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.
We will show how to construct partition and n-point 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.
The algebra of differential operators in noncommutative geometry
17/01/2012 12:00 PM
410
I. Tomasic
Elements of difference algebraic geometry EdGA, II
24/01/2012 12:00 PM
410
I. Tomasic
Elements of difference algebraic geometry EdGA, III
31/01/2012 12:00 PM
410
B. Noohi
Introduction to stacks, I
07/02/2012 12:00 PM
410
B. Noohi
Introduction to stacks, II
14/02/2012 12:00 PM
410
B. Noohi
Introduction to Stacks, III
21/02/2012 12:00 PM
410
B. Noohi
Introduction to Stacks, IV
28/02/2012 12:00 PM
410
S. Majid
Lie theory of finite groups, II
06/03/2012 10:00 PM
G.O. Jones 602
S. Majid
Quantum anomalies and the origin of time (outreach lecture)
13/03/2012 12:00 PM
410
S. Majid
Lie theory of finite groups, III
20/03/2012 12:00 PM
410
R. O'Buachalla
Noncommutative complex geometry, II
08/05/2012 1:00 PM
103
D Sternheimer
Quantum groups and deformation quantization, relations and perspectives
09/05/2012 1:00 PM
103
D. Sternheimer
A conjectural quantum (space-time) origin for internal symmetries
15/05/2012 1:00 PM
103
M. Bassett
Connes-Bost model
22/05/2012 1:00 PM
103
B. Lewis
Primitive spectrum of nilpotent universal enveloping algebras, I
29/05/2012 1:00 PM
103
B. Lewis
Primitive spectrum of nilpotent universal enveloping algebras, II
18/09/2012 1:00 PM
203 Maths Building
R. 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 sub-manifolds 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 PM
B11
Shahn Majid
Braided approach to quantum groups, I
09/10/2012 1:00 PM
203
Shahn Majid
Braided approach to quantum groups, II
16/10/2012 1:00 PM
203
Shahn Majid
Braided approach to quantum groups, III
13/11/2012 12:00 PM
203
K. Ardakov
Introduction to geometric representation theory, I
20/11/2012 12:00 PM
203
K. Ardakov
Introduction to geometric representation theory, II
30/11/2012 11:30 PM
B17
K. Ardakov
Introduction to geometric representation theory, III
14/12/2012 11:00 AM
B11
Shahn Majid
Introduction to quantum geometric representations of quantum groups
15/01/2013 12:00 PM
203
W. Tao
Duality for generalised differentials on quantum groups and Hopf quivers
28/01/2013 11:00 AM
103
B. Noohi
Artin-Zhang noncommutative projective schemes, I
The Artin-Zhang 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 AM
103
B. Noohi
Artin-Zhang noncommutative projective schemes, II
The Artin-Zhang 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 AM
103
B. Noohi
Artin-Zhang noncommutative projective schemes, III
The Artin-Zhang 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 AM
203
E. Beggs (Swansea)
From homotopy to Ito calculus and Hodge theory
14/03/2013 4:00 PM
410
G. Ginot (Paris)
From Deligne conjecture in Hochschild cohomology to Gerstenhaber-Schack 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_2-algebra (an algebra over the dimension 2-little disk). Its higher generalization asserts the existence of an E_{n+1}-algebra structure on the Hochschild cochain complex of E_n-algebras. Gerstenhaber-Schack 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 Gerstenhaber-Schack one.
27/03/2013 11:00 AM
410
Re O'Buachalla (Prague)
Noncommutative Holomorphic Vector Bundles
15/10/2013 12:00 PM
203
S Majid
Reconstruction and quantisation of Riemannian manifolds
23/10/2013 2:00 PM
MLT
A Fink
The Hopf algebra of matroid valuations
29/10/2013 11:00 AM
203
S Ramgoolam
Quiver gauge theory combinatorics and permutation topological field theories
12/11/2013 11:00 AM
203
M.E. Bassett
Finite connected spaces and Boolean algebra
19/11/2013 11:00 AM
203
W. Tao
Noncommutative differentials on cotangent spaces
26/11/2013 11:00 AM
203
E.J. Beggs (Swansea)
Semiquantisation functor and Poisson geometry, II
03/12/2013 11:00 AM
203
H.-L. Huang (Shandong)
On finite pointed tensor categories
10/12/2013 11:00 AM
203
A. 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 space-valued 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 PM
M203
Thomas 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 PM
M203
Thomas 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 PM
M203
Simona Paoli (Leicester)
A model of weak 2-categories 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 2-category is the classical notion of bicategory. In this talk I present a new model of a weak 2-category 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 2-categories and weak 2-groupoids 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 PM
M203
Jeffrey Giansiracusa (Swansea)
G-equivariant open-closed TCFTs
Open 2d TCFTs correspond to cyclic A-infinity algebras, and Costello showed that any open theory has a universal extension to an open-closed 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 G-equivariant generalization of this theorem, meaning that the surfaces are now equipped with principal G-bundles. Equivariant Hochschild homology and a new ribbon graph decomposition of the moduli space of surfaces with G-bundles are the principal ingredients. This is joint work with Ramses Fernandez-Valencia.
