# 'Mathematical Physics, Geometry and Combinatorics Group

The research being pursued by members of the 'Mathematical Physics, Geometry and Combinatorics group embraces a broad spectrum of topics of current interest, with particular focus on random matrix theory, quantum information theory, mathematical foundations of quantum mechanics, mesoscopic disordered systems, statistical mechanics, graph theory, matroid theory, and infinite dimensional Riemannian geometry, algebraic geometry and orthogonal polynomials. The list below highlights a number of the research activities being undertaken, and lists the names of the various members of the group who are active in these areas.

For more detailed descriptions of our research as well as other relevant information (collaborations, lists of publications, projects, conference presentations, and so on) please follow the links to the web pages of individual group members.

## Research Themes

**The Tutte polynomial and related graph polynomials (Chun, Hall, Noble)**

The Tutte polynomial is one of a wide family of graph polynomials that link a range of graph invariants as diverse as the partition function of the Potts model from statistical physics, the Jones polynomial from knot theory, and the chromatic polynomial. Current research at Brunel in this direction includes an exploration of the combinatorial structures in ribbon graphs analogous to the matroid structure in graphs, with a particular focus on ribbon graph polynomials. The picture illustrates some of the evaluations of the Tutte polynomial..

**The computational complexity of graph and matroid problems. (Chun, Hall, Noble)**

A key question in computational complexity is that of determining which combinatorial problems are tractable (i.e., can be solved reasonably quickly on a computer even as the size of the input grows) and which are intractable (ie, large instances are very unlikely to be solvable in a reasonable period of time in general). Recent work at Brunel has investigated the intractability threshold of the frequency assignment problem, where radio frequencies are assigned to transmitters to keep interference to an acceptable level while minimizing spectrum usage. One of the intractability proofs involved the gadget depicted.

**Random matrix theory (Rodgers, Savin, Smolyarenko)**

Matrices with random entries arise in diverse areas of investigation, including in particular physics, statistics, and finance. The celebrated Bohigas-Giannoni-Schmit conjecture relates spectral statistics of quantum or wave chaotic systems to that of large random matrices, obeying only the fundamental intrinsic symmetries of the problem. Consequently, random matrix theory (RMT) provides a mathematically elegant and powerful description of universal (i.e., independent of microscopic details) statistical properties of physical systems ranging from nuclei and quantum chromodynamics in particle physics, to nanostructures, microwave cavities, and the acoustics of engineering structures. Current research in the Department of Mathematics at Brunel focuses on various aspects and applications of RMT, in particular universality and parametric statistics, the supersymmetry method, non-Hermitian extensions, and so on. Brunel has established an international conference series on RMT and its applications, which has been running annually since 2005.

**Structural matroid theory (Chun, Hall)**

A matroid is a combinatorial structure generalizing the concept of linear independence in vector spaces. Matroids arise naturally in many areas of mathematics, including linear algebra, transversal theory, field extensions, and in several different ways, graph theory. Many concepts—for instance, connectivity—can be generalized to matroids. Research at Brunel University in matroid theory is focussed on proving chain and splitter theorems, which are tools to aid inductive proofs. Chain theorems show that a matroid satisfying a certain connectivity criterion contains a smaller matroid satisfying the same criterion; splitter theorems show if a matroid contains a smaller matroid and both satisfy a certain connectivity criterion then, except in some well-described set of small cases, one may move from the larger matroid to the smaller matroid by making small changes to the matroid at each stage and preserving the connectivity criterion at each stage.

**Resonance scattering and transport in wave chaotic systems (Savin, Smolyarenko)**

Scattering and transport of waves though systems with chaotic dynamics is an exciting field which has a number of important applications (eg, quantum dots, optical microstructures, and mechanical nanoparticles). The most salient feature of such open systems is the set of resonances that correspond to the decaying states formed at the intermediate stage of a scattering event. Mathematically, they can be described in terms of an eigenvalue problem of some non-Hermitian operators corresponding to the relevant S matrix. An active topic of current research is to develop and apply the scattering formalism and appropriate generalisation of RMT for such systems. In particular, the obtained analytic predictions for complex impedances and reflection coefficients in open wave chaotic systems at finite absorption were recently tested by experimental groups in Maryland (USA), Marburg (Germany) and Warsaw (Poland). The related topic is to quantify quantum transport in such systems by developing the novel approach based on merging group-theoretical (Selberg's integral) and combinatorial methods. Studying non-orthogonality of resonance states in open systems, the associated enhanced sensitivity to perturbations, and their links with parametric dynamics of resonances is yet another promising direction of current research, with further potential applications (e.g., to random lasers, to fidelity in quantum computing, and so on).

