Quantum Algebras
The seminar is normally organised by Shahn Majid weekly or fortnightly during term time. We have both informal study group talks and external invited ones  any suggestions or offers welcome. All are welcome.

DateRoomSpeakerTitle

M203Matt Fayers (QMUL)TBA

14/10/2009 2:00 PMM203J. LopezIntroduction to crystal bases I

21/10/2009 2:00 PMM203J. KlimBicrossproduct Hopf quasigroups associated to subgroupsSeminar series:

28/10/2009 1:00 PMM203J. LopezIntroduction to crystal bases IISeminar series:

11/11/2009 1:00 PMM203S. MajidTwisting of algebras and nonassociative Riemannian geometry

18/11/2009 1:00 PM203J. LopezCrystal basis of U(sl_2)Seminar series:

09/12/2009 1:00 PM203J. LopezLusztig completion and the integral form of a reductive algebraic groupSeminar series:

16/12/2009 1:00 PMM203J. LopezTBASeminar series:

03/02/2010 1:00 PM203J. Klim (QMUL)Integration and Fourier Theory on Hopf quasigroupsSeminar series:

10/02/2010 1:00 PM203S. Majid (QMUL)Lie theory of finite simple groups

24/02/2010 1:00 PM203S. Majid (QMUL)Lie theory of finite simple groups, II

10/03/2010 1:00 PM203E.J. Beggs (Swansea)Noncommutative complex structures

02/06/2010 2:00 PM203A. Zuevsky (NUI Galway)Vertex operator algebras on Riemann surfacesSeminar series:
We will show how to construct partition and npoint functions for vertex operator (super) algebras on higher genus Riemann surfaces. As a result a way to generate modular forms, twisted elliptic functions, generalized triple Jacobi and Fay's identities will be revealed.

11/10/2010 4:00 PM203D. Gurevich (Valenciennes)Advances in Braided GeometrySeminar series:

18/10/2010 4:00 PM203B. Zilber (Oxford)Discrete models for quantum particles and model theory

01/11/2010 3:00 PM203R. O'Buachalla (QMUL)Quantum CP^n

08/11/2010 3:00 PM203A. Cox (City)Diagrammatic KazhdanLusztig theory and the walled Brauer algebra

15/11/2010 3:00 PM203Ivan Tomasic (QMUL)Multiplicity in difference geometry

22/11/2010 3:00 PM203Shahn Majid (QMULNoncommutative geometry of graphsSeminar series:Quantum Algebras

17/01/2011 3:00 PM203A. Doering (Oxford)The Spectral Presheaf associated with a von Neumann algebra

14/02/2011 3:00 PM203Minhyong Kim (UCL)Nonabelian cohomology in Diophantine geometry

08/03/2011 4:00 PM203E.J. Beggs (Swansea)Noncommutative line bundles and the Thom constructionSeminar series:

28/03/2011 4:00 PM203T. Brzezinski (Swansea)Bundles over quantum real projective space

07/06/2011 4:00 PM203T. Ivanova (AUBG)Quadratic algebras, YangBaxter equations and ArtinShelter regularity

21/07/2011 4:00 PM203Ryszard Nest (Copenhagen)BaumConnes conjecture and quantum groupsSeminar series:

17/10/2011 3:00 PM410B. NoohiIntroduction to 2Categories, I

24/10/2011 2:00 PM410B. NoohiIntroduction to 2Categories, II

07/11/2011 2:00 PM103B. NoohiIntroduction to 2Categories, III

14/11/2011 2:00 PM103S. Majid2groups

23/11/2011 3:00 PM410R. O' BuachallaNoncommutative complex geometry

05/12/2011 2:00 PM103E.J. Beggs (Swansea)The algebra of differential operators in noncommutative geometry

17/01/2012 12:00 PM410I. TomasicElements of difference algebraic geometry EdGA, II

24/01/2012 12:00 PM410I. TomasicElements of difference algebraic geometry EdGA, III

