Quantum Algebras

DateRoomSpeakerTitle

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.

24/06/2019 1:00 PMGraduate Centre, Room GC205P. Xu (Penn State)Atiyah class and Todd class of dg manifolds
Exponential maps arise naturally in the contexts of Lie theory and of smooth manifolds. The infinite jets of these exponential maps are related to the PoincaréBirkhoffWitt isomorphism and the complete symbols of differential operators. We will discuss how these exponential maps can be extend to the context of dg manifolds. As an application, we will describe a natural Linfinity structure associated with the Atiyah class of a dg manifold.

23/08/2019 12:00 PMB.R. 3.02 (Bancroft Rd)D. BarNatan (Toronto)Everything around sl_{2+}^€ is DoPeGDO. So what?
Abstract. I'll explain what "everything around" means: classical and quantum $m, \Delta, S, {\rm Tr}, R, C, \theta$ as well as $P,\phi, J, D\!\!\!\!D$ and more, and all of their compositions. What DoPeGDO means: the category of Docile Perturbed Gaussian Differential Operators. And what $sl_{2+}^\epsilon$ means: a solvable approximation of the semisimple Lie algebra $sl_2$.
Knot theorists should rejoice because all this leads to very powerful and wellbehaved polytimecomputable knot invariants. Quantum algebraists should rejoice because it's a realistic playground for testing complicated equations and theories.
This is joint work with Roland van der Veen and continues work by Rozansky and Overbay.

21/05/2019 1:00 PMQueens' Building, Room: W316A. Schenkel (Nottingham)Higher structures in algebraic quantum field theory
Algebraic quantum field theory (AQFT) is a wellestablished framework to axiomatize and study quantum field theories on Lorentzian manifolds, i.e. spacetimes in the sense of Einstein’s theory of general relativity. The “traditional" AQFTs appearing in the literature are only 1categorical algebraic structures, which turns out to be insufficient to capture the important examples given by quantum gauge theories. In this talk I will give a rather nontechnical overview of our recent works towards establishing a higher categorical framework for AQFT. I will also provide a sketch how examples of such higher categorical theories can be constructed from (linear approximations of) derived stacks and how they relate to the BRST/BV formalism.

22/03/2019 3:00 PMQueens' Building, Room: W316G. Ginot (Paris 13)ChernSimons theory, String Topology and the Derived Character Variety
Abstract:
This is joint work in progress with O. Gwilliam and M. Zeinalian. String topology arised as a higher dimensional generalisation of GoldmanTuraev Lie bialgebra structure on free loops on a surface which is closely related to the Poisson algebra of functions on character varieties. Our aim is to consider a higher (and derived) version of this relation relating string topology and quantization of ChernSimons field theory.

05/02/2019 1:00 PMQueens' Building, Room: W316Aryan GhobadiHopf monads

26/02/2019 1:00 PMQueens' Building, Room: W316Ryan AzizDifferential calculi by supercobosonisation

29/01/2019 1:00 PMQueens' Building, Room: W316Anna PacholDigital quantum geometries

22/01/2019 1:00 PMQueens' Building, Room: W316Shahn MajidQuantum Riemannian geometry and Hawking effect on the integers

08/01/2019 1:00 PMQueens' Building. Room: W316Edwin Beggs (Swansea)Quantum geodesics and positive maps on C*algebras

13/12/2018 1:00 PMQueens' Building, Room: W316Liam WilliamsPoisson principal bundles, II

06/12/2018 1:00 PMQueens' Building, Room: W316Andrzej Sitarz (Krakow)The Standard Model and Noncommutative Geometry 2.0

25/10/2018 1:00 PMQueens' Building, Room: W316Ryan AzizSupercodoublebosonisation and quantum differentials

18/10/2018 1:00 PMGraduate Centre, Room: GC205Shahn MajidQuantum Gravity on four points

15/11/2018 1:00 PMGraduate Centre, Room: GC205Rauan AkylzhanovHormanderMihlin L^pmultiplier theorems on noncommutative spaces
Abstract
(joint work with Michael Ruzhansky)
We construct a Fouriertype formalism on von Neumann algebras. In this setting, we establish Paleytype inequalities on semifinite von Neumann algebras. Using these inequalities in combination with quantum group version of Pontryagin duality, we obtain a simple and elegant proof of Ho ̈rmander Mihlin L^pmultiplier theorem. As a particular case, we recover [2]. If time permits, we shall discuss a general H ̈ormanderMihlin Lpmultiplier theorem on semifinite von Neumann algebras.

