Edwin Beggs - personal home page

My research is somewhere in the region of Algebra, Computer Science, Geometry and Theoretical Physics. One connecting interest is mathematical physics, but my approach to the subject is more related to pure mathematics. I am a generalist rather than a specialist, and frequently end up learning things as I go along.
Edwin with Taiko drum

A Taiko drum - I am not much good on these, usually I play Djembes.

Edwin's publications - Google Scholar
Edwin's official Swansea University page
Wales Mathematical Physics - Physical Mathematics Zoom Seminar

This year I am teaching the modules for Mathematics
MA-111 - Foundations of Algebra
MA-212 - Groups and Rings
and supervising projects for the Computer Science MSc, including Quantum Computation.

I have been involved in outreach events such as the Urdd eisteddfod (Builth Wells) June 2018 and in June 2020 I made the 40 minute lecture `The geometry of curved spaces' for the program `Bridging Mathematics live lectures for Year 13 students' for the Further Maths Support Program Wales (which, given the circumstances, was actually a recorded lecture).

I am currently Academic Integrity representative and Ethics representative for the Maths Dept, sit on the College of Science Corporate Responsibility committee, and am a fire warden and first aider in the Computational Foundry.

I was awarded a Diploma in Computing (Open University) in December 2001 and Fellowship of the HEA in 2020.

As a student I worked for three summers with Prof. Sir William Hawthorne at the Whittle Laboratory in Cambridge on computer simulations of fluid flow in turbomachinery. The first summer I was employed by Cummins, and the last two by Rolls Royce aeroengines.

Journal of Physics A: Mathematical and Theoretical

An editor of a special issue on Noncommutative Geometry and Physics - this will be looking for papers shortly. Note that serious Physics content will be required, not just Mathematics.

Photo of book cover Quantum
      Rieman nian Geometry

Edwin Beggs & Shahn Majid (Queen Mary London)
Quantum Riemannian Geometry - published March 2020.

Supercomputing Wales: A New Decade of Supercomputing - Wales Millennium Centre, Cardiff,
Member of expert panel discussion on Quantum Computing. Friday 24th January 2020, 1 to 2:30

Noncommutative geometry
Geodesic path in 2 by 2 matrices
The state space of the algebra of 2 by 2 complex matrices forms a ball in 3D real space, and the pure states are the boundary, which is a 2-sphere. This is a picture of a 2D real plane intersecting the state space, with the yellow circle marking the pure states. The blue curve is the path of a geodesic in state space, for a given calculus, connection and initial vector field on the algebra.

There are many ways to extend the ideas of classical differential geometry to a noncommutative world. Our view is that there is no clear answer as to which of these is correct, given that many of them have their own rich pure mathematical theory. However, if we were to think about what should ideally be noncommutative differential geometry, we might identify the following considerations. There should be a broad collection of examples of interest across different branches of mathematics. Noncommutative geometry should reduce to classical geometry as a special case, though some aspects of the theory may become trivial in the classical case. Most constructions in classical differential geometry should have noncommutative geometry analogues. Last but not least, as geometry originated as a practical subject, there should be applications, which historically has meant applications in physics and applied mathematics.
The beginning of the preface of the book (see above).
    I shall not go much into research done before the writing of the book unless I consider it a current research interest, the reader may look there for details. After the publication of the book I turned to the dynamics of noncommutative systems. Classically this often reduces to the time variation of a point on a set, so the first place to look in the noncommutative world is for time evolution of a state of an algebra A. For a C* algebra the states are given by Hilbert C*-modules for the algebra A, and by the KSGNS construction a path of states is given by a C(R)-A Hilbert C*-bimodule. Given a calculus on the algebra and a connection on the algebra and bimodule, it turns out that the usual theory of bimodule connections (see book) gives the velocity of the path and a simple form of the geodesic velocity equation, generalising the classical differential equation for the velocity of a geodesic. It is then quite simple to give examples of geodesics on algebras like matrices or the Hopf algebra of functions on a group.
   Edwin Beggs, Noncommutative geodesics and the KSGNS construction, Journal of Geometry and Physics Volume 158, December 2020, 103851