20/01/2015 1:00 PM
410
W. Tao
Noncommutative differentials on tangent bundles from bicrossproduct construction
03/02/2015 1:00 PM
513
M. Sadrzadeh (CS)
Multilinear algebraic semantics for natural language (and some quantum connections)
10/02/2015 1:00 PM
513
A. Harju (Helsinki)
Spectral triples on orbifolds
17/02/2015 1:00 PM
410
A. Harju (Helsinki)
Quantum orbifolds
03/03/2015 1:00 PM
513
S. Ramgoolam (Physics)
Four dimensional CFT as two dimensional SO(4,2)-invariant topological field theory
10/03/2015 1:00 PM
410
D. 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 PM
410
open discussion
Linear logic, shuffle algebras and 2-vector spaces
This is an open discussion for anyone who wants to talk about the topics in the title, such as in the two papers
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 cross-fertilization between the field of network theory and quantum gravity.
In this blackboard talk, I will present recent results on network geometry showing that self-organized Complex Quantum Network Manifolds in d>2 are scale-free, 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 simplices of dimension d.
08/12/2015 10:00 AM
Maths 410
S. Majid
Hodge duality as Fourier transform
03/02/2016 3:00 PM
103
D. Tourigny (DAMTP, Cambridge)
Deformed Hamiltonian vector fields
10/02/2016 3:00 PM
103
L. Williams
Courant algebroids and Poisson-Riemannian geometry
17/02/2016 3:00 PM
103
A. Pachol
From Hopf algebras and algebroids to quantum spacetimes
09/03/2016 3:00 PM
Maths 103
Shahn Majid
Noncommutative geometry over finite fields
16/03/2016 3:00 PM
103
C. Fritz (Sussex)
Noncommutative nonassociative models of spacetime
05/05/2016 12:00 PM
Maths 103
D. 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 so-called 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 PM
Maths 103
C. Lomp (Porto)
Integral Calculus on Quantum Exterior Algebras
Hom-connections and associated integral forms have been introduced and studied by T.Brzezinski as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection 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 hom-connection. We specialise our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. 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 n-space.
22/06/2016 1:00 PM
Maths 103
E.J. Beggs (Swansea)
A differential graded category in noncommutative geometry
18/07/2016 1:00 PM
203
P. Osei (Perimeter Institute)
Quantum isometry groups and semidualisation in 3d gravity
06/10/2016 3:00 PM
203
Pierre Bieliavsky (Université catholique de Louvain)
Noncommutative surfaces in higher genera
We construct a version of noncommutative surfaces analogous to the well-known 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 non-formal in the sense that the deformed product of any two smooth functions is again a smooth function-rather than a formal power series as in formal star-product theory. The deformation consists in a real one-parameter 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 star-algebra. In particular they extend to the space of square integrable distributions as an algebra of Hilbert-Schmidt operators. A quantization in the usual sense represent them as sub-algebras of bounded operators acting on the projective discrete series representations of SL(2,R).
