Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar\'e equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle

Vlasov moment flows and geodesics on the Jacobi group

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered in arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.Comment: 45 page

The Dynamics of a Rigid Body in Potential Flow with Circulation

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor.Comment: 28 pages, 2 figures; v2: typos correcte

Shifted Symplectic Structures

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks. We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures. Our main existence theorem states that for any derived Artin stack $F$ equipped with an $n$-shifted symplectic structure, the derived mapping stack $\textbf{Map}(X,F)$ is equipped with a canonical $(n-d)$-shifted symplectic structure as soon a $X$ satisfies a Calabi-Yau condition in dimension $d$. These two results imply the existence of many examples of derived moduli stacks equipped with $n$-shifted symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We also show that Lagrangian intersections carry canonical (-1)-shifted symplectic structures.Comment: 52 pages. To appear in Publ. Math. IHE

Turbulence and Holography

We examine the interplay between recent advances in quantum gravity and the problem of turbulence. In particular, we argue that in the gravitational context the phenomenon of turbulence is intimately related to the properties of spacetime foam. In this framework we discuss the relation of turbulence and holography and the interpretation of the Kolmogorov scaling in the quantum gravitational setting.Comment: 19 pages, LaTeX; version 2: reference adde

Three-dimensional modeling of melt flow and interface shape in the industrial liquid-encapsulated Czochralski growth of GaAs

The heat transport in the melt and in the crystal including the interface shape was numerically investigated by local, fully three-dimensional (3D), time-dependent simulations for an industrially sized liquid-encapsulated Czochralski setup for growing GaAs crystals with 150 mm diameter. The thermal boundary conditions for the local 3D simulations were obtained from global quasi-steady 2D simulations. It was found that the type of thermal boundary conditions (fixed temperature or heat flux) has a strong influence on the 3D results of the interface shape and on the melt convection. Furthermore, a high sensitivity of the interface deflection on the temperature at the bottom wall of the crucible is observed. For a control of the interface shape the use of two types of magnetic fields (horizontal and vertical) was considered. It was found that a horizontal magnetic field has a bigger influence on the interface shape than a vertical magnetic field

Large modification of crystal-melt interface shape during Si crystal growth by using electromagnetic Czochralski method (EMCZ)

Large-diameter, high-quality Si wafers are required for further advance of ultra large-scale integrated circuit (ULSI) device processing. Therefore, a new crystal growth technique is needed to obtain large-diameter, high-quality Si crystals containing homogeneously distributed oxygen and reduction of grown-in defect density in the concentration required for ULSI device processing. To address this requirement, we developed a new crystal growth technique using electromagnetic force (EMF), which we call the electromagnetic Czochralski (EMCZ) method. Using the EMCZ method, we were able to grow defect-free Si crystals of 200 mm diameter with a higher pulling rate rather than those grown under the conventional CZ crystal growth conditions. High-speed pulling of defect-free crystals by the EMCZ method is due to large modifications of the crystal-melt interface. In this method, the interface shape is largely modified to the upward convexly toward the crystal. The large upward convex shape of the crystal-melt interface during EMCZ growth results from the temperature distribution at the interface by the controlled melt flow generated by the EMF. We confirmed this large modification of interface shape by experiments and numerical simulations. The mechanism of this modification of interface shape is discussed from the viewpoint of melt flow around the growing interface