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- ItemLongtime behavior for a generalized Cahn-Hilliard system with fractional operators(Messina : Accademia Peloritana dei Pericolanti, 2020) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn-Hilliard system, with possibly singular potentials, that we have recently investigated in the paper Well-posedness and regularity for a generalized fractional Cahn-Hilliard system. More precisely, we study the ω-limit of the phase parameter y and characterize it completely. Our characterization depends on the first eigenvalues λ1≥0 of one of the operators involved: if λ1>0, then the chemical potential μ vanishes at infinity and every element yω of the ω-limit is a stationary solution to the phase equation; if instead λ1=0, then every element yω of the ω-limit satisfies a problem containing a real function μ∞ related to the chemical potential μ. Such a function μ∞ is nonunique and time dependent, in general, as we show by an example. However, we give sufficient conditions for μ∞ to be uniquely determined and constant.
- ItemCharacterization of Silicon Crystals Grown from Melt in a Granulate Crucible(Warrendale, Pa : TMS, 2020) Dadzis, K.; Menzel, R.; Juda, U.; Irmscher, K.; Kranert, C.; Müller, M.; Ehrl, M.; Weingärtner, R.; Reimann, C.; Abrosimov, N.; Riemann, H.The growth of silicon crystals from a melt contained in a granulate crucible significantly differs from the classical growth techniques because of the granulate feedstock and the continuous growth process. We performed a systematic study of impurities and structural defects in several such crystals with diameters up to 60 mm. The possible origin of various defects is discussed and attributed to feedstock (concentration of transition metals), growth setup (carbon concentration), or growth process (dislocation density), showing the potential for further optimization. A distinct correlation between crystal defects and bulk carrier lifetime is observed. A bulk carrier lifetime with values up to 600 μs on passivated surfaces of dislocation-free parts of the crystal is currently achieved.
- ItemOptimization of the Epitaxial Growth of Undoped GaN Waveguides in GaN-Based Laser Diodes Evaluated by Photoluminescence(Warrendale, Pa : TMS, 2020) Netzel, C.; Hoffmann, V.; Einfeldt, S.; Weyers, M.Non-intentionally doped c-plane GaN layers are generally employed as p-side waveguide layers in violet/blue-emitting laser diodes. The recombination and diffusion of charge carriers in the p-side GaN waveguide influence the injection efficiency of holes into the InGaN quantum wells of these devices. In this study, the non-radiative recombination and the diffusivity in the [000-1] direction for charge carriers in such GaN layers are investigated by the photoluminescence of buried InGaN quantum wells, in addition to the GaN photoluminescence. The vertical charge carrier diffusion length and the diffusion constant in GaN were determined by evaluating the intensity from InGaN quantum wells in different depths below a top GaN layer. Additionally, the intensity from the buried InGaN quantum wells was found to be more sensitive to variations in the non-radiative recombination rate in the GaN layer than the intensity from the GaN itself. The study enables conclusions to be drawn on how the growth of a p-side GaN waveguide layer has to be optimized: (1) The charge carrier diffusivity in the [000-1] direction at device operation temperature is limited by phonon scattering and can be only slightly improved by material quality. (2) The use of TMGa (trimethylgallium) instead of TEGa (triethylgallium) as a precursor for the growth of GaN lowers the background silicon doping level and is advantageous for a large hole diffusion length. (3) Small growth rates below 0.5 μm/h when using TMGa or below 0.12 μm/h when using TEGa enhance non-radiative recombination. (4) A V/III gas ratio of 2200 or more is needed for low non-radiative recombination rates in GaN.
