My Research
Results

My Research
Results

Research outline

I'm interested in almost
all aspects of pure mathematics and mathematical
structures needed to address fundamental problems in
theoretical physics. A fuller narrative account is
below but here as some of my favourite results, some
of them with collaborators.

1980s
An explanation of the confinement of quarks using
infinitesimal holonomy and the background field method

Among the first to
introduce the use of loop variables as functionally
Fourier dual to gauge fields.

Introduced quantum Born
reciprocity (or
position-momentum/observer-observed interchange)
as a key input for quantum gravity

Pioneered the modern theory
of quantum groups introducing one of the two
main types (the bicrossproducts associated to Lie
group factorisations)

Introduced the centre
of a monoidal category (as part of my more general
duality construction) and an algebraic formulation of
Drinfelds double as a `double cross product’

1990s
Introduced a new kind of braided algebra done with
knots and tangles (the systematic theory of hopf
algebras in braided categories with main theorems such
as reconstruction and bosonisation)

Introduced the first
successful model of quantum spacetime with
quantum group symmetry, a model which was later shown
to predict variable speed of light (this is currently
tested by a satellite in orbit)

Introduced cogravity: the
idea of curvature in momentum space (as equivalent to
quantum spacetime via quantum Fourier transform)

Pioneered the `quantum
groups approach to noncommutative geometry’ including
first to introduce quantum group gauge theory and
frame bundles

Constructed Drinfeld
quantum groups by an inductive construction on braided
categories. Part of this is a canonical braided-Hopf
algebras associated to an object in a braided category

Classified differential
calculi on all main classes of quantum groups

Showed that the famous
Octonions are in fact associative when viewed in a
certain monoidal category

Introduced a theory of
braided-Lie algebras generating Drinfeld quantum
groups

Introduced cocycle twists
of module algebras (extending Drinfeld twist of
quantum groups)

2000s
No-go theorem forcing invariant differential calculi
on the main quantum groups to have extra dimensions or
else be nonassociative

Found the quantum
Riemannian and complex structure on the quantum
q-sphere

Found the quantum Riemanian
structure on finite group S3 (its Ricci curvature is
constant)

Harmonic analysis on
quantum spacetime to find variable speed of light
prediction (previously had been speculated)

Found braided-Hopf algebra
structure on the Fomin-Kirillov algebra relating
noncommutative geometry of Sn to classical cohomology
of the flag variety

Gave a construction for the
Fock space of anyons using braided categories

Related braided-Lie
algebras to differential calculi on quantum groups

Introduced notion of
complex conjugation (`bar’) for monoidal categories

2010s Advanced a previous
bimodule-connection approach to noncommutative
differential geometry with first solved model on
quantum spacetime

First actual construction
of noncommutative-geometric or quantum black hole

Introduced Lie theory of
finite groups and applied it to Killing form and
coverings

Introduced duality for
differential structures on quantum groups

New point of view of
Riemannian geometry as cocycle extension data for the
exterior algebra on the manifold

Introduced quantisability
conditions for a classical metric to come from a
quantum one (example: quantum spacetime forces
Bertotti-Robinson solution of Einstein’s equations)

2020s Classified Digtal quantum groups in low dimension (and digital quantum geometries)

Solved first actual models of quantum gravity on quantum spacetime (on
the fuzzy sphere and on Z_n, earlier on Z_2xZ_2)

Fuller research statement

My earliest works were in
high-energy particle physics at Harvard. I
introduced ideas such as `infinite spin’
regularization ([30-31] in my list of publications)
as a way to handle the infinities in quantum field
theory, an infinitesimal proof of quark confinement
using the background field method [32], and a Fourier
duality between loops and gauge fields on a manifold
[37]. I also worked on Yang-Mills gauge theory
[33-34].