14/10/2016 1:00 PM
203
Ivan Tomasic
Cohomology of difference algebraic groups, I
28/10/2016 1:00 PM
203
Ivan Tomasic
Cohomology of difference algebraic groups, II
01/11/2016 1:00 PM
410
A. Pachol
Classification of noncommutative differentials in two and three dimensions
18/11/2016 1:00 PM
203
I. Tomasic
Cohomology of difference algebraic groups, III
02/12/2016 1:00 PM
410
Greg Ginot (Paris)
Quantization of bialgebras via deformation theory
We will discuss a new proof of Etingof-Kazdhan 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) dg-bialgebras 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 PM
203
Ivan Tomasic
Paths in digraphs, difference algebra and entropy
17/02/2017 3:00 PM
G.O.Jones LG1
Joao Martins
Categorification of the Knizhnik-Zamolodchikov connection via infinitesimal 2-braidings
I will report on a long term joint project with Lucio Cirio and Florian Schaetz on categorifications of the Knizhnik-Zamolodchikov connection via infinitesimal 2-braidings. In particular, I will describe a categorification of the Drinfeld-Kohno Lie algebra of chord diagrams in the realm of a differential crossed module of horizontal 2-chord diagrams. I will also explain how this categorified Lie algebra arises from a linearization (called an infinitesimal braided 2-category) of the axioms defining a braided monoidal 2-category.
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 2-braidings and differential crossed modules. Advances in Mathematics, Volume 277, 4 June 2015, Pages 426-491 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 PM
W316
Nicola Gambino
Operads in 2-algebraic geometry
21/03/2017 2:00 PM
W316
O.A. Laudal (Oslo)
Gravity induced from quantum `time-space'
28/03/2017 2:00 PM
W316
Ryan Aziz
Codouble bosonisation
09/05/2017 2:00 PM
W316
R 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 PM
W316
R. Akylzhanov (Imperial)
Smooth dense subalgebras on compact quantum groups
11/07/2017 2:00 PM
W316
Robert 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 non-trivial 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 Etingof--Ostrik et al.
22/01/2013 12:00 PM
203
W. Tao
Duality for generalised differentials on quantum groups and Hopf quivers, II
27/01/2015 1:00 PM
410
S. Majid
Emergence of Bertotti-Robinson 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 Bertotti-Robinson metric, i.e. a solution of Einstein’s equations with cosmo- logical constant and a non-null 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 PM
Queens E303
G. 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 space-time. Here we define complex quantum network geometries and manifolds, describing the underlying structure of growing simplicial complexes. These networks grow according to a non-equilibrium 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 (d-1)-faces of a d-dimensional 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 scale-free i.e. they are characterized by large inhomogeneities of degrees like most complex networks. From the self-organized evolution of CQNM quantum statistics emerge spontaneously. We define the generalized degrees associated with the δ-faces of the d-dimensional CQNMs, and we show that the statistics of these generalized degrees can either follow Fermi-Dirac, Boltzmann or Bose-Einstein distributions depending on the dimension of the δ-faces.
28/11/2011 4:00 PM
103
I. Tomasic
Elements of difference algebraic geometry E-\sigma GA, I
05/12/2016 12:00 PM
B.R. 4.01 (Bancroft Rd)
Ping Xu (Penn State)
Kontsevich-Duflo theorem for Lie pairs
The Kontsevich-Duflo theorem asserts that, for any complex manifold X, the Hochschild-Kostant-Rosenberg 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 Kontsevich-Duflo 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 PM
W316
Shahn Majid
Quantum 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 Levi-Civita connection for any metric with un-directed edge weights on a square graph. We find a 1-parameter family of quantum-Levi-Civita connections and a proposal for an Einstein-Hilbert 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 quantum-geometric graph Laplacian in both signatures.
21/11/2017 2:00 PM
W316
Re 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, Dolbeault-Dirac, 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 anti-holomorphic Dolbeault-Dirac 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 non-zero index, and so, non-zero K-homology 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 D-series quantum groups, namely the odd and even dimensional quantum quadrics.
05/12/2017 2:00 PM
LK1 (Lock Keeper's Cottage)
Johan Noldus
Generally covariant quantum mechanics
23/01/2018 11:00 AM
Queens' Building, Room W316
Shahn Majid
Introduction 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 AM
Queens' Building, Room W316
Shahn Majid
Quantum 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 AM
Queens' Building, Room W316
E.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 AM
Queens' Building, Room W316
Shahn Majid
Quantum 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 AM
W316, Queens' Building
Shahn Majid
Quantum Riemannian Geometry, V
This week I will start to cover the two most well-known `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 PM
Graduate Centre, Room GC205
P. 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 Poincaré--Birkhoff--Witt 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 L-infinity structure associated with the Atiyah class of a dg manifold.