31/01/2012 12:00 PM410B. NoohiIntroduction to stacks, I

07/02/2012 12:00 PM410B. NoohiIntroduction to stacks, II

14/02/2012 12:00 PM410B. NoohiIntroduction to Stacks, III

21/02/2012 12:00 PM410B. NoohiIntroduction to Stacks, IV

28/02/2012 12:00 PM410S. MajidLie theory of finite groups, II

06/03/2012 10:00 PMG.O. Jones 602S. MajidQuantum anomalies and the origin of time (outreach lecture)

13/03/2012 12:00 PM410S. MajidLie theory of finite groups, III

20/03/2012 12:00 PM410R. O'BuachallaNoncommutative complex geometry, II

08/05/2012 1:00 PM103D SternheimerQuantum groups and deformation quantization, relations and perspectives

09/05/2012 1:00 PM103D. SternheimerA conjectural quantum (spacetime) origin for internal symmetries

15/05/2012 1:00 PM103M. BassettConnesBost model

22/05/2012 1:00 PM103B. LewisPrimitive spectrum of nilpotent universal enveloping algebras, I

29/05/2012 1:00 PM103B. LewisPrimitive spectrum of nilpotent universal enveloping algebras, II

18/09/2012 1:00 PM203 Maths BuildingR. Gallego Torrome (Sao Paulo)Differential Geometry of Generalized Higher Order Fields
Given a base manifold M, a generalized higher order field is determined by fields along the lift of smooth curves on M (or along higher dimensional parameterized submanifolds of M) to a higher order jet bundle J^k_0(M). In this talk, we elaborate this notion, developing the fundamental tools of the differential geometry of generalized higher order fields. Then the de Rham cohomology and integration theory of generalized higher order differential forms is presented. Metrics of maximal acceleration are explained in detail, as examples of generalized higher order fields.

02/10/2012 2:00 PMB11Shahn MajidBraided approach to quantum groups, I

09/10/2012 1:00 PM203Shahn MajidBraided approach to quantum groups, II

16/10/2012 1:00 PM203Shahn MajidBraided approach to quantum groups, III

13/11/2012 12:00 PM203K. ArdakovIntroduction to geometric representation theory, I

20/11/2012 12:00 PM203K. ArdakovIntroduction to geometric representation theory, II

30/11/2012 11:30 PMB17K. ArdakovIntroduction to geometric representation theory, III

14/12/2012 11:00 AMB11Shahn MajidIntroduction to quantum geometric representations of quantum groups

15/01/2013 12:00 PM203W. TaoDuality for generalised differentials on quantum groups and Hopf quivers

28/01/2013 11:00 AM103B. NoohiArtinZhang noncommutative projective schemes, I
The ArtinZhang approach to noncommutative projective geometry is based on the observation that the the geometry of a scheme X is captured by the abelian category A of (quasi)coherent sheaves on X. So, one can turn things around and define a noncommutative scheme as an abelian category A satisfying certain conditions. I will briefly discuss some of the basic ideas involved in this approach.

04/02/2013 11:00 AM103B. NoohiArtinZhang noncommutative projective schemes, II
The ArtinZhang approach to noncommutative projective geometry is based on the observation that the the geometry of a scheme X is captured by the abelian category A of (quasi)coherent sheaves on X. So, one can turn things around and define a noncommutative scheme as an abelian category A satisfying certain conditions. I will briefly discuss some of the basic ideas involved in this approach.

11/02/2013 11:00 AM103B. NoohiArtinZhang noncommutative projective schemes, III
The ArtinZhang approach to noncommutative projective geometry is based on the observation that the the geometry of a scheme X is captured by the abelian category A of (quasi)coherent sheaves on X. So, one can turn things around and define a noncommutative scheme as an abelian category A satisfying certain conditions. I will briefly discuss some of the basic ideas involved in this approach.