**Complex Networks (Rodgers, Smolyarenko)**

Numerous problems arising in a variety of social and natural sciences (including, for example, biology, sociology, economics, finance, and information systems) can be cast in terms of assemblies of interacting agents. In recent years, network theory, built on the basis of graph theory, became a powerful tool for the study of such systems. In the language of network theory, agents are represented as nodes of a graph, while edges encode interactions. Almost universally, experimentally observed large networks exhibit common topological features such as low average path lengths (“small-world” property), and power-law distributions of the number of nearest neighbours of a node (node degrees). The research being pursued at Brunel is directed at modelling and explaining these topological features of large networks by studying probabilistic models of network growth.

**Mathematical foundations of quantum theory (Brody, Hughston, Jevtic, Virmani)**

The theories that we use to describe the physical world can broadly be divided into two categories, “classical” and “quantum”. Quantum theory has a mathematical structure that exhibits many remarkable properties and at the same time gives rise at a microscopic level to features that are often at first consideration counterintuitive. Over the last two decades the issues associated with quantum foundations have emerged from what was previously regarded by many as a quiet backwater to become one of the most important areas of investigation in modern science. A number of different branches of mathematics are involved in a crucial way, including various areas of analysis, geometry, group theory, and probability theory. Some of the areas in quantum foundations currently under investigation at Brunel include (a) non-Hermitian extensions of quantum theory, (b) dynamic models for state-vector reduction, (c) use of the theory of Lévy processes in the development of models for the interaction of a quantum system with a noisy environment, (d) the construction of non-quantum theories to mimic the behaviour of various quantum systems, and (e) the theory of quantum entanglement and its applications.

**Mathematical theory of quantum information (Anwar, Brody, Hughston, Jevtic, Virmani)**

Quantum Information theory examines what happens to information processing if we perform it not by using conventional “bits” and logical operations, but instead by using “quantum bits” that are units of information stored in quantum systems (such as the energy levels of atoms, or the polarisations of photons). Quantum information processing has the potential be much more powerful than conventional methods, and a great deal of effort is being made with a view to the realisation of this possibility. One of the topics being pursued by members of the MPC group at Brunel in relation to quantum information is that of understanding just what it is that brings this power, and how it can be damaged by imperfections in the non-ideal systems that we may access in real laboratories. Other topics under investigation include (a) the optimisation of quantum measurements in various scenarios, (b) the development of quantum information theory in a relativistic setting, (c) the mathematical theory of generalised measurements and quantum state transformations, (d) the theory of informationally complete measurements.

**Geometric methods for classical and quantum mechanics (Brody, Hughston, Meier)**

Geometric methods serve as powerful tools for explaining the behaviour of physical systems, both classical and quantum. For example, in a program of research that we like to call "geometric quantum mechanics'', methods of algebraic geometry, symplectic geometry and complex manifold theory are used to provide a novel basis for the description and explanation of quantum phenomena. Physical characteristics of quantum systems can be represented by various geometrical objects that are preferentially identified in the manifold of pure quantum states or in the tangent bundle of that space. This allows one to gain a deeper understanding of quantum mechanical notions such as entanglement, measurement, and uncertainty. From a mathematical perspective one can view quantum theory as an exploration of (a) the algebraic geometry of complex projective space when this space has been enriched with the Kählerian geometry of the Fubini-Study metric, and (b) the profound relation of this geometry to various aspects of modern probability theory. According to this view the various algebraic sub-varieties of complex projective space, along with various other geometrical entities, admit a systematic interpretation as elements of the physical theory.

New insights can be gained concerning the measurement problem, quantum information, quantum control, constrained quantum systems, and quantum statistical mechanics. In particular, the geometrical approach offers surprising insights concerning (a) generalised measurements, including so-called SIC-POVMs, and (b) quantum systems in thermal equilibrium. The geometrical approach also acts as a natural point of departure for other topics currently under investigation such as (a) foundational issues in quantum theory, and (b) the development of physical theories that extend quantum mechanics in various ways. The latter include, for example: stochastic models for the collapse of the wave function; nonlinear models of the Kibble-Weinberg type; and novel approaches toward a better understanding of the interrelationship of the geometry of quantum theory and the geometry of space-time, building on and generalising various well-established constructions coming from Penrose's twistor theory.

