Grassmann geometries in infinite dimensional homogeneous spaces and an application to reflective submanifolds

Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. The solutions are shown to correspond to various special submanifolds, depending on which homogeneous space U/L one projects to. We apply the construction to a question which generalizes, to the context of reflective submanifolds of arbitrary symmetric spaces, the problem of isometric immersions of space forms with negative extrinsic curvature and flat normal bundle. For this problem, we prove that the only cases where local solutions exist are the previously known cases of space forms, in addition to constant curvature Lagrangian immersions into complex projective and complex hyperbolic spaces. We also prove non-existence of global solutions in the compact case. The solutions associated to other reflective submanifolds correspond to special deformations of lower dimensional submanifolds. As an example, we obtain a special class of surfaces in the 6-sphere.Comment: 31 pages. Minor revision. Some notational changes, comments added. Section 6.5 has been added. Section 8.1 rewritte

Correlation Does Not Imply Correlation: A Thesis on Causal Influence and Simpson’s Paradox

In our data-driven world, it has become commonplace to attempt to findcausal relationships. One of the themes of this thesis is to show methods ofdetermining causation. The second theme follows a saying in mathematics, correlation does not imply causation . We will also discuss situations wherecorrelation does not even imply correlation itself. These cases are describedby Simpson’s paradox in an exploration of different areas of mathematicsand computer coding