[1] R. Akylzhanov, S. Majid, and M. Ruzhansky. Smooth dense subalge bras and Fourier multipliers on compact quantum groups. Comm. Math. Phys., 362(3):761–799, Sep 2018.

[2] R. Akylzhanov and M. Ruzhansky. L^pL^q multipliers on locally compact groups. arXiv:1510.06321, submitted to Journal of Functional Analysis, 2017.

[3] L. Grafakos and L. Slav ́ıkov ́a. A sharp Version of the Ho ̈rmander multiplier theorem. Int. Math. Res. Not. IMRN, 2017.


01/11/2018 1:00 PMQueens' Building, Room: W316Liam WilliamsPoisson principal bundles

19/07/2018 12:00 PMRoom 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.

28/06/2019 2:00 PMBancroft Road, Room 4.01John Barrett (Nottingham)Finite spectral triples for the fuzzy torus
Abstract: Finite real spectral triples are defined to characterise the noncommutative geometry of a fuzzy torus. The geometries are the noncommutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four unique spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus. This is joint work with James Gaunt.

04/10/2019 11:00 AMMathematical Sciences Building, Maths seminar roomTomasz Brzezinski (Swansea)Towards noncommutative bundles with homogeneous fibres

18/10/2019 11:00 AMMathematical Sciences Building, Maths seminar roomShahn MajidQuantum geometry and logic

01/11/2019 11:00 AMMathematical Sciences Building, Maths seminar roomJ. Argota QuirozQuantum field theory on R x Z_n

15/11/2019 11:00 AMMathematical Sciences Building, Maths seminar roomJ. Argota QuirozQuantum field theory on R x Z_n, Part II

03/12/2019 1:00 PMGraduate Centre Building GC114J. Lukierski (Wroclow)Quantum deformations of twistors

29/11/2019 11:00 AMMathematics Building, Room MB503A. GhobadiBialgebroids and graphs

28/01/2020 12:00 PMMaths Building Seminar Room MB503Shahn MajidQuantum geodesics in quantum mechanics

17/02/2021 2:00 PMZoom Seminar (standard access details, contact S. Majid if need them)Ginestra BianconiEmergent hyperbolic geometry and dynamics
Abstract: Simplicial complexes naturally describe discrete topological spaces and when their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks. In this talk, we present nonequilibrium statistical mechanics approaches to model large simplicial complexes. We propose the simplicial complex model of Network Geometry with Flavor and we explore the hyperbolic nature of its emergent geometry and the interesting result that quantum statistics describe their random topological structure. Finally, we reveal the rich interplay between Network Geometry with Flavor and dynamics. In particular, we discuss the critical properties of higherorder percolation and diffusion investigated using a realspace renormalization group approach.

07/02/2020 11:00 AMMathematical Sciences Building, Room MB503S. Willerton (Sheffield)Hopf Monads, Hopf algebras and diagrammatics
[Joint work with Christos Aravanis.]
Inside the derived category of a complex manifold there is a certain Lie algebra object with a universal enveloping algebra U, the latter can be defined in terms of a push forward and pull back along the diagonal embedding. This is formally analogous to the group algebra of a finite group in its representation category (which you might want to think of in terms of the (co)end construction for the representation category). I will show how this can be used to equip U with the structure of a Hopf algebra which acts on the whole category, using the technology of Hopf monads (due to Bruguiere, Lack and Virelizier). The key to the existence of the antipode is the projection formula: the fact that the projection formula holds has an appealing description in terms of surface diagrams (one dimension up from string diagrams). 
11/02/2020 12:00 PMMathematical Sciences Building, Room MB503R. WilsonSome potential applications of SL(4,R)
Since 1928, quantum mechanics has been based on the Dirac group SL(2,C)
of 2x2 complex matrices with determinant 1. If one allows the complex
structure to vary, one obtains the group SL(4,R) of all 4x4 real matrices
with determinant 1. I show how this larger group allows one to model
a great deal more of fundamental physics, possibly including a quantum
theory of gravity. 
25/02/2020 12:00 PMMathematical Sciences Building, Room: MB503Shahn MajidQuantum geometry and probability

07/04/2020 12:00 PMVideo seminar  contact s.majid@qmul.ac.uk to registerEdwin Beggs (Swansea)Quantum geodesic deviation

10/03/2020 12:00 PMMathematical Sciences Building, Room MB503Julio Argota QuirozQFT on R x Z_n, Part III: curved spacetime

19/05/2020 12:00 PMVideo seminar  contact s.majid@qmul.ac.uk for detailsE. Lira TorresQuantum Geometry of the Fuzzy Sphere