13/03/2013 11:00 AM203E. Beggs (Swansea)From homotopy to Ito calculus and Hodge theory

14/03/2013 4:00 PM410G. Ginot (Paris)From Deligne conjecture in Hochschild cohomology to GerstenhaberSchack conjecture
The original Deligne conjecture (which has many proofs) asserts that the Hochschild chain complex of an associative algebra carries a structure of an E_2algebra (an algebra over the dimension 2little disk). Its higher generalization asserts the existence of an E_{n+1}algebra structure on the Hochschild cochain complex of E_nalgebras. GerstenhaberSchack have introduced an interesting cochain complex to study deformations of bialgebras. They conjectured that this complex has the structure of an analogue of a Poisson algebra (up to homotopy) whose bracket is of degree 2. We will explain how the Deligne conjecture gives a solution to the GerstenhaberSchack one.

27/03/2013 11:00 AM410Re O'Buachalla (Prague)Noncommutative Holomorphic Vector Bundles

15/10/2013 12:00 PM203S MajidReconstruction and quantisation of Riemannian manifolds

23/10/2013 2:00 PMMLTA FinkThe Hopf algebra of matroid valuations

29/10/2013 11:00 AM203S RamgoolamQuiver gauge theory combinatorics and permutation topological field theories

12/11/2013 11:00 AM203M.E. BassettFinite connected spaces and Boolean algebra

19/11/2013 11:00 AM203W. TaoNoncommutative differentials on cotangent spaces

26/11/2013 11:00 AM203E.J. Beggs (Swansea)Semiquantisation functor and Poisson geometry, II

03/12/2013 11:00 AM203H.L. Huang (Shandong)On finite pointed tensor categories

10/12/2013 11:00 AM203A. Doering (Oxford)Towards Duality Theory for Noncommutative Operator Algebras
Noncommutative Geometry is largely motivated by the idea that there should exist noncommutative spaces forming geometric counterparts to noncommutative (operator) algebras, just as compact Hausdorff spaces are in dual equivalence to unital commutative C*algebras. Yet, concrete examples of noncommutative spaces are rather rare. In this talk, I will show that to each unital C*algebra or von Neumann algebra one can associate a spectral presheaf that is a direct generalisation of the Gelfand spectrum to the noncommutative case. There is a contravariant functor from the category of unital C*algebras to a category of compact Hausdorff spacevalued presheaves containing the spectral presheaves. Moreover, the spectral presheaf of a von Neumann algebra with no type I_2 summand contains enough information to reconstruct the algebra. For unital C*algebras, partial reconstruction results exist. If time permits, I will also sketch some applications of the spectral presheaf in foundations of quantum physics, where it originally arose.

04/02/2014 3:00 PMM203Thomas Coyne (QMUL)An application of homotopy theory to stacks
I will introduce some basic ideas from topology and show how they can be applied to categories other than the category of topological spaces. Eventually, I hope to say something about applying some of these ideas to topological stacks.

25/02/2014 3:00 PMM203Thomas Coyne (QMUL)Model categories
I will define what a model category is. We shall see that they have many similar properties to those found in homological algebra or topology. I shall show that we can use these `homological algebra techniques' in a number of other categories.

04/03/2014 3:00 PMM203Simona Paoli (Leicester)A model of weak 2categories and of categories of fractions
Higher categorical structures arise commonly in several areas of mathematics, such as homotopy theory, algebraic geometry and mathematical physics. The prototype in dimension 2 of a weak 2category is the classical notion of bicategory. In this talk I present a new model of a weak 2category consisting of a certain class of double categories, called weakly globular double categories. This model offers several advantages, and in particular gives rise to a new construction of the category of fractions of a category, which avoids the size issues of the classical category of fractions.
The talk begins with a gentle introduction to weak 2categories and weak 2groupoids as they arise in homotopy theory, followed by some background on double categories. I will then introduce weakly globular double categories and illustrate their equivalence with bicategories, and their use in defining a weakly globular double category of fractions. This is joint work with Dorette Pronk.

