Our research in this area encompasses the following:
Algebraic topology and homotopy theory
Algebraic topology studies geometric objects with algebraic methods and thus brings together methods from different areas of mathematics. Homotopy theory is a way of describing notions of equivalence between objects and functions, which can be done in both a very geometric, hands-on fashion or using more abstract, axiomatic tools found around category theory.
The related research in Kent lies at the more algebraic end of the spectrum, including model categories, stable homotopy theory, homological algebra, and algebraic geometry and mathematical physics.
Cluster algebras and mirror symmetry
As a relatively new class of commutative algebras, cluster algebras have a multitude of connections with diverse areas of mathematics and theoretical physics. Mirror symmetry was originally introduced as a duality between two seemingly different physical theories. In mathematical terms, it is a conjectural relation between a given geometric object, or variety, and its so-called ‘mirror’, which has proven extremely useful to solve difficult questions in various areas of mathematics.
Cluster algebras
Cluster algebras are a new class of commutative algebras which were introduced by Fomin and Zelevinsky around 2000. Since then they have become one of the fastest growing areas of algebra, due to their multitude of connections with diverse areas of mathematics and theoretical physics, ranging from Lie theory and Teichmuller theory to solvable lattice models in statistical mechanics and quantum field theory.
Rather than being defined directly by generators and relations that are given from the outset, cluster algebras are obtained from an initial set of variables (a seed) by a recursive procedure known as mutation. At each step, mutation produces a new set of variables (a cluster) from the previous one via a nonlinear rational transformation. The corresponding cluster algebra is the algebra generated by all cluster variables obtained from all possible sequences of mutations applied to the initial seed. One of the remarkable features of cluster algebras is the Laurent property: although they are generated by birational iterations, all of the clusters turn out to be given by Laurent polynomials in the seed variables with integer coefficients.
Research at Kent is concerned with recurrences defined by cluster algebras and their connection with discrete integrable systems, more general iterations that display the Laurent phenomenon, as well as the noncommutative case i.e. quantum cluster algebras, and their application to Representation Theory of quantum algebras and Total Positivity.
Mirror symmetry
Originally discovered by physicists, mirror symmetry emerged as a mathematical concept in the early 1990s with the work of Candelas, de la Ossa, Green, and Parkes. In mathematical terms, mirror symmetry is a conjectural relation between two geometric objects, which has proven extremely useful to solve difficult questions in various areas of mathematics, such as enumerative geometry. It uses tools from algebraic geometry, symplectic geometry, category theory, representation theory, and mathematical physics.
Saying that two varieties are ‘mirror dual’ means that certain invariants of the symplectic structure of the first variety match certain invariants of the complex structure of the second. As discovered by Maxim Kontsevich in 1994, this relationship between geometric objects can also be expressed on the level of categories, leading to the homological mirror symmetry programme.
Research on mirror symmetry at Kent is mostly focused on mirror symmetry for varieties with an action of an algebraic group and homological mirror symmetry.
Invariant theory
The theory of groups and invariants is the mathematical language for analyzing symmetries.
Objects or phenomena of interest and their properties are described mathematically in terms of solutions of systems of equations, involving polynomial functions that depend on chosen coordinates. Changes of coordinates are then described by a suitable transformation group, acting on the system and its ingredients. The functions which are unchanged by that group action, the “invariants”, reveal the objective nature and the underlying symmetries of the studied phenomena. It is, therefore, a major goal of invariant theory to provide general principles to find all such invariants for a given group, describe their properties and perform efficient computations.
Historically, researchers first looked at functions with real or complex coefficients, but more recent developments ask for invariants over more general coefficients, including modular fields. In that situation, many results in “classical invariant theory” are known to be false, in which case the researcher looks for appropriate replacements.
Major open questions which are addressed at Kent include:
- When is a modular ring of invariants a polynomial ring, and if it is not, how “far away” is it from being one?
- What is the relationship between the properties of a representation of a group and the algebraic/geometric properties of the corresponding ring of invariants
Quantum algebra
Quantum algebra is an area of mathematics where we study algebras that are deformations of classical objects such as commutative polynomial algebras, permutation groups, and general linear groups. These quantum algebras are often studied through the geometric intuition that term from the classical objects.
Quantised coordinate rings
One of the main ideas in the study of geometric objects is that their properties are completely encoded in the functions on them. In algebraic geometry, one studies geometric spaces defined by polynomial equations, and so the polynomial functions on a space characterise this space.
These functions have a nice structure: you can add and multiply polynomial functions, and two polynomial functions always commute (fg = gf), so that polynomial functions on a space form a commutative ring.
