I will give an overview of recent results obtained in joint work with Yoshimichi Ueda on the structure and the rigidity of arbitrary free product von Neumann algebras. First, I will explain that in any free product von Neumann algebra, any amenable von Neumann subalgebra that has a diffuse intersection with one of the free components is necessarily contained in this free component. This result completely settles the problem of maximal amenability inside free product von Neumann algebras. Then I will present new Kurosh-type rigidity results for free product von Neumann algebras. Namely, I will explain that for any family of nonamenable factors belonging to a large class of (possibly type III) factors including nonprime factors, nonfull factors and factors with a Cartan subalgebra, the corresponding free product von Neumann algebra with respect to arbitrary states retains the cardinality of the family as well as each factor up to unitary conjugacy, after permutation of the indices.

\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
We give an essentially self-contained proof of the fact that a certain
$p$-adic power series
$$
\Psi= \Psi_p(T) \in T + T^{2}\Z[[T]]\;,
$$
which trivializes the addition law of the formal group of Witt
$p$-covectors $\widehat{\rm CW}_{\Z}$, is $p$-adically entire and
assumes values in $\Z_p$ all over $\Q_p$. We also carefully examine its
valuation and Newton polygons. We will recall and use the isomorphism
between the Witt and hyperexponential groups over $\Z_p$, and the
properties of $\Psi_p$, to show that, for any
perfectoid field extension $(K,|\,|)$ of $(\Q_p,|\,|_p)$, and to a
choice of a pseudo-uniformizer $\varpi = (\varpi^{(i)})_{i \geq 0}$ of
$K^\flat$, we can associate a continuous additive character
$\Psi_{\varpi}: \Q_p \to 1+K^{\circ \circ}$, and we will give a formula
to calculate it. The character $\Psi_{\varpi}$ extends the map $x
\mapsto \exp \pi x$, where
$$\pi := \sum_{i\geq 0} \varpi^{(i)} p^i + \sum_{i<0}
(\varpi^{(0)})^{p^{-i}} p^i \in K\;.
$$
I will also present numerical computation of the first coefficients of
$\Psi_p$, for small $p$, due to M. Candilera.

We study variations of GIT quotients of log pairs (X,D) where X is a hypersurface of some fixed degree and D is a hyperplane section. GIT is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. Any two such compactifications are related by birational transformations. We describe an algorithm to study the stability of the Hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. Finally, we relate this compactifications with the (conjectural) moduli space of log K-semistable pairs.

This is work in progress with Patricio Gallardo (University of Georgia).

Appearances not withstanding this is a talk about rational curves because they're the cause of the failure. Similarly, since homotopy groups are constant on the fibres of topological fibrations, a counterexample has to be in positive or mixed characteristic, and the specific one which I'll discuss is bi-disc quotients over Spec
Z. The example also has considerable logical implications for studying boundedness of rational curves on surfaces of general type, i.e. it cannot be implied by any theorem in $ACF_0$. Conversely, and more substantially, this boundedness can be proved uniformly in sufficiently large primes $p$ in $ACF_0$ provided the surface enjoys $c_1^2> c_2$.