11/03/2014 3:30 PMM203Jeffrey Giansiracusa (Swansea)Gequivariant openclosed TCFTs
Open 2d TCFTs correspond to cyclic Ainfinity algebras, and Costello showed that any open theory has a universal extension to an openclosed theory in which the closed state space (the value of the functor on a circle) is the Hochschild homology of the open algebra. We will give a Gequivariant generalization of this theorem, meaning that the surfaces are now equipped with principal Gbundles. Equivariant Hochschild homology and a new ribbon graph decomposition of the moduli space of surfaces with Gbundles are the principal ingredients. This is joint work with Ramses FernandezValencia.

20/01/2015 1:00 PM410W. TaoNoncommutative differentials on tangent bundles from bicrossproduct construction

03/02/2015 1:00 PM513M. Sadrzadeh (CS)Multilinear algebraic semantics for natural language (and some quantum connections)

10/02/2015 1:00 PM513A. Harju (Helsinki)Spectral triples on orbifolds

17/02/2015 1:00 PM410A. Harju (Helsinki)Quantum orbifolds

03/03/2015 1:00 PM513S. Ramgoolam (Physics)Four dimensional CFT as two dimensional SO(4,2)invariant topological field theory

10/03/2015 1:00 PM410D. Schäppi (Sheffield)Universal weakly Tannakian categories
Classical Tannaka duality is a duality between groups and their categories of representations. It answers two basic questions: can we recover the group from its category of representations, and can we characterize categories of representations abstractly? These are often called the reconstruction problem and the recognition problem. In the context of affine group schemes over a field, the recognition problem was solved by Saavedra and Deligne using the notion of a (neutral) Tannakian category.
In my first talk on Monday [in the Algebra seminar] I explained how this theory can be generalized to the context of certain algebraic stacks and their categories of coherent sheaves (using the notion of a weakly Tannakian category). Today I will talk about work in progress to construct universal weakly Tannakian categories and some of their applications. The aim is to interpret various constructions on stacks (for example fiber products) in terms of the corresponding weakly Tannakian categories.

17/03/2015 1:00 PM410open discussionLinear logic, shuffle algebras and 2vector spaces
This is an open discussion for anyone who wants to talk about the topics in the title, such as in the
two papershttp://www.site.uottawa.ca/%7Ephil/papers/shuf.pdf(link is external)
http://math.ucr.edu/home/baez/2rep [PDF 1,183KB]
All are welcome

02/06/2015 2:00 PM203E.J. Beggs (Swansea)On the construction of spectral triples

21/07/2015 2:00 PM103B. Jurco (Charles U, Prague)Higher gauge theory

24/11/2015 10:00 AMMaths 410Shahn MajidNew construction of braidedLie algebras

01/12/2015 10:00 AMMaths 410G. BianconiNetwork geometry
Recently, network geometry and topology are gaining increasing interest in the context of complexity science.
Progress in this field is expected to have relevance for a number of applications, including a new generation of routing protocols, data mining techniques, advances in the theoretical foundations of network clustering, and the development of a geometric information theory of networks. It is also believed that network geometry could provide a theoretical framework for establishing crossfertilization between the field of network theory and quantum gravity.In this blackboard talk, I will present recent results on network geometry showing that selforganized Complex Quantum Network Manifolds in d>2 are scalefree, i.e. they are characterized by a very heterogeneous degree distributions like most complex networks. This networks can be mapped to quantum network states, and their quantum nature is revealed by the emergence of quantum statistics characterizing the statistical properties of their structure.
A generalization of these networks is constituted by Network Geometry with Flavour, providing a general framework to understand the interplay between dimensionality and quantum statistics in these growing networks formed by simplicies of dimension d.