23/08/2019 12:00 PM
B.R. 3.02 (Bancroft Rd)
D. Bar-Natan (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 semi-simple Lie algebra $sl_2$.
Knot theorists should rejoice because all this leads to very powerful and well-behaved poly-time-computable 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 PM
Queens' Building, Room: W316
A. Schenkel (Nottingham)
Higher structures in algebraic quantum field theory
Algebraic quantum field theory (AQFT) is a well-established 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 1-categorical 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 non-technical 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 PM
Queens' Building, Room: W316
G. Ginot (Paris 13)
Chern-Simons 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 Goldman-Turaev 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 Chern-Simons field theory.
05/02/2019 1:00 PM
Queens' Building, Room: W316
Aryan Ghobadi
Hopf monads
26/02/2019 1:00 PM
Queens' Building, Room: W316
Ryan Aziz
Differential calculi by super-cobosonisation
29/01/2019 1:00 PM
Queens' Building, Room: W316
Anna Pachol
Digital quantum geometries
22/01/2019 1:00 PM
Queens' Building, Room: W316
Shahn Majid
Quantum Riemannian geometry and Hawking effect on the integers
08/01/2019 1:00 PM
Queens' Building. Room: W316
Edwin Beggs (Swansea)
Quantum geodesics and positive maps on C*-algebras
13/12/2018 1:00 PM
Queens' Building, Room: W316
Liam Williams
Poisson principal bundles, II
06/12/2018 1:00 PM
Queens' Building, Room: W316
Andrzej Sitarz (Krakow)
The Standard Model and Noncommutative Geometry 2.0
25/10/2018 1:00 PM
Queens' Building, Room: W316
Ryan Aziz
Super-co-double-bosonisation and quantum differentials
18/10/2018 1:00 PM
Graduate Centre, Room: GC205
Shahn Majid
Quantum Gravity on four points
15/11/2018 1:00 PM
Graduate Centre, Room: GC205
Rauan Akylzhanov
Hormander-Mihlin L^p-multiplier theorems on noncommutative spaces
Abstract
(joint work with Michael Ruzhansky)
We construct a Fourier-type formalism on von Neumann algebras. In this setting, we establish Paley-type inequalities on semi-finite 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^p-multiplier theorem. As a particular case, we recover [2]. If time permits, we shall discuss a general H ̈ormander-Mihlin Lp-multiplier theorem on semi-finite 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^p-L^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 PM
Queens' Building, Room: W316
Liam Williams
Poisson principal bundles
19/07/2018 12:00 PM
Room W316, Queens' Building
C. 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 8C-dimensional 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 PM
Queens' Building, Room W316
Sina 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 2-categories and bicategories J. Pure Appl. Algebra, 218(6):1034–1074, 2014.
[Joh93] Peter Johnstone. Fibrations and partial products in a 2-category. Applied Categorical Structures, 1(2):141–179, 1993.
[Str74] Ross Street. Fibrations and Yoneda’s lemma in a 2-category. 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 AM
W316, Queens' Building
Shahn Majid
Quantum Riemannian Geometry, VI
Last of the series based on the forthcoming book of this title. I will try to cover noncommutative black-holes.
28/06/2019 2:00 PM
Bancroft Road, Room 4.01
John Barrett (Nottingham)
Finite spectral triples for the fuzzy torus
Abstract: Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative 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.
Zoom Seminar (standard access details, contact S. Majid if need them)
Ginestra Bianconi
Emergent 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 non-equilibrium 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 higher-order percolation and diffusion investigated using a real-space renormalization group approach.
07/02/2020 11:00 AM
Mathematical Sciences Building, Room MB-503
S. 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 PM
Mathematical Sciences Building, Room MB-503
R. Wilson
Some 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 PM
Mathematical Sciences Building, Room: MB503
Shahn Majid
Quantum geometry and probability
07/04/2020 12:00 PM
Video seminar -- contact s.majid@qmul.ac.uk to register
Edwin Beggs (Swansea)
Quantum geodesic deviation
10/03/2020 12:00 PM
Mathematical Sciences Building, Room MB-503
Julio Argota Quiroz
QFT on R x Z_n, Part III: curved space-time
19/05/2020 12:00 PM
Video seminar - contact s.majid@qmul.ac.uk for details
E. Lira Torres
Quantum Geometry of the Fuzzy Sphere
30/09/2020 2:00 PM
Zoom seminar -- email s.majid@qmul.ac.uk to register
Francesco Toppan (CBPF, Rio de Janeiro)
A theoretical test of Z_2xZ_2-graded parastatistics
The recent surge of interest in Z2xZ2-graded invariant mechanics poses the challenge of understanding the physical consequences of a Z2xZ2-graded symmetry. Non-trivial physics can be detected in the multiparticle sector, being induced by the Z2xZ2-graded parastatistics obeyed by the particles.
