Abstract:
The European XFEL, currently under construction, will produce a coherent X-ray pulse every 222 ns in pulse trains of up to 2700 pulses. In conjunction with the fast 2D area detectors currently under development, it will be possible to perform X-ray Photon Correlation Spectroscopy (XPCS) experiments on sub-microsecond timescales with non-ergodic systems. A case study for the Adaptive Gain Integrating Pixel Detector (AGIPD) at the European XFEL employing the intensity autocorrelation technique was performed using the detector simulation tool HORUS. As optimum results from XPCS experiments are obtained when the pixel size approximates the (small) speckle size, the presented study compares the AGIPD (pixel size of (200 $\upmu$m)$^2$) to a possible apertured version of the detector and to a hypothetical system with (100 $\upmu$m)$^2$ pixel size and investigates the influence of intensity fluctuations and incoherent noise on the quality of the acquired data. The intuitive conclusion that aperturing is not beneficial as data is 'thrown away' was proven to be correct for low intensities. For intensities larger than approximately 1 photon per (100 $\upmu$m)$^2$ aperturing was found to be beneficial, as charge sharing effects were excluded by it. It was shown that for the investigated case (100 $\upmu$m)$^2$ pixels produced significantly better results than (200 $\upmu$m)$^2$ pixels when the average intensity exceeded approximately 0.05 photons per (100 $\upmu$m)$^2$. Although the systems were quite different in design they varied in the signal to noise ratio only by a factor of 2-3, and even less in the relative error of the extracted correlation constants. However the dependence on intensity showed distinctively different features for the different systems.

Abstract:
We present the results of a computational X-ray cross correlation analysis (XCCA) study on two dimensional polygonal model structures. We show how to detect and identify the orientational order of such systems, demonstrate how to eliminate the influence of the "computational box" on the XCCA results and develop new correlation functions that reflect the sample's orientational order only. For this purpose, we study the dependence of the correlation functions on the number of polygonal clusters and wave vector transfer $q$ for various types of polygons including mixtures of polygons and randomly placed particles. We define an order parameter that describes the orientational order within the sample. Finally, we determine the influence of detector noise and non-planar wavefronts on the XCCA data which both appear to affect the results significantly and have thus to be considered in real experiments.

Abstract:
We demonstrate the potential of X-ray free-electron lasers (XFEL) to advancethe understanding of complex plasma dynamics by allowing for the first time nanometer and femtosecond resolution at the same time in plasma diagnostics. Plasma phenomena on such short timescales are of high relevance for many fields of physics, in particular in the ultra-intense ultra-short laser interaction with matter. Highly relevant yet only partially understood phenomena may become directly accessible in experiment. These include relativistic laser absorption at solid targets, creation of energetic electrons and electron transport in warm dense matter, including the seeding and development of surface and beam instabilities, ambipolar expansion, shock formation, and dynamics at the surfaces or at buried layers. We demonstrate the potentials of XFEL plasma probing for high power laser matter interactions using exemplary the small angle X-ray scattering technique, focusing on general considerations for XFEL probing.

Abstract:
We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given to those admitting a compatible connection with completely skew torsion.

Abstract:
I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization.

Abstract:
Jun-Muk Hwang and Ngaiming Mok have proved the rigidity of irreducible Hermitian symmetric spaces of compact type under Kaehler degeneration. I adapt their argument to the algebraic setting in positive characteristic, where cominuscule homogeneous varieties serve as an analogue of Hermitian symmetric spaces. The main result gives an explicit (computable in terms of Schubert calculus) lower bound on the characteristic of the base field, guaranteeing that a smooth projective family with cominuscule homogeneous generic fibre is isotrivial. The bound depends only on the type of the generic fibre, and on the degree of an invertible sheaf whose extension to the special fibre is very ample. An important part of the proof is a characteristic-free analogue of Hwang and Mok's extension theorem for maps of Fano varieties of Picard number 1, a result I believe to be interesting in its own right.

Abstract:
We give a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in question are expressed in terms of elementary Jordan matrices and integers with values in $\{-1,0,1\}$ related to signatures of quadratic forms naturally associated to the symplectic matrix.

Abstract:
The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,\bar{\Omega})$, having chosen a given reference Lagrangian $V$. Paths of symplectic endomorphisms of $(\R^{2n},\Omega_0)$ are viewed as paths of Lagrangians defined by their graphs in $(W=\R^{2n}\oplus \R^{2n},\bar{\Omega}=\Omega_0\oplus -\Omega_0)$ and the reference Lagrangian is the diagonal. Robbin and Salamon give properties of this generalized Conley-Zehnder index and an explicit formula when the path has only regular crossings. We give here an axiomatic characterization of this generalized Conley-Zehnder index. We also give an explicit way to compute it for any continuous path of symplectic matrices.

Abstract:
We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive $S^1$-equivariant symplectic homology can be computed; it is generated by those orbits. We prove a "Viterbo functoriality" property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive $S^1$-equivariant symplectic homologies and morphisms compose nicely. These properties allow us to give a proof of Ustilovsky's result on the number of non isomorphic contact structures on the spheres $S^{4m+1}$. They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in $\mathbb{R}^{2n}$. We extend this result to some hypersurfaces in some negative line bundles.

Abstract:
We give here a self contained and elementary introduction to the Conley-Zehnder index for a path of symplectic matrices. We start from the definition of the index as the degree of a map into the circle for a path starting at the identity and ending at a matrix for which 1 is not an eigenvalue. We prove some properties which characterize this index using normal forms for symplectic matrices obtained from geometrical considerations. We explore the relations to Robbin-Salamon index for paths of Lagrangians. We give an axiomatic characterization of the generalization of the Conley-Zehnder index for any continuous path of symplectic matrices defined by Robbin and Salamon.