08/12/2015 10:00 AMMaths 410S. MajidHodge duality as Fourier transform

03/02/2016 3:00 PM103D. Tourigny (DAMTP, Cambridge)Deformed Hamiltonian vector fields

10/02/2016 3:00 PM103L. WilliamsCourant algebroids and PoissonRiemannian geometry

17/02/2016 3:00 PM103A. PacholFrom Hopf algebras and algebroids to quantum spacetimes

09/03/2016 3:00 PMMaths 103Shahn MajidNoncommutative geometry over finite fields

16/03/2016 3:00 PM103C. Fritz (Sussex)Noncommutative nonassociative models of spacetime

05/05/2016 12:00 PMMaths 103D. Gurevich (Valenciennes)Quantum matrix algebras and braided Yangians
By quantum matrix algebras I mean algebras related to quantum groups and close in a sense to that Mat(m). These algebras have numerous applications. In particular, by using them (more precisely, the socalled reflection equation algebras) we succeeded in defining partial derivatives on the enveloping algebras U(gl(m)). This enabled us to develop a new approach to Noncommutative Geometry: all objects of this type geometry are deformations of their classical counterparts. Also, with the help of the reflection equation algebras we introduced the notion of braided Yangians, which are natural generalizations of the usual ones and have similar properties.

01/06/2016 2:00 PMMaths 103C. Lomp (Porto)Integral Calculus on Quantum Exterior Algebras
Homconnections and associated integral forms have been introduced and studied by T.Brzezinski as an adjoint version of the usual notion of a connection in noncommutative geometry. Given a flat homconnection on a differential calculus (Omega, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the noncommutative de Rham complex (in the sense of Brzezinski et al.). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat homconnection. We specialise our study to the case where an ndimensional differential calculus can be constructed on a quantum exterior algebra over an Abimodule. Criteria are given for free bimodules with diagonal or upper triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum nspace.

22/06/2016 1:00 PMMaths 103E.J. Beggs (Swansea)A differential graded category in noncommutative geometry

18/07/2016 1:00 PM203P. Osei (Perimeter Institute)Quantum isometry groups and semidualisation in 3d gravity

06/10/2016 3:00 PM203Pierre Bieliavsky (Université catholique de Louvain)Noncommutative surfaces in higher genera
We construct a version of noncommutative surfaces analogous to the wellknown noncommutative torus. More precisely, we define an associative deformed multiplication of the algebra of smooth functions on any compact surface of negative constant curvature. The deformation is nonformal in the sense that the deformed product of any two smooth functions is again a smooth functionrather than a formal power series as in formal starproduct theory. The deformation consists in a real oneparameter smooth family of associative products whose infinite jet at the value zero of the parameter defines an associative formal star product directed by the Kaehler two form. For any value of the parameter the deformed algebra admits a natural topology which endows it with the structure of a Frechet algebra. Each of these noncommutative Frechet algebras carries a trace defined by the usual integral on the surface. Moreover, these algebras are tracial w.r.t. the trace form, in the sense that the trace of the deformed product of two functions equals the integral of the point wise multiplication of these functions. The deformed algebra when equipped with the complex conjugation also turns into a staralgebra. In particular they extend to the space of square integrable distributions as an algebra of HilbertSchmidt operators. A quantization in the usual sense represent them as subalgebras of bounded operators acting on the projective discrete series representations of SL(2,R).

14/10/2016 1:00 PM203Ivan TomasicCohomology of difference algebraic groups, I

28/10/2016 1:00 PM203Ivan TomasicCohomology of difference algebraic groups, II

01/11/2016 1:00 PM410A. PacholClassification of noncommutative differentials in two and three dimensions

18/11/2016 1:00 PM203I. TomasicCohomology of difference algebraic groups, III

02/12/2016 1:00 PM410Greg Ginot (Paris)Quantization of bialgebras via deformation theory
We will discuss a new proof of EtingofKazdhan quantization theorem via an approach to deformation quantization of Lie bialgebras similar to Kontsevich/Tamarkin formality for quantization of Poisson manifolds. The idea is based on a relationship in between deformation complexes of (homotopy) dgbialgebras and those of $E_2$algebras and on a proof of a formality theorem conjectured by Kontsevich. This is joint work with Sinan Yalin.

