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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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 simplicies of dimension d.
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.
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.
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).
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.
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.
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 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.
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.
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.
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.
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 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.
1st of Graduate Lecture Course based on version 1.0 of the forthcoming book with Beggs of the same title.
2nd of Graduate Lecture Course based on version 1.0 of forthcoming textbook with Beggs of this title
3rd of Graduate Lecture Course based on Version 1.0 of our forthcoming book of this 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.
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.
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. 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.
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.
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.
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.
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.
Last of the series based on the forthcoming book of this title. I will try to cover noncommutative black-holes.
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.
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.
[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).
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.