Witten's influential invariants for links in 3-manifolds given in terms of a non-rigorous Feynman path integral have been rigorously defined first by Reshetikhin and Turaev. Their combinatorial definition based on the axioms of topological quantum field theory is expected to have an asymptotic expansion in view of the perturbation theory of Witten's path integral with leading order term (the semiclassical approximation) given by formally applying the method of stationary phase. Furthermore, the terms in this asymptotic expansion are expected to be well-known classical invariants like the Chern-Simons invariant, spectral flow, the Rho invariant and Reidemeister torsion. I will present new results on the expansion for finite order mapping tori, whose leading order terms we identified with classical topological invariants. Joint with Jørgen E. Andersen.
In the first part of the talk we outline the basic ideas of synthetic approach to differential geometry. The main idea of this approach, which originates from considerations of Sophus Lie is very simple: All geometric constructions are performed within a suitable base category in which space forms are objects. In the second part we indicate how a synthetic method could be employed in the context of Noncommutative Differential Geometry. This talk is addressed to mathematicians who have some very basic familiarity with general category theory culture and are familiar with elementary concepts of geometry and algebra. The aim is to explain synthetic approach to commutative and noncommutative geometry on two examples of geometric notions. First we explain all categorical ingredients that enter the synthetic definition of a principal bundle (in classical geometry) and then we show that noncommutative generalisation of this definition yields in particular principal comodule algebras or faithfully flat Hopf-Galois extensions.
In the first lecture I will explain what is the problem of quantum gravity from a physics and a mathematics perspective. In the second lecture I will concentrate on the spin foam approach and explain its basic features.
This will be a very introductory (and informal) mini-course on string theory. No prior exposure to string theory will be expected. Although I will use physicists language, the course is mainly addressed at math-students / postdocs who are welcome to help me translating expressions into math-language during the lectures. I will try to make at least some connection to higher gauge fields and also say a few words on how non-commutative geometry arises from open strings. In the first session I will concentrate on the bosonic string, while in the second I intend to discuss the superstring.
In the first lecture I will explain what is the problem of quantum gravity from a physics and a mathematics perspective. In the second lecture I will concentrate on the spin foam approach and explain its basic features.
This will be a very introductory (and informal) mini-course on string theory. No prior exposure to string theory will be expected. Although I will use physicists language, the course is mainly addressed at math-students / postdocs who are welcome to help me translating expressions into math-language during the lectures. I will try to make at least some connection to higher gauge fields and also say a few words on how non-commutative geometry arises from open strings. In the first session I will concentrate on the bosonic string, while in the second I intend to discuss the superstring.
In this talk, we will present a graphical category in terms of certain planar braid-like diagrams. The definition of this category is inspired by the representation theory of Hecke algebras of type A (which are certain deformations of the group algebra of the symmetric group). The Heisenberg algebra (in infinitely many generators), which plays an important role in the description of certain quantum mechanical systems, injects into the Grothendieck group of our category, yielding a "categorification" of this algebra. We will also see that our graphical category acts on the category of modules of Hecke algebras and of general linear groups over finite fields. Additionally, other algebraic structures, such as the affine Hecke algebra, appear naturally.
We will assume no prior knowledge of Hecke algebras or the Heisenberg algebra. The talk should be accessible to graduate students. This is joint work with Anthony Licata and inspired by work of Mikhail Khovanov.
In my talk I will first remind everyone of the relation between quantum , the -Schur algebra and colored HOMFLY homology. After that I will explain how these objects and the relation between them have been categorified.
I will describe how to characterize the Seifert graphs of link diagrams, and how, using the notion of partial duality in ribbon graphs, this leads to a generalization of the classical result that a plane graph is Eulerian if and only if its geometric dual is bipartite. This is joint work with Iain Moffatt and Natalia Virdee.
(Joint with Lucio Simone Cirio, Max Planck Institute for Mathematics) In the context of higher gauge theory, we categorify the Knizhnik-Zamolodchikov connection in the configuration space of particles in the complex plane by means of a differential crossed module of (totally symmetric) horizontal 2-chord diagrams, categorifying the 4-term relation.
We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, inside which we formulate the notion of infinitesimal 2-R matrix, an infinitesimal counterpart of some of the relations satisfied by braid cobordisms.
Gel'fand-Naimark duality (1943) between the categories of unital commutative -algebras and compact Hausdorff spaces is a key insight of 20th century mathematics, providing an enormously useful bridge between algebra on the one hand and topology and geometry on the other. Many generalisations and related dualitites exist, in logical, localic and constructive forms. Yet, all this is for commutative algebras (and distributive lattices of projections, or opens), while in quantum theory and in a large variety of mathematical situations, noncommutative algebras play a central role. A good, generally useful notion of spectra of noncommutative algebras is still lacking. Clearly, such spectra will be of considerable interest for physics and Noncommutative Geometry. I will report on recent progress towards defining spectra of noncommutative operator algebras, mostly for von Neumann algebras. This work comes from the approach using topos theory to reformulate quantum physics (C. Isham, AD), where a presheaf or sheaf topos is assigned to each noncommutative operator algebra, together with a distinguished spectral object. It will be shown that this assignment is functorial, and that the spectral object determines the algebra up to Jordan isomorphisms (J. Harding, AD). Progress on characterising the action of the unitary group of an algebra - relating to Lie group and Lie algebra aspects - is presented. Moreover, recently established connections with Zariski geometries from geometric model theory will be sketched (B. Zilber, AD). This is joint work with John Harding, Boris Zilber, and Chris Isham.
I will describe an extended (2-categorical) topological QFT with target 2-category consisting of C*-algebras and bimodules. The construction is explained as factorizable into a classical field theory valued in groupoids, and a quantization functor, as in the program of Freed-Hopkins-Lurie-Teleman. I will explain the Lagrangian action functional in terms of cohomological twisting of the groupoids in the classical part of the theory, and describe how this is incorporated into the quantization functor. This project is joint work with Derek Wise.
In 1989 François Jaeger showed that the the Kauffman polynomial of a link can be obtained as a weighted sum of HOMFLYPT polynomials on certain links associated to . In this talk I will explain how to use a version of Jaeger's theorem to stablish a connection between the and the -Schur algebras. I will then present a subcategory of the Schur category which categorifies the algebra (joint with E. Wagner).
Recent work on higher gauge theory suggests the presence of 'higher symmetry' in superstring theory. Just as gauge theory describes the physics of point particles using Lie groups, Lie algebras and bundles, higher gauge theory is a generalization that describes the physics of strings and membranes using categorified Lie groups, Lie algebras and bundles. In this talk, we will summarize the mathematics of a higher gauge theory. We then show how to construct the categorified Lie algebras relevant to superstring theory by a systematic use of the normed division algebras. At the end, we will touch on how this leads to a categorified supergroup extending the Poincare supergroup in the mysterious dimensions where the classical superstring makes sense — 3, 4, 6 and 10.