In the 1980s, developing out of ideas in physics, Drinfeld, Fadeev, Jimbo, etc. introduced new rings, called quantised coordinate rings. These rings appear to share all the structure of rings of functions on geometric spaces, except that the multiplication is noncommutative. And so they are studied as if they were rings of functions (except for the noncommutativity); the guiding principle is the quest for the “noncommutative geometry”.
Research at Kent is concerned with the representation theory of quantised coordinate rings appearing in Lie theory such as quantum matrices, quantum flag varieties, and their quantum Schubert cells. One of the main features of our research is our strong combinatorial point of view on this topic.
Poisson algebras
Poisson algebras first appeared in the work of Siméon-Denis Poisson two centuries ago when he was studying the three-body problem in celestial mechanics. Since then, Poisson algebras have been shown to be connected to many areas of mathematics and physics, and so, because of their wide range of applications, their study is of great interest for both mathematicians and theoretical physicists.
One way to approach Poisson algebras is via quantisation. To give the reader a bit of the flavour of this subject, consider the simple example of the quantum plane A=k<x, y > generated by two indeterminates x,y subject to the relation xy=qyx.
If q≠ 0,1, then this algebra is noncommutative.
However, when q=1, this algebra becomes the commutative polynomial ring P in two variables. Thus we will think of A as a noncommutative deformation of P. Even better, one can show that the algebra A gives rise to a Poisson bracket on P by a “semi-classicalisation process”.
Examples of Poisson varieties that arise as semiclassical limits include flag varieties and their Schubert cells.
Research at Kent is concerned with the representation theory of Poisson algebras as well as the study of (potential) links between noncommutative algebras and their Poisson counterparts.
Complex reflection groups and their deformations
A complex reflection group is a group of matrices with complex coefficients generated by pseudo-reflections (elements whose vector space of fixed points is a hyperplane); these groups were completely classified by Shephard and Todd.
Deformations of these groups (known as cyclotomic Hecke and Cherednik algebras) play an important role in knot theory, categorification of highest weight representations of quantum groups, symplectic geometry, and algebraic combinatorics.
Research at Kent is concerned with developing tools for understanding the simple representations of these algebras over fields of arbitrary characteristic.
Representation Theory
Representation Theory was introduced at the end of the nineteenth century thanks to the work of Frobenius, Burnside, and Schur. It was then concerned with the study of properties of abstract groups via their representations as linear transformations of vector spaces. This allowed the study of abstract groups through methods from Linear Algebra. This powerful idea was then extended to other mathematical structures such as associative algebras, Hopf algebras or Lie algebras. Since its introduction, it is a story of rich innovations and applications to other topics in mathematics and to all sciences.
Representation theory of infinite dimensional algebrasOne of the fundamental (yet very basic) questions is to classify the irreducible representations of a given infinite dimensional algebra A over a field; this problem is often quite difficult (if not impossible).
A now-standard approach to this problem, proposed by Dixmier, is to study the annihilators of the simple left-modules over A, i.e. the annihilators of the irreducible representations of A. These are the so-called primitive ideals.
Research at Kent is concerned with primitive ideas of quantum algebras (e.g. quantum flag varieties and related algebras) and their Poisson analogues. We developed combinatorial tools to study those ideas (e.g. Cauchon diagrams, automaton) and established strong links with Total Positivity, a theory that plays an increasing role in Theoretical Physics. More recently, we study the representation theory of Poisson algebras via methods from differential algebra and model theory of differential fields.
Representation theory of symmetric groups
The representation theory of symmetric groups has been studied for over a century using ideas from geometry, categorical Lie theory, and combinatorics.
One of the most difficult open problems in this field is to understand how certain representations (obtained from induction and tensor product) decompose into their simple constituents. The coefficients describing this decomposition are called the plethysm and Kronecker coefficients. These coefficients arise in many places across mathematics; in particular, they form the centrepiece of a new approach that seeks to settle the celebrated P versus NP problem.
Research at Kent focuses on obtaining a combinatorial understanding of these coefficients. We have classified the maximal and minimal constituents of arbitrary plethysm products as well as the Kronecker products whose constituents are multiplicity-free.
Applications of representation theory
Representation theory is very important in high energy physics. For example, representations of SU(3) are the essential ingredient of Quantum Chromo Dynamics (QCD) and shed light on the zoo of particles produced in accelerators such as LHC at CERN. Representations of SU(2)xSU(2) that are invariant under certain discrete symmetries arise in the Skyrme model when calculating ground and excited states of atomic nuclei.
Representation theory also has numerous interactions with algebraic geometry, in particular via the theory of complex algebraic groups. Applications include the study of mirror symmetry for varieties with actions of algebraic groups.