I will explain how Dirac operators provide precious information about geometric and algebraic aspects of representations of real Lie groups. In particular, we obtain an explicit realisation of representations, leading terms in the asymptotics of characters and a precise connection with nilpotent orbits.

TQFTs lie at the intersection of maths and theoretical physics. Topologically, they are a recipe for calculating an invariant of manifolds by cutting them into elementary pieces; physically, they describe the evolution of the state of a particle. These two viewpoints allow physical intuition to be harnessed to shed light on topological problems, including understanding the topology of 4-manifolds and calculating geometric invariants using topology.

Recent results have provided classifications of certain types of TQFTs as algebraic structures. After reviewing the definition of TQFTs and giving some diagrammatic examples, I will give informal arguments as to how these classifications arise. Finally, I will show that in many cases these algebras are in fact free, and give an explicit classification of them in this case.
 

The Oberwolfach Research Institute for Mathematics (Mathematisches Forschungsinstitut Oberwolfach/MFO) was founded in late 1944 by the Freiburg mathematician Wilhelm Süss (1895-1958) as the „National Institute for Mathematics“. In the 1950s and 1960s the MFO developed into an increasingly international conference centre.

The aim of my project is to analyse the history of the MFO as it institutionally changed from the National Institute for Mathematics with a wide, but standard range of responsibilities, to an international social infrastructure for research completely new in the framework of German academia. The project focusses on the evolvement of the institutional identity of the MFO between 1944 and the early 1960s, namely the development and importance of the MFO’s scientific programme (workshops, team work, Bourbaki) and the instruments of research employed (library, workshops) as well as the corresponding strategies to safeguard the MFO’s existence (for instance under the wings of the Max-Planck-Society). In particular, three aspects are key to the project, namely the analyses of the historical processes of (1) the development and shaping of the MFO’s workshop activities, (2) the (complex) institutional safeguarding of the MFO, and (3) the role the MFO played for the re-internationalisation of mathematics in Germany. Thus the project opens a window on topics of more general relevance in the history of science such as the complexity of science funding and the re-internationalisation of the sciences in the early years of the Federal Republic of Germany.

George Soules [1] introduced a set of vectors $r_1,...,r_N$ with the remarkable property that for any set of ordered numbers $\lambda_1\geq\dots\geq\lambda_N$, the matrix $\sum_n \lambda_nr_nr_n^T$ has nonnegative off-diagonal entries. Later, it was found [2] that there exists a whole class of such vectors - Soules vectors - which are intimately connected to binary rooted trees. In this talk I will describe the construction of Soules vectors starting from a binary rooted tree, and introduce some basic properties. I will also cover a number of applications: the inverse eigenvalue problem, equitable partitions in Laplacian matrices and the eigendecomposition of the Clauset-Moore-Newman hierarchical random graph model.

[1] Soules (1983), Constructing Symmetric Nonnegative Matrices
[2] Elsner, Nabben and Neumann (1998), Orthogonal bases that lead to symmetric nonnegative matrices

We present a self-contained construction of the Euclidean $\Phi^4$ quantum
field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we
consider an approximation of the stochastic quantization equation on
$\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and
side length $M$. We introduce an energy method and prove tightness of the
corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow
\infty$. We show that every limit point satisfies reflection positivity,
translation invariance and nontriviality (i.e. non-Gaussianity). Our
argument applies to arbitrary positive coupling constant and also to
multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano
Gubinelli.

Quantum Yang-Mills theory is an important part of the Standard model built
by physicists to describe elementary particles and their interactions. One
approach to this theory consists in constructing a probability measure on an
infinite-dimensional space of connections on a principal bundle over
space-time. However, in the physically realistic 4-dimensional situation,
the construction of this measure is still an open mathematical problem. The
subject of this talk will be the physically less realistic 2-dimensional
situation, in which the construction of the measure is possible, and fairly
well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the
distribution of a stochastic process with values in a compact Lie group (for

example the unitary group U(N)) indexed by the set of continuous closed
curves with finite length on a compact surface (for example a disk, a sphere
or a torus) on which one can measure areas. It can be seen as a Brownian
motion (or a Brownian bridge) on the chosen compact Lie group indexed by
closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills
measure is constructed, and describe it without assuming any prior
familiarity with the subject. I will then present a set of results obtained
in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel,
Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N
tends to infinity of the Yang-Mills measure constructed with the unitary
group U(N).

A dedicated search of the CMB sky, driven by implications of conformal
cyclic cosmology (CCC), has revealed a remarkably strong signal, previously
unobserved, of numerous small regions in the CMB sky that would appear to be
individual points on CCC's crossover 3-surface from the previous aeon, most
readily interpreted as the conformally compressed Hawking radiation from
supermassive black holes in the previous aeon, but difficult to explain in
terms of the conventional inflationary picture.