In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$
holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The
asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical
(ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.  

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In
particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

In this talk, we consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results

Risk-indifference pricing is proposed as an alternative to utility indifference pricing, where a risk measure is used instead of a utility based preference. In this, we propose to include the possibility to change the attitude to risk evaluation as time progresses. This is particularly reasonable for long term investments and strategies. 

Then we introduce a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. The risk-indifference pricing system is studied from the point of view of its properties as a convex price system. We tackle questions of time-consistency in the risk evaluation and the corresponding prices. This analysis provides a new insight also to time-consistency for ordinary dynamic risk-measures.

Our techniques and results are set in the representation and extension theorems for convex operators. We shall argue and finally provide a setting in which fully-dynamic risk-indifference pricing is a well set convex price system.

The presentation is based on joint works with Jocelyne Bion-Nadal.