Starting with
my 1988 PhD thesis, I helped pioneer the modern theory
of ‘quantum groups’ or Hopf algebras, introducing one
of the two main classes of these objects, namely the
‘bicrossproduct’ quantum groups C[M]
U(g) associated to Lie
group factorisations X = GM [35-39]. These have become
increasingly important, most recently in applications
to quantum spacetime where my model [128,150] based on
them could provide the first experimental tests of
quantum gravity, see expositions [13-14,24-25]. I also
worked on the other class of ‘quasitriangular’ quantum
groups Uq(g) of Drinfeld and Jimbo q-deforming the
enveloping algebras of simple Lie algebras g. I have
results on the Drinfeld quantum double construction,
including its formulation[37] as a double cross
product H
H which is now more or
less standard in algebra circles. At the end of the
1980’s I was among those developing the connection
between quantum groups and braided categories. Results
included the interpretation of q-dimensions as the
categorical rank[44], relations of that with the
theory of modular functions[54,55], an important
functor [51] from modules of a quasitriangular Hopf
algebra to modules of its double (so-called `crossed
modules’), and generalised Tannaka-Krein
reconstruction theorems for quasi-quantum groups and
‘braided groups’ or Hopf algebras in braided
categories [48-49,58,64]. Another important result was
a duality operation for monoidal categories [40,47,76]
a special case of which was later called the `centre’
by V.G. Drinfeld and connected with the quantum double
(in a famous letter to me after reading the preprint
version of [47]). A later application was the
quantum double of a quasi-quantum group found in
[140].

A large body of
my subsequent work introduced and studied these
braided groups and ‘braided geometry’ as the deeper
structure behind q-deformations [59-108]. I showed
that every quasitriangular quantum group has a braided
version by `transmutation’[64-65] and that every
braided group of a certain type has an associated
ordinary quantum group by `bosonisation’[66], both
constructions are now standard in algebra circles.
Among results was a new inductive construction [103]
of all quantum groups Uq(g) for g simple by repeatedly
adjoining additive braided groups such as the
n-dimensional additive quantum plane Aq . For example
Uq(sln+1) = Aq
Uq(sln)
Aq . Iterating this
provides a new ‘block decomposition’ of any Uq(g) into
a product of braided spaces and a Cartan part. The
construction included Lusztig’s construction of Uq(g),
and also gave a fresh approach to the structure theory
of semisimple Lie algebras themselves. Other braided
groups related to ‘noncommutative exterior algebras’
of finite groups [160] and are conjecturally related
to Lusztig canonical bases as well as to geometry in
characteristic p. Braided groups were also shown to
underly the theory of nonrelativistic anyons [115] and
to provide a generalisation of cyclic cohomology
appropriate to q-deformation examples [116]. Braided
group computations are best done by a new kind of
`braided algebra’ consisting of algebra operations
`wired up’ like the wiring in a computer except that
there is a nontrivial braiding operation each time one
`wire’ jumps over the other. I was the first to
seriously use such methods for algebra [58,65-66],
which represent a new kind of `braided logic’ which
might also provide the right semantics for quantum
computers.

Also using tensor category
ideas, in 1999 I introduced[109] a now well-known
approach to the nonassociative Octonions algebra, as
associative in a nontrivial monoidal category of
modules over a certain quasi-Hopf algebra structure on
(Z_2)^3. It then becomes by a theorem of Mac Lane ‘as
good as’ associative in the sense that all linear
algebra constructions can be done as usual with
bracketing inserted afterwards via an associator. I
recently[118-119] applied this to a theory of
algebraic Moufang loops (such as the unit octonions)
and obtained a new description of the structure
constants of the Lie algebra g2.