24/01/2017 2:00 PM203Ivan TomasicPaths in digraphs, difference algebra and entropy

17/02/2017 3:00 PMG.O.Jones LG1Joao MartinsCategorification of the KnizhnikZamolodchikov connection via infinitesimal 2braidings
I will report on a long term joint project with Lucio Cirio and Florian Schaetz on categorifications of the KnizhnikZamolodchikov connection via infinitesimal 2braidings. In particular, I will describe a categorification of the DrinfeldKohno Lie algebra of chord diagrams in the realm of a differential crossed module of horizontal 2chord diagrams. I will also explain how this categorified Lie algebra arises from a linearization (called an infinitesimal braided 2category) of the axioms defining a braided monoidal 2category.
This talk is based on:
T Kohno: Higher holonomy of formal homology connections and braid cobordisms. J. Knot Theory Ramifications, 25, 1642007 (2016)
L. S. Cirio and J Faria Martins: Infinitesimal 2braidings and differential crossed modules. Advances in Mathematics, Volume 277, 4 June 2015, Pages 426491
L. S. Cirio and J Faria Martins: Categorifying the Knizhnik–Zamolodchikov connection. Differential Geometry and its Applications, Volume 30, Issue 3, June 2012, Pages 238–261. 
28/02/2017 2:00 PMW316Nicola GambinoOperads in 2algebraic geometry

21/03/2017 2:00 PMW316O.A. Laudal (Oslo)Gravity induced from quantum `timespace'

28/03/2017 2:00 PMW316Ryan AzizCodouble bosonisation

09/05/2017 2:00 PMW316R Hazrat (Western Sydney University)Steinberg algebras
Steinberg algebras were introduced 4 years ago and they cover many algebras constructed in a combinatorial manner, such as Leavitt path algebras which arise from directed graphs. We introduce these algebras and state some of their main structural properties proved recently, including their irreducible representations and the Morita theory which gives a unified approach to equivalence of path algebras coming from symbolic dynamics.

30/05/2017 2:00 PMW316R. Akylzhanov (Imperial)Smooth dense subalgebras on compact quantum groups

11/07/2017 2:00 PMW316Robert Laugwitz (Rutgers)Categorical Modules over the Relative Monoidal Center
I will explain how a comodule algebra over a bialgebra is also a comodule algebra over its Drinfeld double in a nontrivial way. Working with modules, this result holds when working in braided monoidal categories, and is hence valid for the double bosonization of Majid. A special case recovers a result by Lu showing that the Heisenberg double is a comodule algebra over the Drinfeld double. From a categorical point of view, this construction is part of a bigger picture of how to construct categorical modules over the relative monoidal center, generalizing work of EtingofOstrik et al.

22/01/2013 12:00 PM203W. TaoDuality for generalised differentials on quantum groups and Hopf quivers, II

27/01/2015 1:00 PM410S. MajidEmergence of BertottiRobinson metric from noncommutative spacetime
We show that a hypothesis that spacetime is quantum with coordinate algebra [x_i , t] = λ x_i , and spherical symmetry under rotations of the x_i, essentially requires in the classical limit that the spacetime metric is the BertottiRobinson metric, i.e. a solution of Einstein’s equations with cosmo logical constant and a nonnull electromagnetic field. We also describe the noncommutative geometry and the full moduli space of metrics that can emerge as classical limits from this algebra.

