Unintended low energy thermal or mechanical stimuli can lead to the accidental ignition of explosive materials. During such events, described as ‘insults’ in the literature, ignition of the explosive is caused by localised regions of high temperature known as ‘hot spots’. We develop a model which helps us to understand how highly localised shear deformation, so-called shear banding, acts as a mechanism for hot spot generation. Through a boundary layer analysis, we give a deeper insight into how the additional self heating caused by chemical reactions affects the initiation and development of shear bands,  and highlight the key physical properties which control this process.

We study random (simple) graphs with given vertex degrees, in the sparse case where the average degree is bounded. Assume also that the second moment of the vertex degree is bounded. The standard method then is to use the configuration model to construct a random multigraph and condition it on
being simple.

This works well for results of the type that something holds with high probability, or that something converges in probability, but it does not immediately apply to convergence in distribution, for example asymptotic normality. (Although this has been done by special arguments in a couple of cases, by Janson and Luczak and by Riordan.) A typical example is the recent result by Barbour and Röllin on asymptotic normality of the size of the giant component of the multigraph (in the supercritical case); it is an obvious conjecture that the same results hold for the random simple graph.

We discuss two new approaches to this, both based on old methods. Both apply to the size of the giant component, using rather minor special arguments.

One approach uses the method of moments to obtain joint convergence of the variable of interest together with the numbers of loops and multiple edges
in the  multigraph.

The other approach uses switchings to modify the multigraph and construct a simple graph. This simple random graph will not have a uniform distribution,
but almost, and this is good enough.

Roughly speaking, a commutative-by-finite Hopf algebra is a Hopf
algebra which is an extension of a commutative Hopf algebra by a
finite dimensional Hopf algebra.
There are many big and significant classes of such algebras
(beyond of course the commutative ones and the finite dimensional ones!).
I'll make the definition precise, discuss examples
and review results, some old and some new.
No previous knowledge of Hopf algebras is necessary.
 

For a polynomial $h(x)$ in $F[x]$, where $F$ is any field, let $A$ be the
$F$-algebra given by generators $x$ and $y$ and relation $[y, x]=h$.
This family of algebras include the Weyl algebra, enveloping algebras of
$2$-dimensional Lie algebras, the Jordan plane and several other
interesting subalgebras of the Weyl algebra.

In a joint work in progress with Samuel Lopes, we computed the Hochschild
cohomology $HH^*(A)$ of $A$ and determined explicitly the Gerstenhaber
structure of $HH^*(A)$, as a Lie module over the Lie algebra $HH^1(A)$.
In case $F$ has characteristic $0$, this study has revealed that $HH^*(A)$
has finite length as a Lie module over $HH^1(A)$ with pairwise
non-isomorphic composition factors and the latter can be naturally
extended into irreducible representations of the Virasoro algebra.
Moreover, the whole action can be understood in terms of the partition
formed by the multiplicities of the irreducible factors of the polynomial
$h$.
 

In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof. 

In the sequel we will apply this to extend local deformations of closed $G_2$ structures, and to construct 
$C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds. 

This is joint work with Christian Bär (Potsdam).

We recall the impact of stabilizing a 4-manifold with $S^2 \times S^2$. The corresponding local situation concerns knots in the 3-sphere which bound (nullhomotopic) discs in a stabilized 4-ball. We explain how these discs arise, and discuss bounds on the minimal number of stabilizations needed. Then we compare this minimal number to the 4-genus.
This is joint work with A. Conway.

In my talk I will discuss modular properties of false theta functions. Due to a wrong sign factor these are not directly seen to be modular, however there are ways to repair this. I will report about this in my talk.

Given a "random" polynomial over the integers, it is expected that, with high probability, it is irreducible and has a big Galois group over the rationals. Such results have been long known when the degree is bounded and the coefficients are chosen uniformly at random from some interval, but the case of bounded coefficients and unbounded degree remained open. Very recently, Emmanuel Breuillard and Peter Varju settled the case of bounded coefficients conditionally on the Riemann Hypothesis for certain Dedekind zeta functions. In this talk, I will present unconditional progress towards this problem, joint with Lior Bary-Soroker and Gady Kozma.