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 ${\varphi}^{4}$ 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 ${\varphi}^{4}$ 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.

Aleksandar Mikovic, Universidade Lusófona and Grupo de Física Matemática 2-BF Theories

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.

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.