Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller

The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasiconformality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Grötzsch, Lavrentieff, Ahlfors and Teichmüller, which are the 20th-century founders of the theory

Thurston's metric on Teichm\"uller space and the translation lengths of mapping classes

We show that the Teichm\"uller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichm\"uller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances for mapping classes for their actions on Teichm\"uller spaces equipped with their arc metrics

Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory

International audienceIn the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coordinates. At the basis of the construction are certain maps with controlled Lipschitz constants between right-angled hyperbolic hexagons having three non-consecutive edges of the same size. Using these maps, we obtain Lipschitz-minimizing maps between hyperbolic particular pairs of pants and, more generally, between some hyperbolic sufaces of finite type with arbitrary genus and arbitrary number of boundary components. The Lipschitz-minimizing maps that we contruct are distinct from Thurston's stretch maps

Grassmannians,Calibrations and Five-Brane Intersections

We present a geometric construction of a new class of hyper-Kahler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kahler geometry with torsion associated with the NS-5-brane along quaternionic planes in quaternionic k-space, \bH^k. We find the moduli space of these geometries and show that it can be constructed using the bundle space of the canonical quaternionic line bundle over a quaternionic projective space. We also investigate several special cases which are associated with certain classes of quaternionic planes in \bH^k. We then show that the eight-dimensional geometries we have found can be constructed using quaternionic calibrations. We generalize our construction to superpose the same four-dimensional hyper-Kahler geometry with torsion along complex planes in \bC^{2k}. We find that the resulting geometry is Kahler with torsion. The moduli space of these geometries is also investigated. In addition, the applications of these new geometries to M-theory and sigma models are presented. In particular, we find new solutions of IIA supergravity with the interpretation of intersecting NS-5-branes at Sp(2)-angles on a string and show that they preserve 3/32, 1/8, 5/32 and 3/16 of supersymmetry. We also show that two-dimensional sigma models with target spaces the above manifolds have (p,q) extended supersymmetry.Comment: 39 pages, phyzzx; a previously undetermined fraction of supersymmetry has now been fixed; a table has been replaced; version submitted for publication in CM

Solitons in (1,1)-supersymmetric massive sigma model

We find the solitons of massive (1,1)-supersymmetric sigma models with target space the groups $SO(2)$ and $SU(2)$ for a class of scalar potentials and compute their charge, mass and moduli space metric. We also investigate the massive sigma models with target space any semisimple Lie group and show that some of their solitons can be obtained from embedding the $SO(2)$ and $SU(2)$ solitons.Comment: Phyzzx.tex, 32 pp, 3 fig

Instantons at Angles

We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group $Sp(k)$ and are examples of Hermitian-Einstein connections on \bE^{4k}. We show that the eight-dimensional solutions preserve 3/16 of the ten-dimensional N=1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the longitudinal intersecting 5-branes on a string at angles configuration of M-theory.Comment: 17 pages, phyzzx, many changes and a new section was adde

Refugees, trauma and adversity-activated development

The nature of the refugee phenomenon is examined and the position of mental health professionals is located in relation to it. The various uses of the word 'trauma' are explored and its application to the refugee context is examined. It is proposed that refugees' response to adversity is not limited to being traumatized but includes resilience and Adversity-Activated Development (AAD). Particular emphasis is given to the distinction between resilience and AAD. The usefulness of the 'Trauma Grid' in the therapeutic process with refugees is also discussed. The Trauma Grid avoids global impressions and enables a more comprehensive and systematic way of identifying the individual refugee's functioning in the context of different levels, i.e. individual, family, community and society/culture. Finally, I discuss implications for therapeutic work with refugees