Geometrical methods can help one understand common traits in mechanical systems that seem at first sight very different from one another. A celebrated example was discovered in 1966 by V. I. Arnold: both the motion of a rigid body and the evolution of an ideal incompressible fluid can be described by geodesics (length-minimizing curves) on certain spaces. While the mathematical details vary between the two systems, the geometric nature of the evolution—geodesic motion—is shared by both. If a mechanical system exhibits symmetries, one can eliminate the corresponding degrees of freedom. The question of how exactly this can be done has led to the mathematical theory of symmetry reduction. One of the topics in this active area of research currently being pursued at Brunel is concerned with mechanical systems whose variational descriptions depend on higher-order derivatives of the evolution curve. A number of first-order symmetry reduction methods have been extended to these systems, such as the so-called Euler-Poincaré reduction for systems on Lie groups. For example, an interesting problem in quantum control has been studied using these methods: a quantum mechanical system is steered along a smooth trajectory through a number of target states, in such a way that the Hamiltonian satisfies a “least change” constraint. Geometrically, this can be interpreted as an optimal curve-fitting problem on complex projective space.

**Birational geometry and topology of singular Fano 3-folds (Kaloghiros)**

The basic objects studied in algebraic geometry are algebraic varieties; these are geometric shapes defined by polynomial equations in an ambient affine or projective space. In birational geometry, two varieties X and Y are identified if they become isomorphic after removing small subsets. Informally, X and Y are the same “up to elementary surgery". Conjecturally, the Minimal Model Program (MMP) shows that, up to elementary surgery, every variety can be constructed out of "building blocks" of pure geometric type; these have: (1) positive curvature (Fano varieties), (2) zero curvature (Calabi-Yau varieties), (3) negative curvature (varieties of general type). These pure geometric types correspond intuitively to the geometry of the sphere, of the Euclidian plane and of the hyperbolic plane. Understanding varieties of pure geometric type is crucial in classification problems, or when studying geometric properties of general algebraic varieties.

In dimension 3, the MMP is completely known: up to elementary surgery, every manifold X is either a family of mildly singular Fano varieties, or is a family of Calabi-Yau varieties over a base of general type. Mildly singular Fano varieties are thus one of the possible end products of the MMP on manifolds of dimension 3. The interplay between their birational geometry and their topology is very rich. For example, the number of "essentially different" surfaces that can lie on such a 3-fold is a topological invariant and can be bounded using birational geometry. Conversely, topology helps us understand which singular Fano 3-folds are the same up to surgery. This is a very delicate and classical problem in algebraic geometry, where the question of which Fano 3-folds are rational (birational to projective space of dimension 3) has been studied for a long time.

**Geometry of uniruled Calabi-Yau varieties (Kaloghiros)**

Calabi-Yau varieties can be thought of as having flat curvature. Birational geometers often think of these as being ‘just beyond Fano varieties’, but whereas one can classify Fano varieties, and understand birational maps (elementary surgeries) between them, this is much harder in general for Calabi-Yau varieties. An important class of examples of Calabi-Yau varieties occurs as “slices” of Fano varieties. One can hope to obtain some information about birational maps between such Calabi-Yau varieties by studying birational maps between the Fano varieties in which they live. Explicitly, one considers elementary surgeries of Fano varieties that preserve the “volume” of some Calabi-Yau slice. These pairs of varieties (Fano varieties and their Calabi-Yau slices) have important applications in theoretical physics.

**Polynomial inequalities and non-asymptotic bounds for orthogonal polynomials and special functions (Krasikov)**

The global behaviour of even very classical orthogonal polynomials such as the Laguerre and Jacobi polynomials is only partially understood. There are many open questions related to the extremal values of such polynomials and their zeros, as well as for other standard special functions. Even less is known about discrete orthogonal polynomials, such as the Krawtchouk polynomials which play an important role in the graph and coding theory. More generally, these questions are related to the theory of hyperbolic polynomials (and analogues entire functions like sine or Bessel function) – real polynomials with only real zeros – a topic under investigation at Brunel.

**Combinatorics, graph theory, and reconstruction conjectures (Krasikov)**

Reconstruction conjectures are among the oldest and most important open questions in graph theory and combinatorics. The edge reconstruction conjecture of Harary, for example, asks whether a graph is defined up to isomorphism by its edge deleted subgraphs. Problems of this type are often related to many other important and interesting mathematical questions from algebra, analysis and number theory.

**Bounds on the size of error correcting codes (Krasikov)**

Error correcting codes are used to protect information (usually represented as bits) from error by encoding it in longer strings of bits that encode the same information in multiple ways. It is important to understand how small a code can be (measured by the length of the encoded strings) while protecting against a given error rate and encoding a given amount of information. There is a large gap between upper and lower bounds on the size of error-correcting codes given by an explicit construction or just as a random set of points in the Hamming space. Tools from almost all branches of mathematics have been used in attempt to close the gap, however the problem has remained largely unsolved. Brunel researchers are working on developing new techniques that may improve existing bounds.