L4

Suppose that we want to minimise the L-infinity norm of the Laplacian of a function (or a similar quantity) under Dirichlet boundary conditions. This is a convex, but not strictly convex variational problem. Nevertheless, it turns out that it has a unique solution, which is characterised by a system of PDEs. The behaviour is thus quite different from the better-known first order problems going back to Aronsson. This is joint work with N. Katzourakis (Reading).
 

Abstract regular polytopes are finite quotients of Coxeter complexes
with string diagram, satisfying a natural intersection property, see
e.g. [MMS2002]. They arise in a number of geometric and group-theoretic
contexts. The first class of such objects, beyond the
well-understood examples coming from finite and affine Coxeter groups,
are locally toroidal cases, e.g.  extensions of quotients of the affine
F_4 complex [3,3,4,3].  In 1996 P.McMullen & E.Schulte constructed a
number of examples of locally toroidal abstract regular polytopes of
type [3,3,4,3,3], and conjectured completeness of their list. We
construct counterexamples to the conjecture using a Y-shaped
presentation for a subgroup of the Monster, and discuss various
related questions.
 

Non-abelian gauge theories underly particle physics, including collision processes at particle accelerators. Recently, quantum scattering probabilities in gauge theories have been shown to be closely related to their counterparts in gravity theories, by the so-called double copy. This suggests a deep relationship between two very different areas of physics, and may lead to new insights into quantum gravity, as well as novel computational methods. This talk will review the double copy for amplitudes, before discussing how it may be extended to describe exact classical solutions such as black holes. Finally, I will discuss hints that the double copy may extend beyond perturbation theory. 

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under the only assumption of $L^1$ weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary $L^1$ vorticity. Relations with previously known notions of solutions are shown.

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.

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.

We will discuss some recent results with Martin Bridson about 
Sidki's construction X(G). In particular, if G is a finitely presented
group then X(G) is a finitely presented group. We will discuss as well the
result that if G has polynomial isoperimetric function and the maximal
metabelian quotient of G is virtually nilpotent then X(G) has polynomial
isoperimetric function. Part of the arguments we will use have homological
nature.