that include solar magneto-seismology, dynamics of the solar atmosphere, solar wind, magnetosphere and Space Weather. This Centre is one of the largest and most dynamic solar and solar system physics research groups in the country and is well renowned internationally.


Statistics

Sheffield has a proud tradition of research and teaching in statistics (and probability), dating back to the early 1950s under Geoffrey Jowett and Hilda Davies. In 1965 Professor Joe Gani was appointed as the first professor and head of the new Department of Probability and Statistics which separated from the Mathematics Departments. He established an MSc course and PhD programme which have now developed into three MSc courses and a large PhD group covering a wide range of areas including many joint projects with other university departments. The group has a seminar series (joint with probability) with external invited speakers, and regular informal research meetings, led by members of the group.

Linked with the group is the Statistical Services Unit, which provides a comprehensive range of services to industry, commerce and the public services, including consultancy, courses and the development of computer software.

Research in statistics includes: experimental design; genetic epidemiology; nonparametric regression; medical statistics and surveillance methods; theory and applications of Bayesian statistics, with particular interests in archaeology, ecology, health economics, time series analysis, prior elicitation, and analysing uncertainty in complex computer models.


Topology

The work of the Topology group is focused on stable homotopy theory, and equivariant versions for spectra with action of a compact Lie group. One major theme is elliptic cohomology, which has rich connections with the work of the Number Theory group. Another theme involves the commutative algebra of generalised cohomology rings, particularly in an equivariant context, which again interacts strongly with the Algebra group.

A third body of work involves the categorical foundations of homotopy theory, in terms of model categories and their derived categories (which are often triangulated). The group has worked extensively on highly structured ring spectra and their modules, which also fall under this heading.