07/07/2015 2:00 PMQueens E303G. Bianconi (QMUL)Complex Quantum Network Geometries
Networks are topological and geometric structures used to describe systems as different as the Internet, the brain or the quantum structure of spacetime. Here we define complex quantum network geometries and manifolds, describing the underlying structure of growing simplicial complexes. These networks grow according to a nonequilibrium dynamics. Their temporal dynamics is a classical evolution describing a given path of a path integral defining the quantum evolution of quantum network states. The quantum network states are characterized by quantum occupation numbers that can be mapped respectively to the nodes, and (d1)faces of a ddimensional simplicial complexes. We show that these networks follow quantum statistics and that they can undergo structural phase transitions where the geometrical properties of the networks change drastically.
One class of these type of networks are Complex Quantum Network Manifolds (CQNM) constructed from growing simplicial complexes of dimension d. Here we show that in d=2 CQNM are homogeneous networks while for d>2 they are scalefree i.e. they are characterized by large inhomogeneities of degrees like most complex networks. From the selforganized evolution of CQNM quantum statistics emerge spontaneously. We define the generalized degrees associated with the δfaces of the ddimensional CQNMs, and we show that the statistics of these generalized degrees can either follow FermiDirac, Boltzmann or BoseEinstein distributions depending on the dimension of the δfaces. 
28/11/2011 4:00 PM103I. TomasicElements of difference algebraic geometry E￼\sigma GA, I

05/12/2016 12:00 PMB.R. 4.01 (Bancroft Rd)Ping Xu (Penn State)KontsevichDuflo theorem for Lie pairs
The KontsevichDuflo theorem asserts that, for any complex manifold X, the HochschildKostantRosenberg map twisted by the square root of the Todd class of the tangent bundle of X is an isomorphism of associative algebras form the sheaf cohomology group H•(X,∧TX) to the Hochschild cohomology group HH•(X). We will show that, beyond the sole complex manifolds, the KontsevichDuflo theorem extends to a very wide range of geometric situations describable in terms of Lie algebroids and including foliations and actions of Lie groups on smooth manifolds.

17/11/2017 2:58 PMW316Shahn MajidQuantum gravity on a square graph
Noncommutative Riemannian geometry can be applied in principle to any bidirected graph, with the metric viewed as assigning weights to each arrow. We completely solve for a quantum LeviCivita connection for any metric with undirected edge weights on a square graph. We find a 1parameter family of quantumLeviCivita connections and a proposal for an EinsteinHilbert action that does not depend on the parameter. The minimum of this action or `energy' is precisely the rectangular case where parallel edges have the same weight. We also allow negative weights corresponding to a Minkowski signature time direction and we look at the eigenvalues of the quantumgeometric graph Laplacian in both signatures.

21/11/2017 2:00 PMW316Re O'Buachalla (Warsaw)Spectral Triples and Quantum Homogeneous Kähler Spaces
The notion of a noncommutative Kähler structure was recently introduced as a framework in which to understand the metric aspects of Heckenberger and Kolb's remarkable covariant differential calculi over the cominiscule quantum flag manifolds. Many of the fundamental results of classical Kähler geometry are shown to follow from the existence of such a structure, allowing for the definition of noncommutative Lefschetz, Hodge, DolbeaultDirac, and Laplace operators. In this talk we will discuss how a Kähler structure can be used to complete a calculus to a Hilbert space, and show that when the calculus is of so called ladder type, the holomorphic and antiholomorphic DolbeaultDirac operators give spectral triples. Moreover, we show how Euler characteristics can be used to calculate the indexes of the Dirac operators, presenting the possibility of doing index calculations using noncommutative generalisations of classical vanishing theorems. The general theory will be applied to quantum projective space where a direct noncommutative generalisation of the Kodaira vanishing theorem allows us to show that both Dirac operators have nonzero index, and so, nonzero Khomology class. Time permitting, we will show how full Hilbert C*modules can also be constructed from a Kähler structure, and discuss conjectured examples from the B and Dseries quantum groups, namely the odd and even dimensional quantum quadrics.

05/12/2017 2:00 PMLK1 (Lock Keeper's Cottage)Johan NoldusGenerally covariant quantum mechanics

23/01/2018 11:00 AMQueens' Building, Room W316Shahn MajidIntroduction to quantum Riemannian geometry, I
1st of Graduate Lecture Course based on version 1.0 of the forthcoming book with Beggs of the same title.