   Of course this raised more problems and questions. There was a definition of the reality of a vector field relative to a state (yes, noncommutative vector fields exist and are not just derivations) which was required to maintain normalisation of a time varying state. These noncommutative vector fields had divergences, again relative to a state. The definition of geodesic (or rather we shall be strict and say autoparallel path) was given in terms of a connection. Now in classical Riemannian geometry we simply plug in the Levi Civita connection and everything is fine. In noncommutative geometry for a given Riemannian metric there may or may not be a Levi Civita connection. However the most annoying aspect is that classically there is a variational derivation for the geodesic equation, and this is a problem on the noncommutative side. It is not that variational approaches do not work in noncommutative geometry, e.g. there is a very nice variational approach to the Laplacian in  many cases. The variational approach to the noncommutative geodesic equation fails because some terms pile up on the boundary of the integral of the velocity squared terms, and there is no obvious way to get them to behave - in  other words, there may be a nice trick to make the whole thing work. There is also the question of whether there is a natural metric on the state space, after the manner of Marc Rieffel's paper Compact Quantum Metric Spaces, for which the geodesic motion has nice properties like a uniform Lipschitz constant.

   There was an obvious direction to take the noncommutative geodesic work in. In General Relativity test particles move along geodesics. Therefore (insert massive disclaimers and whatever here) quantum particles ought to move along `quantum' geodesics. The first try at this hugely over-ambitious and likely wrong guess was asking whether in quantum mechanics (not field theory) we could make sense of time evolution as geodesic motion. And here, at least, the answer is yes, and is given in the following paper by Shahn Majid and myself. Interestingly, and in accord with several other quantum systems, we effectively have to take a central extension of the original algebra (or in our case, its calculus) to get things to work.

   <<We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus depending on the Hamiltonian and a flat quantum connection with torsion on it such that a quantum formulation of autoparallel curves (or `geodesics') reduces to Schrodinger's equation. The connection is compatible with a natural quantum symplectic structure and associated generalised quantum metric. A remnant of our approach also works on any symplectic manifold where, by extending the calculus, we can encode any hamiltonian flow as `geodesics' for a certain connection with torsion which is moreover compatible with an extended symplectic structure. Thus we formulate ordinary quantum mechanics in a way that more resembles gravity rather than the more well-studied idea of formulating geometry in a more quantum manner. We then apply the same approach to the Klein Gordon equation on Minkowski space with a background electromagnetic field, formulating quantum `geodesics' on the relevant relativistic Heisenberg algebra. Examples include a proper time relativistic free particle wave packet and a hydrogen-like atom. >>
   Edwin Beggs & Shahn Majid, Quantum geodesics in quantum mechanics, arXiv:1912.13376 [math-ph]

   In classical differential geometry much work goes into the definition of the Riemann curvature tensor, and then defining the Ricci tensor is just a contraction of indices. In noncommutative geometry there is also a Riemann curvature tensor, but we have a problem with Ricci. There are good cases where Ricci can be defined (see the book and Shahn Majid's web page) but there is no sensible general definition. It may be related to divergences of noncommutative vector fields and variational methods. The importance of Ricci is that it enters in Einstein's field equations for General Relativity. It might be thought that a noncommutative version of General Relativity would only be relevant near the Planck length or in other extreme circumstances, but this is not necessarily the case. Any quantum system which has a measurable gravitational effect in principle quantises gravity, and if that gravitational field can be made dependent on entangled states then we may have interesting quantum gravitational effects. Current experiments are heading in a direction where performing such gravitational measurements are not so unthinkable as it once was (e.g. Entanglement between two spatially separated atomic modes), especially if we can use quantum interference to measure gravitational fields. Experimental physicists are very ingenious and I would never bet against them. :)

Ever since writing a paper with Paul Smith (Univ. of Washington) on Non-commutative complex differential geometry (Journal of Geometry and Physics 72, 2013) I have been hoping for progress to be made on a noncommutative version of Serre's GAGA, géométrie algébrique et géométrie analytique. Classically this is the fact that complex manifolds (insert conditions) are the same as complex algebraic varieties (insert conditions), and it leads to a greater understanding of these objects than if we only had one way of looking at them. It would be of great interest to have a noncommutative version of GAGA. I am very glad to see that progress is being made in this direction, with just a couple of the relevant papers being Positive Line Bundles Over the Irreducible Quantum Flag Manifolds by Fredy Díaz García, Andrey Krutov, Réamonn Ó Buachalla, Petr Somberg, Karen R. Strung and A Kodaira Vanishing Theorem for Noncommutative Kahler Structures by Réamonn Ó Buachalla, Jan Stovicek, Adam-Christiaan van Roosmalen.