The other large body of my
work since the 1990s is the pioneering of what I call
the ‘quantum groups approach to
noncommutative differential geometry’ [122-192], including models of quantum spacetime.
In fact, while the Uq (g) quantum groups arose from
quantum integrable systems and statistical physics,
the bicrossproduct ones arose specifically from the
search for a generalisation of geometry to situations
where both gravitational effects (curvature) and
quantum effects are present, the so-called ‘Planck
scale’. From a mathematical point of view this amounts
to extending differential geometry to allow a
noncommutative ‘ring of functions’. The main problem
is that usual ideas of topological spaces, sheaf
theory and local trivialisations do not work well when
the ‘coordinate rings’ are noncommutative. A solution
to this problem appeared in 1992 in work [127]: we
developed purely algebraic replacements for these
ideas sufficient to define principal bundles and
connections (with quantum group fibre) on
noncommutative algebras, and sufficient also to
include nontrivial examples such as a q-monopole over
a noncommutative q-sphere. In key works [146,155] I
extended this to noncommutative Riemannian geometry on
potentially any unital algebra using a quantum frame
bundle approach with quantum group fibre. In more
recent work on ‘formalism’ I have looked at the notion
of ∗-structure needed to define real forms [178] and
at Riemannian geometry based on ∗-compatible `bimodule
connections’ [182]. This extends the frame bundle
theory to a slightly more general theory based on
vector bundles (as projective modules) and hence
closer towards Alain Connes’ approach to
noncommutative geometry modelled on the Dirac operator
without reference to quantum frame bundles. There are
also links between Connes’ approach and bicrossproduct
quantum groups [173]. My early results include
classification theorems for bicovariant differential
calculi both on q-deformation quantum groups in 1998
[138], quantum doubles [156] and bicrossproduct ones
[145,161]. By now the geometry of quantum groups has
its own Mathematics Subject Classification code 58B32
and featured as an integral part of a 6-month
programme on Noncommutative Geometry that I organised
in 2006 at the Newton Institute along with Alain
Connes and Albert Schwarz.

Several distinctive features have emerged in this quantum groups approach
to noncommutative geometry. The first is applications
to the noncommutative geometry of finite groups even
where the ring of functions is commutative (for
example I found [155] that S3 is ‘Einstein’ while A4
is Ricci flat [161]). This theory is expected to have
deep connections with canonical bases as an extension
of Schur-Weyl duality [160] and in the meantime the
notion of Killing form and `Lie algebra’ provides a
new tool in the theory of finite simple groups. In
[121] it is proved that this Killing form is
non-degenerate in certain cases and in [188] that when
the `Lie algebra’ is of a certain type which includes
Weyl groups of semisimple Lie algebras, the
noncommutative first de Rham cohomology is essentially
trivial with a 1-dimensonal `nonclassical’ part. The
theory also applies finite-dimensional Hopf
algebras[156, 162] such as quantum groups at roots of
unity (which could be viewed as quantizations of
finite groups of Lie type) and potentially also to
finite sets[174-175,180]. The other is a fundamental
anomaly or obstruction to the construction of
associative differential calculi of classical
dimensions and appropriate symmetry. This was
confirmed in a semiclassical analysis [171] for all
standard quantum groups Cq(G) and [181] for all U(g)
regarded as quantizations of g∗, for g simple. These
two theorems imply that in such highly noncommutative
contexts one must either resolve the anomaly with
extra cotangent directions [171] or live with a forced
nonassociative differential geometry [117,181,183].
The first approach can be interpreted as an intrinsic
‘time evolution’ induced by the anomaly and may (or
may not) be connected to the modular automorphism
induced in the operator algebras approach. It is what
I have called `the algebraic origin of time’ [172].
The second approach is more radical but we show in
[171,181,183] how `cochain twists’ may be used to
non-associatively quantize any classical structure
under the action of a symmetry. The use of Drinfeld
twists by cocycles and cochains as a method of
transforming or `quantizing’ algebras had been
introduced by me in the early 1990s and used notably
in [93,109,149]. Some expositions were in [128, 165]
and a recent application was in [176]. More recent
work [170,177] also hints at a deep theory of
`noncommutative complex structures’. The first of
these included among other things a noncommutative
Dolbeault complex for the q-sphere and proved (for
generic q) a q-Borel Weil Bott theorem whereby the
irreducible representations of Uq(su2) are obtained as
holomorphic sections of quantum line bundles
(monopoles) associated to representations of the U(1)
fibre in the q-Hopf fibration. This was used to give a
`geometric Dirac operator’ in the q-sphere in [170]
which is shown in [194] to obey most of the algebraic
side of Connes’ axioms but now geometrically realised.
Noncommutative CP^n and more nontrivial Grassmannians
were obtained in [177] as an approach to
noncommutative twistor theory, recovering in the
process θ-deformed S^4 that had been introduced by
Connes and Landi, as well as noncommutative versions
of several of the key bundles in twistor theory.