30/01/2018 11:00 AMQueens' Building, Room W316Shahn MajidQuantum Riemannian Geometry, II
2nd of Graduate Lecture Course based on version 1.0 of forthcoming textbook with Beggs of this title

06/02/2018 11:00 AMQueens' Building, Room W316E.J. Beggs (Swansea)Quantum Riemannian Geometry, III
3rd of Graduate Lecture Course based on Version 1.0 of our forthcoming book of this title

13/02/2018 11:00 AMQueens' Building, Room W316Shahn MajidQuantum Riemannian Geometry, IV
We continue lectures based on a forthcoming book with the same title. Topics will include quantisation of coadjoint orbits and a discrete version of the same, and quantisations defined by conformal vector felds.

06/03/2018 11:00 AMW316, Queens' BuildingShahn MajidQuantum Riemannian Geometry, V
This week I will start to cover the two most wellknown `quantum spacetimes' in 3 and 4 dimensions. Based on chapter 9 of my forthcoming book and some recent new results.

19/07/2018 11:00 AMRoom W316, Queens' BuildingC. Furey (DAMTP, Cambridge)Division algebras and the Standard Model of elementary particles
It is a little known fact that the division algebras R, C, H, and O can encode much of the behaviour of elementary particle physics. Already by 1937, Arthur Conway had seen how to use the complex quaternions to encode most of the Lorentz representations that we still use in the standard model today. We will extend Conway's results to see how each of the standard model's Lorentz representations arise from a generalized notion of ideals within the algebra. Finally, we will show how the 8Cdimensional complex octonions can yield the behaviour of not one, but three generations of quarks and leptons, as seen by the strong force. This talk will assume as little background knowledge as is reasonably possible; all are welcome.

05/06/2018 1:00 PMQueens' Building, Room W316Sina Hazratpour (Birmingham)Fibrations of topoi from refinement of theories
Grothendieck fibrations play an important role in category theory and also in providing semantics of dependent type theories, most notably via comprehension categories .
In the first part of the talk, I review the basics of Grothendieck fibrations for the benefit of those in the audience not already familiar with them. I also review the generalization of Grothendieck fibrations to the setting of bicategories in two different ways: fibrations internal to bicategories [Str80], [Joh93] and fibrations of bicategories [Buc14]. I will show how these two notions of fibrations are linked together by introducing displayed bicategories.
In the second part, I employ this link to show how some of (op)fibrations of topoi arise from refinement (aka extension) of logical theories which are classi fied by topoi in consideration. Important examples of (op)fibred topoi arising this way will be given, in particular I demonstrate how local homeomorphisms of topoi can be obtained as opfibrations. This connection is in line with the conception of topoi as generalized spaces.
References
[Buc14] Mitchell Buckley. Fibred 2categories and bicategories J. Pure Appl. Algebra, 218(6):1034–1074, 2014.
[Joh93] Peter Johnstone. Fibrations and partial products in a 2category. Applied Categorical Structures, 1(2):141–179, 1993.
[Str74] Ross Street. Fibrations and Yoneda’s lemma in a 2category. In Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 104–133. Lecture Notes in Math., Vol. 420. Springer, Berlin, 1974.
[Str80] Ross Street. Fibrations in bicategories. Cahiers de Topologie et G ́eom ́etrie Diff ́erentielle Cat ́egoriques 21(2):111–160, 1980.
[Vic17] Steven Vickers. Arithmetic universes and classifying toposes. Cahiers de Topologie et G ́eom ́etrie Diff ́erentielle Cat ́egoriques 58(4):213–248, 2017.

13/03/2018 11:00 AMW316, Queens' BuildingShahn MajidQuantum Riemannian Geometry, VI
Last of the series based on the forthcoming book of this title. I will try to cover noncommutative blackholes.