Pure Maths Research

Pure Mathematics at Leicester is represented by a group with distinctive and coherent expertise, particularly in the areas of Algebra, Geometry and Topology.

We believe that the most exciting progress comes from the interaction of different fields, particularly pure mathematics and theoretical physics, and the exploration of existing or suspected links is strongly supported. 

Particular interests include dynamical systems and aperiodic orders, representations and cohomology of algebras and groups, including algebraic groups and quantum groups, polynomial functors and functor homology, unstable and stable homotopy theory, classifying and moduli spaces, K-theory, Lie theory, surface theory and conformal geometry, derived categories, operads and operadic algebras with links and applications to other areas of mathematics, such as algebraic geometry, commutative and noncommutative geometry, mathematical physics and combinatorics.

Guidelines on the application procedure for prospective postgraduate students could be found at the following link.

Main Topics

This section gives the main topics that members of the pure mathematics group are working on, as well as a list of those working in each area; many members of the group have expertise across several of the specific problems currently studied:

• Moduli Spaces
• Noncommutative Geometry
• Homotopy Theory
• Homological Algebra
• Differential Geometry and Topology
• Representation Theory
  

Moduli Spaces

The study of Moduli Spaces belongs to some of the most exciting and profound parts of Pure Mathematics; it goes back to the classical works of Riemann.

The geometry of moduli spaces of complex algebraic curves underlies string theory; the modern attempt to unify all four fundamental interactions observed in nature and, in particular, to reconcile gravity and quantum theory.

Whenever one has a mathematical object with structure (Lie, associative or differential graded algebra, algebraic variety, holomorphic bundle, coherent sheaf, foliation, etc.) it often makes sense to include it into a family of objects of similar type. It is further profitable to consider the space of objects or a moduli space as having a topology of its own or even, perhaps, a finer structure ... that of a manifold, or more generally, a stack.

Among moduli spaces of much current interest are those of Riemann surfaces, of algebraic bundles on curves and more general algebraic varieties and tilings. Their study uses the vast arsenal of modern mathematics ranging from K-theory to operads as well as aspects of analysis such as asymptotic expansions of oscillating integrals. (See: Clark, Hunton, Lazarev, Neumann)

Noncommutative Geometry

This relatively new subject came to the fore in the middle of the 1980's, mainly thanks to the fundamental works of Alain Connes and Vladimir Drinfeld (Fields medals: 1982, 1986). The main idea of noncummutative geometry is to treat noncommutative algebras as if they were algebras of function of 'noncommutative spaces'.

A surprising amount of classical geometric techniques and concepts (such as smoothness, homogeneous spaces, the De Rham complex, symplectic structures, etc.) could be carried over in the framework of noncommutative geometry. The noncommutative analogue of a Lie group is called a quantum group; this purely mathematical notion has its roots in mathematical physics, in particular exactly soluble models of statistical mechanics. Quantum groups have various applications in other parts of mathematics, in particular they give rise to invariants of isotopy classes of knots and links.

Noncommutative spaces naturally occur in problems which involve dealing with spaces having bad topology, such as moduli spaces of tilings, of leaves of foliations, and various representation spaces. Noncommutative geometry also figures prominently in the study of operad algebras (such as A-infinity, L-infinity, or C-infinity-algebra) which were introduced by topologists in order to study homotopy invariant structures of topological spaces, finite-dimensional algebras, algebraic groups, quantum groups, and Lie algebras.

These algebras arise in many different settings and particular projects look at cluster algebras, diagram algebras, Schur algebras, self-injective algebras, and derived categories. (See: Lazarev, Mudrov, Snashall)

Homotopy Theory

Homotopy theory was originally conceived as the study of those properties of shapes and figures which do not change under continuous deformations.

Perhaps since the seminal works of Daniel Quillen (Fields medal 1978) on closed model categories in the end of the 1960's there came the realization that homotopy theory is really ubiquitous and is not constrained to geometry alone.

Particularly exciting was the development of the last decade, largely due to the efforts of Maxim Kontsevich (Fields medal 1998), which linked homotopy theory with algebraic geometry through the ideas from theoretical physics culminating in the Homological Mirror Conjecture which gives a conceptual mathematical explanation of many calculations done by string theorists, such as the enumeration of rational curves on algebraic varieties. 

Another recent major development linking homotopy theory to other areas was the introduction by Vladimir Voevodsky (Fields medal 2002) of motivic cohomology theory which allowed him to use the vast array of homotopical techniques to solve a long-standing conjecture of K-theory. (See: Hunton, Neumann, Pirashvili)

Homological Algebra

This area of mathematics provides a common language for experts in topology, algebraic geometry and pure algebra.

Recently it was realized that homological algebra is also fundamental for theoretical physics, particularly those aspects related with conformal field theories and quantum gauge theories and deformation quantization.

A large part of modern algebraic geometry is now formulated in terms of derived categories of coherent sheaves on complex and algebraic varieties which is the language of homological algebra. Other topics of current interest in homological algebra include Hochschild and cyclic homology, differential-graded categories and operadic algebras. (See: Lazarev, Pirashvili, Snashall)

Differential Geometry and Topology

Differential geometry is the language of modern physics as well as an area of mathematical delight. It studies differential manifolds (locally Euclidean spaces) and their properties that reflect the geometric nature of the second derivative, the many aspects of curvature.

In particular, that includes classical studies of the curvature of curves and surfaces. Local arguments both apply and help to study differential equations, global considerations often involve methods of algebraic topology and algebraic geometry. Differential topology is closely related to differential geometry; the main difference is that the former is more concerned with global properties of differentiable manifolds. Differential topology draws heavily from homotopy theory and algebraic topology.

Algebraic topology studies topological spaces and their various homotopy invariants, homotopy and homology group. A large part of algebraic topology is currently concerned with stable homotopy theory. The objects of the stable homotopy category are spectra, or generalized homology theories. Examples of spectra are the Eilenberg-Mac Lane spectrum representing the usual homology, various bordisms and K-theory spectra as well as more recently invented spectra representing elliptic cohomology and topological modular forms.

A particularly important class of spectra corresponds to complex-oriented homology theories, of which the most prominent is the complex bordism theory MU. These theories have associated formal group laws that establishes a link between stable homotopy theory, algebraic geometry, and number theory. This deep and intricate relationship has been responsible for much of the progress in algebraic topology for the last four decades. (See: Clark, Hunton, Leschke, Neumann)

Representation theory

Representation theory is a vast area of algebra that has immediate connections to algebraic and differential geometry and theoretical physics.

Much of the development of modern representation theory was motivated by representations of finite groups. This classical area is still important and continues to provide instructive examples. Another source of motivation is the classification theory of finite-dimensional representations of Lie algebras. The latter provides an extremely rare example of a situation when a complete as well as a highly nontrivial classification could be achieved.

Particularly relevant to the research interests of the group are representations of Lie algebras, finite-dimensional associative algebras, quivers and quantum groups. One of the ways to characterize an algebra is through its categories of modules, or representations. This leads to the notion of Morita equivalence of algebras: Two algebras are Morita equivalent if they have equivalent categories of modules. Morita theory has recently found rich and manifold generalizations, touching upon such areas as stable homotopy theory, derived categories of algebraic varieties and A-infinity-algebras and categories. (See: Baranov, Pirashvili, Snashall)