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.