The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

The minimally displaced set of an irreducible automorphism is locally finite

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

EXTRACTING AND EVALUATING CLUSTERS IN DINSAR DEFORMATION DATA ON SINGLE BUILDINGS

In the past two decades persistent scatterer interferometry (PSI) has become a well understood and powerful method to monitor the deformations of man-made structures. PSI can derive displacement histories of thousands of scattered points on a single building with accuracy of a few millimetre per year, by analysing space-borne SAR data. In this paper, we present a method to cluster PS points on a single building into segments which show the same deformation behavior. The spatial distribution of those clusters gives an insight into the structural behavior of a building. We use dimensionality reduction to visualize the clusters in the deformation space. The comparison of our extracted displacement patterns with ground truth data from precise levelling and 3D tachymetry confirms the plausibility of our remote sensing method.

Deformation Theory of the Trivial mod p Galois Representation for GLn

We study the rigid generic fiber $\mathcal{X}^\square _{\overline \rho }$ of the framed deformation space of the trivial representation $\overline \rho : G_K \to \textrm{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute Galois group of a finite extension $K/\textbf{Q}_p$. Under some mild conditions on $K$ we prove that $\mathcal{X}^\square _{\overline \rho }$ is normal. When $p> n$ we describe its irreducible components and show Zariski density of its crystalline points.

Deformation spaces and normal forms around transversals

Given a manifold $M$ with a submanifold $N$, the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$, and all other fibers are equal to $M$. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold transverse to the foliation. New examples include $L_{\infty }$-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around $N$, in terms of a model structure over $\unicode[STIX]{x1D708}(M,N)$.

Controlling the deformation space of soft membranes using fiber reinforcement

Recent efforts in soft-body control have been hindered by the infinite dimensionality of soft bodies. Without restricting the deformation space of soft bodies to desired degrees of freedom, it is difficult, if not impossible, to guarantee that the soft body will remain constrained within a desired operating range. In this article, we present novel modeling and fabrication techniques for leveraging the reorientation of fiber arrays in soft bodies to restrict their deformation space to a critical case. Implementing this fiber reinforcement introduces unique challenges, especially in complex configurations. To address these challenges, we present a geometric technique for modeling fiber reinforcement on smooth elastomeric surfaces and a two-stage molding process to embed the fiber patterns dictated by that technique into elastomer membranes. The variable material properties afforded by fiber reinforcement are demonstrated with the canonical case of a soft, circular membrane reinforced with an embedded, intersecting fiber pattern such that it deforms into a prescribed hemispherical geometry when inflated. It remains constrained to that configuration, even with an additional increase in internal pressure. Furthermore, we show that the fiber-reinforced membrane is capable of maintaining its hemispherical shape under a load, and we present a practical application for the membrane by using it to control the buoyancy of a bioinspired autonomous underwater robot developed in our lab. An additional experiment on a circular membrane that inflates to a conical frustum is presented to provide additional validation of the versatility of the proposed model and fabrication techniques.