30/09/2020 2:00 PMZoom seminar  email s.majid@qmul.ac.uk to registerFrancesco Toppan (CBPF, Rio de Janeiro)A theoretical test of Z_2xZ_2graded parastatistics
The recent surge of interest in Z2xZ2graded invariant mechanics poses the challenge of understanding the physical consequences of a Z2xZ2graded symmetry. Nontrivial physics can be detected in the multiparticle sector, being induced by the Z2xZ2graded parastatistics obeyed by the particles.
The toy model of the N=4 supersymmetric/ Z2xZ2graded oscillator is discussed. In this setup the oneparticle energy levels and their degenerations are the same for both supersymmetric and Z2xZ2graded versions. In the multiparticle sector a measurement of an observable operator on suitable states can discriminate whether the system under consideration is composed by ordinary bosons/fermions or by Z2xZ2graded particles.
Therefore, Z2xZ2graded mechanics has experimentally testable consequences.The multiparticle sector is encoded in the coproduct of a Hopf algebra defined on a Universal Enveloping Algebra of a graded Lie superalgebra with a braided tensor product. The talk is based on arXiv:2008.11554.

14/10/2020 2:00 PMZoom Seminar (standard Meeting ID and passcode, email S. Majid if need these)Aryan GhobadiSkew Braces as Hopf Algebras in SupLat
Settheoretical solutions of the YangBaxter equation have primarily been studied with group and ring theoretical techniques, via skew braces. Here, we present a new approach by embedding these solution into the category of complete lattices and joinpreserving morphisms, SupLat. This allows us to utilise the usual techniques from Hopf algebras, to study settheoretical YBE solutions. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its remnant and a coquasitriangular structure on H induces a braiding operator on R(H). We also prove that any group with a braiding operator can be realised as the remnant of such a Hopf algebra. It is wellknown that a group with a braiding operator has an induced secondary group structure, which makes it a skew left brace. We demonstrate that the secondary group structure agrees with the projection of transmutation on the coquasitriangular Hopf algebra. Additionally, for any YBE solution, we obtain a Hopf algebra via FRT re construction in SupLat, whose remnant recovers the universal skew brace of the solution.

25/11/2020 2:00 PMZoom seminar (standard details, email s.majid@qmul.ac.uk if need them)S. Ramgoolam (SPA)Kronecker coefficients and lattices of ribbon graphsBipartite ribbon graphs arise in the enumeration of polynomial invariants of tensor variables. They correspond to Belyi maps between twodimensional surfaces and have an elegant combinatoric characterisation in terms of permutation pairs, subject to an equivalence relation. An associative algebra constructed from the permutation pairs has a block decomposition in terms of simple matrix algebras labelled by triples of Young diagrams. For each triple the size of the block is equal to the Kronecker coefficient for the triple.Solvable quantum mechanical models on these algebras are used to realize Kronecker coefficients in terms of the counting of vectors in a lattice of ribbon graphs. This points towards quantum algorithms for the calculation of Kronecker coefficients.

28/10/2020 2:00 PMZoom seminar (email s.majid@qmul.ac.uk if you need meeting login details)Shahn MajidContinuum limit of discrete quantum geometries
The differential calculus on the polygon Z_n is 2D. We answer what this becomes in the coninuum limit as n> infinity, namely a certain 2D calculus on the circle S^1. What happens to the natural graph metric and conneciton is more subtle and needs a scaling limit. I'll also discuss some open problems as to how to extend this to limits of other discrete quantum geometries.

11/11/2020 2:00 PMZoom seminar (email s.majid@qmul.ac.uk if need access details)J ArgotaQuirozDimension jump in noncommutative spacetime models

09/12/2020 2:00 PMZoom seminar (email s.majid@qmul.ac.uk if need access details)A. Pachol (SBCS)Digital Quantum Groups
Talk based on a recent paper JMP 2020: We find and classify all bialgebras and Hopf algebras or “quantum groups” of dimension ≤ 4 over the field F_2 = {0, 1}. We summarize our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras and, among them, 25 Hopf algebras, with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover selfdual and resembles a digital version of u_q(sl_2). We also find a unique selfdual Hopf algebra in one anyonic variable x^4 = 0. For all our Hopf algebras, we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or “universal Rmatrix” structures on our Hopf algebras. These induce solutions of the Yang–Baxter or braid relations in any representation.