With Tomasz Brzezinski (Swansea) I wrote The Serre spectral sequence of a noncommutative fibration for de Rham cohomology,  Acta Mathematica 2005. This set down the definition of a differential sheaf in  noncommutative geometry, and it has properties in line with classical sheaves including (as the title suggests) the Serre spectral sequence. However, there are big things missing, and to explain this I return to classical geometry. John Milnor showed that there are several different differential structures on the topological sphere S^7, but all of these must have the same de Rham cohomology, because that is just the usual topological cohomology of S^7. Now we look at quantum SU2 and at its two fundamentally different differential structures, the 3D left covariant and 4D bicovariant calculus. These have different de Rham cohomologies, although the C* completion of both lots of differentiable functions is the same, Woronowicz's C* algebra quantum SU2. In fact, in noncommutative geometry there is no C* algebra definition of a sheaf, there is just the differential definition. However, morally the use of differentiation in the definition of a noncommutative sheaf is absolutely minimal, it simply is used to say that a section is `locally trivial' if it has zero derivative. There really ought to be an alternative definition of `locally trivial' not using a differential calculus, but what could it be? One idea is that the theory of Quantales by Mulvey etc. may provide a possibility...

On C* algebras, David Evans (Cardiff) and I wrote The real rank of algebras of matrix valued functions (Internat. J. Math 2 (2)). This used the real rank of Brown and Pedersen, which was based on Rieffel's topological stable rank. With Pavle Goldstein (Zagreb) I wrote Maximal abelian subalgebras of On (C.R. math de l'Acad. sci, Canada 21 (1)) on subalgebras of the Cuntz algebra. The paper The braiding for representations of q-deformed affine sl2 with Peter Johnson used complex analysis to analyse the behaviour of the braiding for q on the unit circle in terms of the number theoretic properties of the number τ in the deformation parameter q=e^(2πiτ).

I have been interested in topological algebras for some time, especially as smooth function algebras (for purposes of noncommutative differential geometry) can be expected to be topological subalgebras of C* algebras. In Bruce Blackadar's book on K-theory he uses the more general local C* algebras rather than C* algebras to define K-theory. In this spirit, following the definition of E-theory by Connes and Higson I wrote Strongly asymptotic morphisms on separable metrisable algebras, Jour. Funct. Anal. 177 (1), which generalises Connes & Higson's asymptotic morphisms to a wider class of algebras.

I am also interested in nonassociative phenomena, e.g. Making nontrivially associated modular categories from finite groups, IJMMS  2004, ID 238947, jointly with Mohammed Al-shomrani (King Abdulaziz University). With Ghaliah Alhamzi (Al-Imam Muhammad Ibn Saud University and Swansea) I have written Matrices, Bratteli Diagrams and Hopf-Galois Extensions (arXiv:2009.01577 ) << We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf-Galois extensions (quantum principle bundles) for certain abelian groups>> and we are currently working and on an exponential map for Hopf algebras generalising that for Lie groups.

Integrable systems
Soliton decay
Solitons in the Principal Chiral equation. The space axis is on the bottom left, and the time axis on the right. The function is integrated energy density, the Morse function for the spatial evolution. The incoming soliton on the bottom decays into two, and one of these (the stationary soliton in the middle) in turn merges with another soliton. This could be interpreted as a `Feynman diagram like' interaction between two solitons mediated by another soliton.

     I completed a D.Phil. under the supervision of Graeme Segal at St Catherine's college Oxford on the topic of integrable systems and the principal chiral model in particular, showing that the space and time evolution was given by Morse functions, Solitons in the chiral equation (Comm Math Phys. 1990). Falleh Al-Solamy (King Abdulaziz University) and I used the energy and momentum Morse functions given to illustrate several types of nontrivial soliton interactions in the principal chiral model (see the picture above).
     In Swansea I worked with Peter Johnson, finding the inverse scattering method for solitons in affine Toda fi eld theory in Inverse scattering and solitons in An−1 affine Toda field theories (Nuclear Physics B 1997) and same title part II (Nuclear Physics B 1998), and a group factorisation formula for the symplectic form in Loop groups and the symplectic form for solitons in integrable theories.
     In An electrically charged monopole in quantum electrodynamics I took advantage of the properties of the matrices in the 4+2 dimensional Dirac equation to give a construction of a topological electric monpole in a (higher dimensional) Kaluza-Klein type extension of standard space time. 