- ItemMini-Workshop: Superpotentials in Algebra and Geometry(Zürich : EMS Publ. House, 2020) González, Eduardo; Rietsch, Konstanze; Williams, LaurenMirror symmetry has been at the epicenter of many mathematical discoveries in the past twenty years. It was discovered by physicists in the setting of super conformal field theories (SCFTs) associated to closed string theory, mathematically described by $\sigma$-models. These $\sigma$-models turn out in two different ways: the A-model and the B-model. Physical considerations predict that deformations of the SCFT of either $\sigma$-model should be isomorphic. Thus the mirror symmetry conjecture states that the A-model of a particular Calabi-Yau space $X$ must be isomorphic to the B-model of its mirror $\check{X}$. Mirror symmetry has been extended beyond the Calabi-Yau setting, in particular to Fano varieties, using the so called Landau-Ginzburg models. That is a non-compact manifold equipped with a complex valued function called the \emph{superpotential}. In general, there is no clear recipe to construct the mirror for a given variety which demonstrates the need of joining mathematical forces from a wide range. The main aim of this Mini-Workshop was to bring together experts from the different communities (such as symplectic geometry and topology, the theory of cluster varieties, Lie theory and algebraic combinatorics) and to share the state of the art on superpotentials and explore connections between different constructions.
- ItemAlgebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (hybrid meeting)(Zürich : EMS Publ. House, 2020) Huybrechts, Daniel; Thomas, Richard; Xu, ChenyangThe talks at the workshop and the research done during the week focused on aspects of algebraic geometry in the broad sense. Special emphasis was put on hyperkähler manifolds and derived categories.
- ItemMini-Workshop: (Anosov)$^3$ (hybrid meeting)(Zürich : EMS Publ. House, 2021) Delarue, Benjamin; Pozzetti, Beatrice; Weich, TobiasThree different active fields are subsumed under the keyword Anosov theory: Spectral theory of Anosov flows, dynamical rigidity of Anosov actions, and Anosov representations. In all three fields there have been dynamic developments and substantial breakthroughs in recent years. The mini-workshop brought together researchers from the three different communities and sparked a joint discussion of current ideas, common interests, and open problems.
- ItemMini-Workshop: Computational Optimization on Manifolds (online meeting)(Zürich : EMS Publ. House, 2020) Herzog, Roland; Steidl, GabrieleThe goal of the mini-workshop was to study the geometry, algorithms and applications of unconstrained and constrained optimization problems posed on Riemannian manifolds. Focus topics included the geometry of particular manifolds, the formulation and analysis of a number of application problems, as well as novel algorithms and their implementation.
- ItemMini-Workshop: Analysis of Data-driven Optimal Control (hybrid meeting)(Zürich : EMS Publ. House, 2021) Morris, KirstenThis hybrid mini-workshop discussed recent mathematical methods for analyzing the opportunities and limitations of data-driven and machine-learning approaches to optimal feedback control. The analysis concerned all aspects of such approaches, ranging from approximation theory particularly for high-dimensional problems via complexity analysis of algorithms to robustness issues.
- ItemAnalysis, Geometry and Topology of Positive Scalar Curvature Metrics (hybrid meeting)(Zürich : EMS Publ. House, 2021) Hanke, Bernhard; Sakovich, AnnaThe investigation of Riemannian metrics with lower scalar curvature bounds has been a central topic in differential geometry for decades. It addresses foundational problems, combining ideas and methods from global analysis, geometric topology, metric geometry and general relativity. Seminal contributions by Gromov during the last years have led to a significant increase of activities in the area which have produced a number of impressive results. Our workshop reflected the state of the art of this thriving field of research.
- ItemMini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces (hybrid meeting)(Zürich : EMS Publ. House, 2021) Güneysu, Batu; Keller, Matthias; Kuwae, KazuhiroA Dirichlet form $\mathcal{E}$ is a densely defined bilinear form on a Hilbert space of the form $L^2(X,\mu)$, subject to some additional properties, which make sure that $\mathcal{E}$ can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f_1,f_2)=\int_D (\nabla f_1,\nabla f_2) $ on a domain $D$ in $\mathbb{R}^m$. The main strength of this theory, however, is that it allows also to treat nonlocal situations such as energy forms on graphs simultaneously. In typical applications, $X$ is a metrizable space, and the theory of Dirichlet forms makes it possible to define notions such as curvature bounds on $X$ (although $X$ need not be a Riemannian manifold), and also to obtain topological information on $X$ in terms of such geometric information.