The toy model of the N=4 supersymmetric/ Z2xZ2-graded oscillator is discussed. In this set-up the one-particle energy levels and their degenerations are the same for both supersymmetric and Z2xZ2-graded 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 Z2xZ2-graded particles. Therefore, Z2xZ2-graded 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 PM
Zoom Seminar (standard Meeting ID and passcode, email S. Majid if need these)
Aryan Ghobadi
Skew Braces as Hopf Algebras in SupLat
Set-theoretical solutions of the Yang-Baxter 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 join-preserving morphisms, SupLat. This allows us to utilise the usual techniques from Hopf algebras, to study set-theoretical 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 co-quasitriangular 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 well-known 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 co-quasitriangular 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 PM
Zoom seminar (standard details, email s.majid@qmul.ac.uk if need them)
S. Ramgoolam (SPA)
Kronecker coefficients and lattices of ribbon graphs
Bipartite ribbon graphs arise in the enumeration of polynomial invariants of tensor variables. They correspond to Belyi maps between two-dimensional 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 PM
Zoom seminar (email s.majid@qmul.ac.uk if you need meeting login details)
Shahn Majid
Continuum 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 PM
Zoom seminar (email s.majid@qmul.ac.uk if need access details)
J Argota-Quiroz
Dimension jump in noncommutative spacetime models
09/12/2020 2:00 PM
Zoom 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 self-dual and resembles a digital version of u_q(sl_2). We also find a unique self-dual 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 R-matrix” structures on our Hopf algebras. These induce solutions of the Yang–Baxter or braid relations in any representation.
03/02/2021 2:00 PM
Zoom Seminar (standard access details, contact S. Majid if need them)
Lukas Muller (MPIM, Bonn)
Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations
The goal of this talk is to briefly review the concept of Grothendieck-Verdier categories as introduced by Boyarchenko-Drinfeld based on Barr's notion of a *-autonomous category and explain its relation to low-dimensional 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 PM
Zoom Seminar (usual access details, contact S. Majid if need them)
Paolo Saracco (Brussels)
Connes-Moscovici'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 well-known 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 Lie-Rinehart algebra and, conversely, the universal enveloping algebra of a Lie-Rinehart 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 Connes-Moscovici'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 Connes-Moscovici'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 Connes-Moscovici's bialgebroid.
Time permitting, I will briefly show how representations of a finite-dimensional anchored Lie algebra are naturally related to certain (elementary) bimodules with flat bimodule connection for the first-order differential calculus associated with the given anchored Lie algebra.
03/03/2021 2:00 PM
Zoom seminar (standard access, contact S. Majid if needed)
Shahn Majid
Braided ZX-calculus in quantum computing
We study the algebraic structure behind ZX-calculus 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 PM
Zoom seminar (standard access, contact S. Majid if needed)
Shahn Majid
Discrete Chern-Simons and topological quantum computing
31/03/2021 2:00 PM
Zoom seminar (standard access, contact S. Majid if needed)
T. Weber (U. Bologna)
Cartan calculus and Riemannian geometry on braided commutative algebras
A well-studied 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 Levi-Civita 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 PM
Zoom seminar (usual access, contact S. Majid if needed)
R. van der Veen (Groningen)
Quantum character varieties of knot groups through bottom tangles
The 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 3-manifolds: 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 non-commutative 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 variety
In 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.
30/09/2021 3:00 PM
MB-503
F. Simao
BV equivalence with boundary
Based on arXiv:2109.05268 An extension of the notion of classical equivalence of equivalence in the Batalin–(Fradkin)–Vilkovisky (BV) and (BFV) framework for local La- grangian field theory on manifolds possibly with boundary is discussed. Equivalence is phrased in both a strict and a lax sense, distinguished by the compatibility between the BV data for a field theory and its boundary BFV data, necessary for quantisation. In this context, the first- and second-order formulations of non-Abelian Yang–Mills and of classical mechanics on curved backgrounds, all of which admit a strict BV-BFV description, are shown to be pairwise equivalent as strict BV-BFV theories. This in particular implies that their BV-complexes are quasi-isomorphic. Furthermore, Jacobi theory and one-dimensional gravity coupled with scalar matter are compared as classically-equivalent reparametrisation-invariant versions of classical mechanics, but such that only the latter admits a strict BV-BFV formulation. They are shown to be equivalent as lax BV-BFV theories and to have isomorphic BV cohomologies. This shows that strict BV-BFV equivalence is a strictly finer notion of equivalence of theories.
14/10/2021 3:00 PM
MB-503
E. Lira-Torres (QMUL)
Geometric Dirac operator on the fuzzy sphere
Based on arXiv: 2104.13212 (math.qa)
28/10/2021 3:00 PM
MB-503
Chengcheng Liu (QMUL)
Quantum geodesics on quantum spacetime
18/11/2021 3:00 PM
MB-503
Julio Argota-Quiroz
Quantum Riemannian geometry on graphs
02/12/2021 3:00 PM
MB-503
Xiao Han (QMUL)
On the second cohomology class of bialgebroids
16/12/2021 3:00 PM
MB-503
Edwin Beggs (Swansea)
Quantum geodesics on curved quantum geometries
26/01/2022 3:00 PM
MB-503
Shahn Majid (QMUL)
Quantum geodesics and curvature
09/02/2022 3:00 PM
MB-503
Shahn Majid (QMUL)
Quantum jet bundles, I
This is based on joint work with Francisco Simao, whose talk is now moved to week 8 and will be part II of the same title.
22/02/2022 1:00 PM
MB-503
F. Simao (QMUL)
Quantum Jet Bundles, II
16/03/2022 3:00 PM
MB-503
Aryan Ghobadi and Xiao Han (QMUL)
On the antipode of Hopf algebroids
30/03/2022 3:00 PM
MB-503 and Zoom (contact s.majid@qmul.ac.uk for details)
Anna Pachol (Oslo)
Dispersion relations in κ-noncommutative cosmology
13/04/2022 3:00 PM
MB-503
Leo McCormack (QMUL)
Braided Lie algebras of quantum doubles
25/05/2022 3:00 PM
MB-503
A. Schenkel (Nottingham)
Quantization of derived cotangent stacks
Derived algebraic geometry is a modern and powerful geometric framework which plays an increasingly important role both in the foundations of algebraic geometry and in mathematical physics. It introduces a refined concept of ‘space’, the so-called derived stacks, that is capable to describe correctly geometric situations that are problematic in traditional approaches, such as non-transversal intersections and quotients by non-free group actions. In this talk I will give a brief introduction to derived algebraic geometry and the quantization of (shifted) Poisson structures on derived stacks. I will then discuss in some detail a simple example, namely the quantization of the canonical Poisson structure on the derived cotangent stack T*[X/G] of a quotient stack [X/G].
This talk is based on joint work with Marco Benini and Jon Pridham [arXiv:2201.10225].
01/06/2022 3:00 PM
MB-503
Thomas Weber (Piedmont)
Levi-Civita connections for non-central metrics on quantum groups
In classical Riemannian geometry the Levi-Civita connection is the unique torsion-free connection compatible with a given pseudo-Riemannian metric. Partially, but not solely motivated by quantum gravity, it is one of the goals of noncommutative geometry to give a comprehensive picture of Riemannian geometry for noncommutative algebras. Consequently, the study of Levi-Civita connections on noncommutative spaces has been a fruitful topic throughout the last decades, giving insights via numerous examples, as the noncommutative 2-torus. The corresponding operations are bimodule connections and the considered metrics are central objects. In this talk we propose a more general approach to Levi-Civita connections on quantum groups, based on arbitrary right connections and non-central metrics. The main tool we utilize in our construction is a generalization of the sum of bimodule connections, involving a lifting of Woronowicz braiding. In the presence of a diagonalizable braiding (this includes all matrix quantum groups) we provide an existence and uniqueness theorem for Levi-Civita connections corresponding to central metrics and, more in general, quasi-central metrics. To prepare for our main result we discuss a braided tensor product of rational morphisms and the canonical torsion-free connection on quantum groups determined by the structure constants. The presentation is based on a collaboration with Paolo Aschieri.
08/06/2022 3:00 PM
MB-503
Ulrich Krähmer (Dresden)
Cyclic duality for slice and orbit 2-categories
After a minicouse on simplicial sets I will explain how they can be extended to duplicial and the closely related paracyclic ones and why I and a few others care about these. The main topic will then be one possible answer to the question why these more exotic index categories are self-dual while the simplicial category is not. [Based on joint work with John Boiquaye and Philipp Joram].
04/10/2022 12:00 PM
MB-503
X. Han (QMUL)
Quantum Lie algebroids and the quantum Atiyah sequence
01/11/2022 12:00 PM
MB-503
Ralf Meyer (Goettingen)
Covariance rings for bimodules over rings
Abstract: I am going to discuss a purely algebraic analogue of the Cuntz-Pimsner algebra construction for C*-correspondences between C*-algebras. It is defined for a bimodule over a ring that is projective and finitely generated as a right module. For the bimodule attached to a regular graph, it generalises the Leavitt path algebra. The construction takes place in the bicategory of rings and bimodules, and generalises to homomorphisms into this bicategory. Namely, it merely takes the bicategorical limit of such a homomorphism.
18/10/2022 12:00 PM
MB-503
Chengcheng Liu (QMUL)
Quantum Kaluza-Klein theory and dynamical mass
29/11/2022 12:00 PM
MB-503
T. Brzeziński (Swansea)
Heaps of Connections
We discuss a recently introduced algebraic structure called a truss, which interpolates between rings and braces, and arises naturally in studies of associative multiplications on affine spaces. The truss structure of connections on a module is presented as a specific example.
13/12/2022 12:00 PM
MB-503
S. Majid (QMUL)
Dirac Operator on the Algebra of 2 x 2 matrices, part II
15/11/2022 12:00 PM
MB-503
E. Lira-Torres (QMUL)
Dirac operator on the noncommutative torus and on the algebra of 2 x 2 matrices
31/01/2023 1:00 PM
MB-503
Shahn Majid (QMUL)
Dirac operators associated to quantum metrics
28/02/2023 1:00 PM
MB-503
A. Cowtan (Oxford)
Quantum codes
14/02/2023 1:00 PM
MB-503
Leo McCormack (QMUL)
Braided centres
21/02/2023 1:00 PM
MB-503
G. Bianconi (QMUL)
The Dirac operator on networks and simplicial complexes and its associated field theory
07/03/2023 1:00 PM
MB-503
S. Sierra (Edinburgh)
(Pseudo) orbits of functions on the Virasoro algebra
The Virasoro Lie algebra Vir is one of the most important infinite-dimensional Lie algebras, ubiquitous in representation theory and mathematical physics. In this talk we consider the dual space Vir*. Although Vir famously has no adjoint group, Poisson geometry on Vir* still gives a meaningful notion of a coadjoint orbit, or more accurately pseudo-orbit. We describe the geometry of these pseudo-orbits: each central character corresponds to a single infinite-dimensional pseudo-orbit, and in addition for the trivial character there is a family of finite-dimensional pseudo-orbits parameterised by partitions and points in a finite-dimensional affine space. Each finite-dimensional pseudo-orbit is actually an orbit: a principal homogeneous space for a finite-dimensional solvable algebraic group. We further show how to use the parameterisation of pseudo-orbits to give a complete description of Poisson subvarieties of Vir*.
This is joint work with Alexey Petukhov.
28/03/2023 1:00 PM
MB-503
D. Mugnolo (Hagen)
An introduction to quantum graphs and their spectral geometry
11/04/2023 1:00 PM
MB-503
R. Laugwitz (Nottingham)
Non-semisimple modular categories type Super A and the Links–Gould invariant
This talk is based on joint work with Chelsea Walton and Guillermo Sanmarco. We construct a series of finite-dimensional quantum groups as double bosonizations of Nichols-Woronowicz algebras of type Super A. For an even root of unity, we prove that ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the negative Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. Based on generalized traces, we compute an invariant of knots associated to the four-dimensional simple module of the rank-two quantum group. This knot invariant is related to a specialization of the Links–Gould invariant and can distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials.
04/04/2023 1:00 PM
MB-503
F. Simao (QMUL)
Gauge theory on digraphs
11/10/2023 11:00 AM
MB-502
X. Han (QMUL)
Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids
25/10/2023 1:00 PM
MB-502
Jan Slovak (Masaryk)
On invariant linear differential operators in terms of induced modules
08/11/2023 12:00 PM
MB-502
M. Hanada (QMUL)
Color confinement and holographic emergent geometry
We discuss confinement/deconfinement phase transition by taking the SU(N) matrix model as example. Roughly speaking, confinement means that individual color degrees of freedom are not visible, and hence energy and entropy are O(N^0). In the deconfined phase, all N^2 color degrees of freedom are unlocked and energy and entropy become O(N^2). We point out that the microscopic mechanism of color confinement has a very close connection to Bose-Einstein condensation. Essentially, Einstein considered SU(N) vector model with S_N gauge group, took large-N limit, and observed "confinement", which is Bose-Einstein condensation. From this viewpoint, we can understand a rich phase structure between confined and deconfined phases that has consequences to emergent geometry in the context of holographic duality.
22/11/2023 12:00 PM
502
C. Liu (QMUL)
Quantum Kaluza-Klein theory, II
06/12/2023 12:00 PM
502
F. Simao (QMUL)
Gauge theory on digraphs, II
30/01/2024 12:00 PM
MB-502
S. Majid (QMUL)
Curvature expectation values in baby quantum gravity models
07/02/2024 4:00 PM
MB-503
G. Landi (Trieste)
Solutions to the quantum Yang Baxter equation and related deformations
20/02/2024 12:00 PM
MB-502
S. Quddus (IIS Bangalore)
Action of some discrete groups in non-commutative geometry
It is well known that group action changes the homological, differential, and topological properties of the space. The aim of this talk is to discuss the action of some finite discrete groups on some non-commutative differential and algebraic spaces. We shall first discuss and compare the (co)homological properties of non-commutative and quantum torus then observe how the quotient spaces resulting from the actions of discrete subgroups of SL(2,Z) behave. We shall discuss the flip actions on the non-commutative sphere from homological perspective.
05/03/2024 12:00 PM
MB-502
K. Kumar (QMUL)
Emergence of modified Einstein equation from 1-loop result in IKKT matrix model
We derive the equations of motion that arise from the one-loop effective action for the geometry of 3+1 dimensional quantum branes in the IKKT matrix model. These equations are cast into the form of generalized Einstein equations, with extra contributions from dilaton and axionic fields, as well as a novel anharmonicity tensor Cµν capturing the classical Yang-Mills-type action. The resulting gravity theory approximately reduces to general relativity in some regime, but differs significantly at cosmic scales, leading to an asymptotically flat FLRW cosmological evolution governed by the classical action.
19/03/2024 12:00 PM
MB-501
L. McCormack (QMUL)
Differential calculi on quantum doubles
09/04/2024 12:00 PM
MB-502
E. Beggs (Swansea)
Complex structure on the quantum plane
28/05/2024 12:00 PM
MB-501
K. Kumar (QMUL)
Octonions and the Standard Model
07/10/2024 3:00 PM
MB-503
S Majid
General-relativistic quantum mechanics and nonassociativity
14/10/2024 3:00 PM
MB-503
X. Han
Hopf algebroid of differential operators on an algebra
04/11/2024 3:00 PM
MB-503
Shahn Majid
Black-hole atoms
18/11/2024 3:00 PM
MB-503
B. Ramos Perez (Santiago de Compostela)
Categorical relationships between Hopf braces, Hopf bracoids and 1-cocycles
25/11/2024 3:00 PM
MB-503
G. Bianconi
Gravity from entropy
Gravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a density matrix and to describe the matter fields topologically, according to a Dirac-Kahler description, as the direct sum of a zero-form, a one-form and a two-form. While the geometry of spacetime is defined by its metric, the matter fields can be used to define an alternative metric, the metric induced by the matter fields, which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the G-field, which acts as a set of Lagrangian multipliers, the proposed entropic action reduces to a dressed Einstein-Hilbert action with an emergent small and positive cosmological constant only dependent on the G-field. The obtained equations of modified gravity remains second order in the metric and in the G-field.
31/01/2025 12:00 PM
MB-503
A. Sitarz (Krakow)
Computing the Einstein and torsion tensors for spectral triples
14/02/2025 12:00 PM
MB-503
Shahn Majid
Cosmological atoms on FLRW spacetimes
28/02/2025 12:00 PM
MB-502
Xiao Han
Algebraic jet bundles
14/03/2025 12:00 PM
MB-503
Leo McCormack
Braided Lie algebras of quantum groups
28/03/2025 12:00 PM
MB-503
N. Herceg (Zagreb)
Metric perturbations in noncommutative gravity
We present a brief review of differential geometry deformed by a Drinfeld twist, which is then applied in the setting of black hole metric perturbations to find the modifications to quasinormal-mode spectrum and equations of motion of the gravitational wave due to spacetime noncommutativity.