     After finishing the book (I still prefer not to think about how much checking went into that) I have plans to get back to integrable systems. I would like to cooperate with others on integrable problems, especially using geometric or noncommutative methods. However, I have two specific projects which I would like to do, if indeed they can be done, which is not at all certain.
     Recall that for several soliton equations (including affine Toda, sine Gordon, principal chiral) using the inverse scattering transform the classical phase space of the pure soliton solutions can be visualised as a loop group consisting of meromorphic functions. This contains all the information about the solitons, including momenta, positions and phases. The more general setting of the classical inverse scattering method is as a group factorisation, or doublecross product, the problem is to find such factorisations with Lorentzian symmetry (this itself generalises into a theory of Hopf algebra factorisations as set down by Shahn Majid). The quantum inverse scattering transform gives exact matrices for the quantum scattering of solitons. There have been examples of solitons in higher dimensions with Lorentzian symmetry for some time, e.g. R.S. Ward, Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system. Phys. Lett. A 208: 203-208 (1995). Recently there has been further examples and numerical evidence of higher dimensional solitons, e.g. Y. Zarmi, Static solitons, Lorentz invariance, and a new perspective on the integrability of the sine Gordon equation in (1 + 2) dimensions, Jour. Math. Phys. 54, 013512 (2013).
     The questions I would like to consider are primarily pure mathematical but with obvious applications to integrable systems.

1) Quantum meromorphic groups:  Is there a Hopf algebra type object which has the same relation to quantum affine SU2 as the meromorphic loop group to the analytic SU2 loop group? (Care needs to be taken with the classical limits here.) There is a grassmannian construction of the meromorphic loop group in terms of representations of the analytic loop group, so that may be a place to start. Another place to start is with the inversion of algebra elements (much about this in An Introduction to Noncommutative Noetherian Rings,  Goodearl & Warfield, L.M.S. student texts 16) and then see what this does to the differential geometry. Maybe it is easiest to generalise the problem to loop Hopf algebras using something similar to the braided constructive procedure used by Shahn Majid for Uq(g).

2) Higher dimensional residues and group factorisations: One way to try to find more higher dimensional integrable systems is to use a higher dimensional variety instead of the complex numbers. Meromorphic functions in higher dimensions have singularities along divisors, not just points, and the geometry of the divisors can be rather complicated. An alternative would be to look at higher codimension singularities – to describe this let us stick to a 2D complex manifold. A meromorphic function is one which can be locally written as a ratio of analytic functions which are not both zero simultaneously. Thus for (z, w) ∈ C^2 the function z/w is meromorphic except at the point (0,0). One well known construction is to de-singularise this by blowing up the point – replacing (0, 0) with a copy of CP^1 , based on approaching the singular point on various lines. However, for our purposes we really would like a point singularity, and we recall a construction of the residue of higher codimension singularities going back to Poincare. Following the low dimensional case, we might attempt to implement a group factorisation with a simple transformation of a matrix valued codimension 2 residue at a point encoding a 2+1 dimensional momentum. Uniqueness of the factorisation would be imposed by adding symmetry conditions to the (mostly) meromorphic functions, and these symmetries should be preserved by Lorentz transformation (up to a gauge equivalence at least).

    A highly speculative approach to something which might be related to integrable systems could be the following (could I insert any more conditionals into this sentence?). Chamseddine and Connes have published papers related to the standard model of particle physics and noncommutative geometry, e.g. The Spectral Action Principle, CMP 186 (1997) (this particular paper looks at a coupling to gravity). There has been much discussion about whether this noncommutative model really gives the standard model or not, and before the reader inserts an answer here I would like to say that I am not very interested in this question. To me the curious thing is that such a model would give anything remotely in the right ball park, and that this is either a wild coincidence or something rather significant. Assuming, for the sake of argument, that it is significant, it makes sense to ask if the noncommutative model is in itself an interesting point of view for the study of a classical system. After all, it is a geometric picture (noncommutative, but still geometric), and geometric pictures have been significant in the study of physical systems before. And why not begin with the `classical' theory of such a system, after all that might be easier than the quantum theory. When we are at it, we may as well simplify the system just by cutting the sizes and dimensions down, as we are not solely interested in the standard model. Where does that leave us? I am not at all sure, but it sounds like an interesting adventure...

Computation and Physical systems
Temple of Apollo at Delphi

The Temple of Apollo at Delphi, the site of the Oracle of Delphi

With John Tucker (Swansea) and Felix Costa (Lisbon) I wrote papers on the computational power and complexity of algorithms which had access to physical information. Some of these were also joint with Bruno Loff (Porto), Diogo Poças (Lisbon), Tânia Ambaram and Pedro Cortez. See Computational complexity with experiments as oracles (Proc. Royal Society A 2008) and Experimental computation of real numbers by Newtonian machines (Proc. Royal Society A 2007), Computations with oracles that measure vanishing quantities ( Math. Struct. in Comp. Sci. 2016), Classifying the computational power of stochastic physical oracles (Int. J. Unconvet. Comp. 2017).