Such constructions are not only of
mathematical interest. Other ‘fuzzy’ spheres as
quotients of U(su2) as a coadjoint quantisation arise
in 3D quantum gravity without cosmological constant
[166], while for 3D quantum gravity with cosmological
constant we found [179] q-fuzzy spheres similarly
based on Uq(su2) viewed as a coordinate algebra.
In [185] I used twisting and braided category methods
to introduce noncommutative differential calculi on
Uq(su2) and q-fuzzy spheres, opening the way to the
Riemannian and complex geometry underlying the model
spacetime of full 3D quantum gravity with cosmological
constant and point sources. Meanwhile in [184,186] I
have achieved a long-cherished goal of constructing a
noncommutative Schwarzschild black hole, turning the
idea of `extra dimensions’ in the calculus around to
define the wave operator.

In recent years this approach as focussed on the general side of
`noncommutative differential geometry’ with quantum
groups merely as examples. Thus, the methods above
provide a noncommutative Riemannian geometry of graphs
[187] and of finite commutative Hopf algebras over
finite fields [200]. The semiclassicalisation of the
theory opened up a new field of
`Poisson-Riemannian geometry’ [197] which underlies
and explains the constraints in several models
[190,192,201] where particular classical geometries
are forced by quantisability constraints if they are
to emerge from a noncommutative geometry. The
Poisson-Lie case connecting to quantum groups is in
[193]. I also gave a new point of view on what the
Hodge operator is classically (namely Fourier
transform on the exterior algebra) and used this to
find a canonical one a quantum group [196] . Also a
new point of view on what Riemannian geometry is as
arising from the Leibniz rule in quantum spacetime
[199]. This phase of my work has now largely been completed as a somewhat
coherent `constructive approach' to noncommutative or quantum Riemannian
geometry as captured in my book [5] with Beggs.

My work since the book has turned
to applications of quantum Riemannian geometry, particularly to my long-cherished goal
of actual quantum gravity, for which I have taken a functional integral approach to
actually quantise the space of `quantum' metrics. I have
now solved this on several models, notably [212] on a square Z_2xZ_2, [219]
on a fuzzy sphere and [220] on a polygon Z_n. I've also looked at particle
creation (or the `Hawking effect') on the integer lattice [220] among other results.
Other current works are on quantum groups and quantum Riemannian geometries [202,205,211,221] over the
field F_2 of two elements (the `digital' case) as part of a growing interest in quantum
computing [218]. I am also currently working on geodesics on quantum Riemannian geometry using an approach of Beggs in
which a field of geodesics flows according to a Schroedinger-like equation. In [217] we apply this to
reformulate ordinary quantum mechanics and obtain a new proposal for relativistic quantum mechanics.

Many mathematicians have
explicitly picked up on my work. These include
traditional ‘pure Hopf algebraists’ such as D.
Radford, S. Montgomery, M. Cohen, M. Takeuchi,
category theorists such as B. Pareigis, D. Yetter, V.
Turaev, R. Street, and operator algebraists such as S.
Baaj, G. Skandalis, T. Yamanouchi and more recently A.
Connes. Also geometers such as J-H. Lu and A.
Weinstein and experts in q-special functions such as
T. Koornwinder. In general, my work includes both
abstract theorems about Hopf algebras, the
construction of nontrivial examples and links with
mathematical physics. On the mathematics side, I have
invited contributions to the Notices of the AMS and to
the Princeton Companion of Mathematics [26-27].

In 1993 I was awarded an
international prize for my work, the `Bleuler medal’.
In 1998 I was featured in Faces
of Maths, a photographic study of leading UK
mathematicians. In 1995, I published a 600+ page textbook [1]
explaining some of the foundations of the subject and
which remains a standard reference today. In 1998 I
gave a part III course `Quantum groups' in the pure
maths tripos at Cambridge University; its lecture notes
were published by the LMS[2]. I taught `Noncommutative
Geometry’ several years at the MSc level at Queen Mary
and a couple of years at the London Taught Course
Centre; the 2011 lectures were published in
[19]. Also, since 1992 I have organised a weekly seminar series on
quantum groups and quantum geometry, held some terms
in DPMMS and some terms in DAMTP when I was in
Cambridge and then held in Queen Mary, University of
London since I moved here in 1999. My 2008 popular science collected Book of Essays [4] addresses the true nature of space and
time. My latest book [5] is an 809 page monograph Quantum
Riemannian Geometry in the presitgious Springer Grundlehren Series.