03/02/2021 2:00 PMZoom Seminar (standard access details, contact S. Majid if need them)Lukas Muller (MPIM, Bonn)Cyclic framed little disks algebras, GrothendieckVerdier duality and handlebody group representations
The goal of this talk is to briefly review the concept of GrothendieckVerdier categories as introduced by BoyarchenkoDrinfeld based on Barr's notion of a *autonomous category and explain its relation to lowdimensional topology. I will show that they are equivalent to cyclic algebras over the framed little disk operad with values in a suitably chosen symmetric monoidal bicategory of linear categories. As an application, I will construct consistent systems of handlebody group representations. The talk is based on joint work with Lukas Woike

20/01/2021 2:00 PMZoom Seminar (usual access details, contact S. Majid if need them)Paolo Saracco (Brussels)ConnesMoscovici's bialgebroids as universal enveloping algebras of anchored Lie algebras
From an algebraic perspective, a bialgebroid can be thought of as an extension of the notion of a bialgebra, where the base field has been substituted by a general (not necessarily commutative) algebra. It is wellknown that the space of primitive elements of a bialgebra is a Lie algebra, but what about the space of primitives of a bialgebroid?
If the base algebra is commutative and the bialgebroid is cocommutative, then the primitives form a LieRinehart algebra and, conversely, the universal enveloping algebra of a LieRinehart algebra is a cocommutative bialgebroid over the given commutative base algebra. However, in general, primitives only form what we call an anchored Lie algebra: a Lie algebra acting on the base algebra by derivations.
In this talk we see how the ConnesMoscovici's bialgebroid construction provides in a natural way universal enveloping algebras for anchored Lie algebras. In this way, representations of an anchored Lie algebra correspond to modules over the ConnesMoscovici's bialgebroid. Moreover, we recover an adjunction between the category of bialgebroids over a fixed base A and the category of anchored Lie algebras over A which, under suitable hypotheses, allows us to describe intrinsically those bialgebroids which are isomorphic to a ConnesMoscovici's bialgebroid.
Time permitting, I will briefly show how representations of a finitedimensional anchored Lie algebra are naturally related to certain (elementary) bimodules with flat bimodule connection for the firstorder differential calculus associated with the given anchored Lie algebra. 
03/03/2021 2:00 PMZoom seminar (standard access, contact S. Majid if needed)Shahn MajidBraided ZXcalculus in quantum computing
We study the algebraic structure behind ZXcalculus in quantum computing. An important role is played by an interacting pair of Hopf algebras and we explain the general noncommutative construction of these as well as the braided version.

17/03/2021 2:00 PMZoom seminar (standard access, contact S. Majid if needed)Shahn MajidDiscrete ChernSimons and topological quantum computing

31/03/2021 2:00 PMZoom seminar (standard access, contact S. Majid if needed)T. Weber (U. Bologna)Cartan calculus and Riemannian geometry on braided commutative algebras
A wellstudied and rather explicit framework of noncommutative geometry is given by Drinfel'd twist deformation quantization of differential geometry. In particular, there are deformed versions of the Cartan calculus and Riemannian geometry. I intend to discuss a generalization of this setting to algebras with a triangular Hopf algebra symmetry. My main goals are to explain the construction of the canonical braided Cartan calculus and to prove existence and uniqueness of an equivariant LeviCivita connection on braided commutative algebras. It turns out that Drinfel'd twists correspond to equivalence classes in braided commutative geometry, in the sense that the Drinfel'd functor intertwines all operations. This reveals the equivalence of our initial examples of classical and twisted differential geometry.

14/04/2021 2:00 PMZoom seminar (usual access, contact S. Majid if needed)R. van der Veen (Groningen)Quantum character varieties of knot groups through bottom tanglesThe fundamental group of a knot complement (the knot group) contains a lot of information about the knot, however extracting this information is not so simple. One approach is to study the variety of all representation of the knot group into an algebraic group. The case where the group is SL(2) is especially popular due to its relation to Thurston's geometrization program of 3manifolds: Most knot groups allow a faithful representation into SL(2,C) that is unique up to conjugation.In joint work with Jun Murakami (Arxiv 1812.09539) we proposed a generalization of the construction of the representation variety of a knot group in terms of Majid's theory of braided Hopf algebras. In this work braids play two roles: both as object of study and providing the noncommutative algebra to study it. For any braid and any braided commutative braided Hopf algebra we construct an ideal such that the quotient only depends on the closure of the braid. We argue that this ideal encodes a natural quantization of the representation varietyIn recent work in progress we recast our work in terms of bottom tangles following Habiro. These special tangles provide a convenient graphical calculus for braided Hopf algebras and taking a certain quotient allows a new perspective on skein theory.