John Tucker (PI) and I were awarded the EPSRC grant EP/C525361/1 ‘Foundations of computing with continuous data: algorithms versus experiments with physical systems’, value 258,042 pounds, beginning January 2006 and ending June 2009.

   The key idea is a physical oracle, a development of the oracle used by Alan Turing to indicate a mythical (beyond standard computation) source of information for an algorithm. Physical oracles to the algorithm are axiomatised abstractions, but in the real world they can control and record data from physical experiments. They may be error prone, and they may have a delay in replying to queries (or never reply at all). Interestingly for current developments, a quantum computer can be considered as a physical oracle to a conventional algorithm.
    As far as complexity theory is concerned, the key is how much information the physical oracle can extract from a physical system in a given time. Many experiments can be analysed, such as Brewster's angle for measuring refractive index. The result of reasonable assumptions on such experiments is that the time taken to estimate a quantity to a given error is of order the error to a negative power.
Given an independent coin toss oracle (in theory radioactive decay can provide this), such a physical oracle has the non-uniform probabilistic complexity class BPP//log*. Now, given the hypothetical Planck limits on measurement, this might sound impractical. However, the completely different independent probabilistic oracle (e.g. a fixed finite but repeatable error) gives precisely the same class, and this entirely independent derivation indicates that there may be something fundamental about BPP//log*.

Helmuth von Moltke

`No plan survives contact with the enemy'. Helmuth von Moltke the Elder (1800-1891)

Several piles of wreckage of space probes on the surface of Mars testify just how difficult it is to have automated control systems for highly complex systems in uncertain environments. Fortunately, the accident reports for space probes and aircraft contain a wealth of data on just how things went wrong. However, correcting one error is not really the solution, it is necessary to build in much more transparency and resilience into control systems. John Tucker and I are currently working on a form of object orientated programming for physical or even social systems. The idea is to divide the system into modes and form a geometric model of the modes, which is naturally a simplicial complex. The control is assigned to a mode by using a computable partition of unity. The algorithm for a given mode is written using a generalised many-sorted algebra. By generalised, we mean that it contains not only sets, functions and relations, but also physical oracles (see above). These oracles are used to both measure and control the real world system.

Biology: Games and population dynamics
Moving fronts between populations
In ecology competition between different phenotypes is often expressed in terms of a game, but its payoff coefficients may vary. Consider an island where the direction of the prevailing wind makes one half of the island drier than the other. A bird species has two phenotypes, both of which would be, in isolation, stable equilibria in both dry and wet conditions in the island. However, when considered together, one phenotype does better in dry conditions than the other (i.e. the other phenotype has a semistable equilibrium). In wet conditions things are the other way around, so the other phenotype does better. We start with wet preferring birds on the wet side, and dry preferring birds on the dry side. Now, what happens to the populations in the long term? The answer is that, as long as the island is not too small, we have a stable front dividing the phenotypes which will persist indefinitely. The exact position of the front may shift because of slight variations in weather from year to year, but the basic form will be unchanged. The width of the front will be determined by the diffusion rate of the phenotypes, basically independently of how slowly conditions change across the island (this is why the island should not be too small). Move the front (as in this picture, with space axis bottom right and time axis upper right), and it springs back to its equilibrium position. However, if the conditions vary with time, we may get a catastrophic change at some point...

I am interested in biology in general, and am keen to find problems in biology where mathematics can help, and people who want to collaborate in biology.

Great Auk memorial

The Great Auk Memorial, Papa Westray, Orkney

Every extinct species should have a memorial, it might remind mankind to be more careful in future.

The tragic story of Avery, the West Point cat.

Their buttons had all been polished
and their uniforms had been freshly pressed,
Avery's team could out-mouse the navy anytime.

Such a cat unflappable had never been seen
since Min the Mog played opposite James Dean.
Cadet Avery, the West Point cat.

One night by the Hudson the alarm went out,
there's vermin in the commander's house!
Avery mobilized his platoon without delay.

They routed the rats and massacred the mice
and victory complete was wrought in a trice.
Lieutenant Avery, the West Point cat.

The flags flew and the bugles bugled
when the president came to see around.
Avery's platoon is on guard in the kitchen
for some bandit's been stealing the presidential milk

and eating the meat and shredding the silk.

Colonel Avery, the West Point cat.

Past midnight the door creaked wider
and the squad pounced on the intruder.
Avery escorted the prisoner to the court martial.
Said general to Avery `you have mistaken the thief,
for that is the cat of the commander in chief'.
Dishonorably discharged Avery, the West Point cat.

Avery the cat
Avery is currently in retirement at an undisclosed location in New England.

Graphs by Mathematica.