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--- geometric_algebra [2020/07/10 17:22] – [Articles] pbk
+++ geometric_algebra [2020/07/19 17:23] – [Articles] pbk
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   * [[https://en.wikipedia.org/wiki/Multivector|Multivector]]
   * [[https://en.wikipedia.org/wiki/Paravector|Paravector]]
+  * [[https://en.wikipedia.org/wiki/Outermorphism|Outermorphism]]
   * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]]
   * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]]
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   * [[http://cs.unitbv.ro/~acami/what_kc_can_do.pdf|What the Kähler Calculus can do that other calculi cannot]] (2016) - //Jose G. Vargas//
 Underlied by Clifford algebra of differential forms --- like tangent Clifford algebra underlies the geometric calculus --- it brings about a fresh new view of quantum mechanics. This view arises, almost without effort, from the equation which is in Kähler Calculus what the Dirac equation is in traditional quantum mechanics.
+  * [[https://www.mit.edu/~fengt/282C.pdf|The Atiyah–Singer index theorem]] (2015) - //Dan Berwick-Evans, via Tony Feng//
+Lecture notes about the Atiyah-Singer index theorem.
   * [[http://www.siue.edu/~sstaple/index_files/CODecompAccepted2015.pdf|Clifford algebra decompositions of conformal orthogonal group elements]] (2015) - //G. Stacey Staples, David Wylie//
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   * [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez//
 Robust methods for finding the best rotation aligning two sets of corresponding vectors are formulated in the linear algebra framework, using tools like the SVD for polar decomposition or QR for finding eigenvectors. Those are well established numerical algorithms which on the other hand are iterative and computationally expensive. Recently, closed form solutions has been proposed in the quaternion’s framework, those methods are fast but they have singularities i.e., they completely fail on certain input data. In this paper we propose a robust attitude estimator based on a formulation of the problem in Geometric Algebra. We find the optimal eigen-quaternion in closed form with high accuracy and with competitive performance respect to the fastest methods reported in literature.
+  * [[https://arxiv.org/pdf/1911.08658|Spinors of real type as polyforms and the generalized Killing equation]] (2019) - //Vicente Cortes, Calin Lazaroiu, C. S. Shahbazi//
+We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS_4 space-time.
   * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler//