Quantum Algebras

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.

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.

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.

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.

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.

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.

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].

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.

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].

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.

Based on arXiv: 2104.13212 (math.qa)

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.

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. 

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

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. 

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.

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.

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.

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.

[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).

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.

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.

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.

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.

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.

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.

Last of the series based on the forthcoming book of this title. I will try to cover noncommutative black-holes.

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. 

3rd of Graduate Lecture Course based on Version 1.0 of our forthcoming book of this title

2nd of Graduate Lecture Course based on version 1.0 of forthcoming textbook with Beggs of this title

1st of Graduate Lecture Course based on version 1.0 of the forthcoming book with Beggs of the same title.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

This is an open discussion for anyone who wants to talk about the topics in the title, such as in the
two papers

http://www.site.uottawa.ca/%7Ephil/papers/shuf.pdf(link is external)

http://math.ucr.edu/home/baez/2rep [PDF 1,183KB]

All are welcome

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.

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.

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.

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.

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.

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.

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.

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.