The main goal of these lectures is to explore the intriguing connections, as investigated by Boyd, Deninger, Rodriguez-Villegas, Smyth and others, between Mahler measures of \(A\)-polynomials of hyperbolic \(3\)-manifolds and the special values of \(L\)-functions of Dirichlet characters and elliptic curves.

Our treatment will be mostly expository and proofs will be mostly sketched.

In these lectures we will meet formulas that relate the Mahler measure of certain polynomials to special values of \(L\)-functions. In this talk we will go through the basic definitions and concepts regarding \(L\)-functions. In particular, we will meet the Dirichlet \(L\)-functions, the \(L\)-series of an elliptic curve as well as some of their elementary properties.

The main goal of these lectures is to explore the intriguing connections, as investigated by Boyd, Deninger, Rodriguez-Villegas, Smyth and others, between Mahler measures of \(A\)-polynomials of hyperbolic \(3\)-manifolds and the special values of \(L\)-functions of Dirichlet characters and elliptic curves.

Our treatment will be mostly expository and proofs will be mostly sketched.

I will try and present some concepts from knot theory, such as the coloured Jones polynomial and Kauffman bracket skein modules, so as to explain the content of the AJ-conjecture (or quantum volume conjecture), which is a statement about a non-commutative version of the \(A\)-polynomial. The talk will be mainly based on T. Le, The coloured Jones polynomial and the \(A\)-polynomial of knots, Adv. Math. 207 (2006) 782-804. If time permits, I will comment on work by Gukov et al which refines or categorifies the quantum \(A\)-polynomial.

The main goal of these lectures is to explore the intriguing connections, as investigated by Boyd, Deninger, Rodriguez-Villegas, Smyth and others, between Mahler measures of \(A\)-polynomials of hyperbolic \(3\)-manifolds and the special values of \(L\)-functions of Dirichlet characters and elliptic curves.

Our treatment will be mostly expository and proofs will be mostly sketched.

In the previous talk we gave an overview of the renormalization procedure in Quantum Field Theory. In this lecture we will demonstrate that abstract procedure on a simple explicit example, the so-called $\phi^4$ theory of a single real scalar field. We will illustrate the construction of a renormalized state sum using two different regularization schemes, construct the renormalization group equations, and discuss some of their properties.

In this follow up to last year's talk, we briefly review the cobordism hypothesis that formed the subject of our first part, and then outline its use for the existence and construction of field theories, in particular Chern-Simons theory, as discussed in a 2009 paper of Freed, Hopkins, Lurie and Teleman.

In the previous talk we discussed the renormalization procedure on the example $\phi^4$ scalar field theory. In this lecture we will conclude the analysis of that example, construct the final renormalized state sum, and discuss the renormalization group equations. At the end we will give some final general remarks about renormalization in QFT.

Quantum groups at roots of unity appear as hidden symmetries in some conformal field theories. For this reason I. Todorov has (in 1990s) used coherent state operators for quantum groups to covariantly build the field operators in Hamiltonian formalism. I tried to mathematically found his coherent states by an analogy with the Perelomov coherent states for Lie groups. For this, I use noncommutative localization theory to define and construct the noncommutative homogeneous spaces, and principal and associated bundles over them. Then, in geometric terms, I axiomatize the covariant family of coherent states which enjoy a resolution of unity formula, crucial for physical applications. Even the simplest case of quantum \(\operatorname{SL2}\) is rather involved and the corresponding resolution of unity formula involves the Ramanujan's \(q\)-beta integral. The correct covariant family differs from ad hoc proposed formulas in several published papers by earlier authors.

We will describe 2-BF topological field theories, which are categorical generalization of the BF theories. We will also explain how to construct invariants of manifolds by using 2-BF theory path integrals.

References

1. João Faria Martins, Aleksandar Mikovic,. Lie crossed modules and gauge-invariant actions for 2-BF theories. Adv. Theor. Math. Phys. Volume 15, Number 4, 1059-1084 (2011).
2. Aleksandar Mikovic, Marko Vojinovic, Poincaré 2-group and quantum gravity,. Class. Quant. Grav. 29, 165003 (2012).

Topological field theories are very special in two dimensions: they have been classified and provide a rich class of examples. In this talk I will discuss a new state sum construction for these models that considers not just the topology of surfaces but also their spin structure. Emphasis is given to partition functions: I will detail general properties of these manifold invariants and discuss some non-trivial examples.

A satisfactory marriage between “higher” categories and operator algebras has never been achieved: although (monoidal) C*-categories have been systematically used since the development of the theory of superselection sectors, higher category theory has more recently evolved along lines closer to classical higher homotopy.

We present axioms for strict involutive \(n\)-categories (a vertical categorification of dagger categories) and a definition for strict higher C*-categories and Fell bundles (possibly equipped with involutions of arbitrary depth), that were developed in collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.

In order to treat some very natural classes of examples arising from the study of hypermatrices and hyper-C*-algebras, that would be otherwise excluded by the standard Eckmann-Hilton argument, we suggest a non-commutative version of exchange law and we also explore alternatives to the usual globular and cubical settings.

Possible applications of these non-commutative higher C*-categories are envisaged in the algebraic formulation of Rovelli's relational quantum theory, in the study of morphisms in Connes' non-commutative geometry, and in our proposed “modular” approach to quantum gravity (arXiv: 1007.4094).

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from the category of unital C*-algebras to a category of presheaves that contains the spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures can lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.

We investigate a (contravariant) functor from C*-algebras to toposes and geometric morphisms that generalizes the Gelfand spectrum in the commutative case. The functor produces a locale, presented by means of a Grothendieck topology on an inf-semilattice of 'Gelfand' opens \([U;a]\).

The Fefferman-Graham ambient metric construction, with some technical asterisks, positively resolves the Dirichlet problem for compactification of asymptotically hyperbolic Einstein metrics, the compactification that occurs in the AdS/CFT correspondence. We show that data on the conformal boundary parallel with respect to Cartan's normal conformal connection — which is nearly the same thing as a holonomy reduction of the conformal structure — can be extended (again with an asterisk) to data parallel with respect to a natural connection on a corresponding bundle over the bulk, which in particular enables holographic study of such data. As an application, we use this extension result to construct metrics of exceptional holonomy.