Archive 2018 JMM AMS Special Session on Quaternions – with videos of presentations

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Joint Mathematics Meetings AMS Special Session

Current as of Saturday, January 20, 2018 03:30:07

Joint Mathematics Meetings
San Diego Convention Center and Marriott Marquis San Diego Marina, San Diego, CA
January 10-13, 2018 (Wednesday – Saturday)
Meeting #1135

Associate secretaries:
Georgia Benkart, AMS benkart@math.wisc.edu
Gerard A Venema, MAA venema@calvin.edu

AMS Special Session on Quaternions

  • Wednesday January 10, 2018, 2:15 p.m.-6:00 p.m.
    AMS Special Session on Quaternions, I
    Room 33C, Upper Level, San Diego Convention Center
    Organizers:
    Terrence Blackman, Medgar Evers College, City University of New York
    Johannes Familton, Borough of Manhattan Community College, City University of New York
    jfamilton@bmcc.cuny.edu
    Chris McCarthy, Borough of Manhattan Community College, City University of New York


    • 2:15 p.m.
      Subalgebras of the Split Octonions.
      Lida Bentz, Virginia Tech
      Tevian Dray*, Oregon State University

      Abstract. Subalgebras of the Split Octonions. The proper subalgebras of the octonions O are well-known: the reals R, the complex numbers C, and the quaternions H. The Cayley–Dickson process also yields split cousins of these division algebras, denoted C’, H’, and O’, but each of these algebras admits additional subalgebras that are not, however, Cayley–Dickson algebras. We present a complete classification of such subalgebras, including examples in dimensions 3, 5, and 6.
      (1135-17-368)


    • 2:45 p.m.
      Quaternionic Spin.
      Corinne A. Manogue*, Oregon State University (Physics)
      Tevian Dray, Oregon State University

      Abstract. Quaternionic Spin. The standard Dirac operator in physics involves representations of the Clifford algebra in 3 + 1 spacetime dimensions, usually expressed in terms of 4 × 4 complex matrices. Rewriting these representations over the quaternions naturally leads to a formalism that treats the massive and massless Dirac equations on an equal footing, and that naturally lives in 5 + 1 spacetime dimensions. Extending to the octonions in turn extends the spacetime dimension to 9 + 1; remarkably, solutions to the octonionic Dirac equation must be quaternionic. Some applications to physics will be briefly discussed if time permits.
      (1135-17-367)


    • 3:15 p.m.
      A quaternionic unraveling of the double-twist in three-space.
      David Pengelley*, Oregon State University
      Daniel Ramras, Indiana University – Purdue University Indianapolis

      Abstract. A quaternionic unraveling of the double-twist in three-space. Physicists and mathematicians have long known it is possible to unravel a double-twist in three space, embodied in motions like the Dirac belt trick, Feynman plate trick, or Philippine candle dance. Quaternions can reveal how efficiently and beautifully this can be done, providing both theoretical constraints on the minimal required complexity, and insights into the geometry and level of simplicity possible. You will emerge knowing how to perform the quaternionic unraveling with your hand.
      (1135-20-422)


    • 3:45 p.m.
      Using quaternions to colour the cells of a tiling of H3
      Vi Hart, eleVR, HARC
      Andrea Hawksley, eleVR, HARC
      Elisabetta A Matsumoto, Georgia Institute of Technology
      Henry Segerman*, Oklahoma State University

      Abstract.
      Using quaternions to colour the cells of a tiling of H3 (Hyperbolic 3 Space). As part of a recent project to create a virtual reality experience of movement within H3, we needed some form of decoration to put in the space, so that users would be able to see the effects of their movement. We chose to use the {4,3,6} tiling of H3 by ideal cubes, with six cubes meeting around each edge. This works well, but if all cubes are rendered identically, it is difficult to keep track of movements in which the user moves through multiple cubes. I will describe how we used quaternions to produce an eight-colouring of the {4,3,6} tiling, giving users an improved set of landmarks to track their movement. The colouring arises as a kind of branched cover over the {4,3,3} tiling of S3.
      (1135-57-42)


    • 4:15 p.m.
      Carcinogenesis: does indefinite metric of a split-quaternion give a clue?

      Garri Davydyan*
      , Ottawa Hospital, Ottawa, Canada
      Abstract. Carcinogenesis: does indefinite metric of a split-quaternion give a clue? Preliminary report. Previously it was assumed that a structure of biologic objects can be described by split-quaternions whose vector part represents basis functional mechanisms of biologic systems (negative feedback, positive feedback and reciprocal links). Split-quaternions with the defined basis have a metric signature (- – + +) that determines normal systemogenesis. Each hierarchical level of the system is equipped with the indefinite metric inherited from the previous level during maturation. If an individual living cycle does not support the metric structure, it causes a pathologic regulation between hierarchical levels. In the frames of this concept, transformation of the indefinite metric to the positive definite one is a major systemic mechanism for cancer development.
      (1135-53-350)


    • 4:45 p.m.
      A Brief introduction to Clifford Algebras.
      Johannes C. Familton*, Borough of Manhattan Community College

      Abstract. A Brief introduction to Clifford Algebras. Those who are contributing papers to the special session on quaternions are familiar with quaternions, and/or the applications of quaternions to their work. A few are also familiar with Clifford Algebra. Last year I did a presentation “The Joining of Quaternions with Grassmann algebras: William Kingdon Clifford”. This year I will give a very basic mathematical introduction to the subject of Clifford Algebra (Geometric Algebra as Clifford called it). It is hoped that this talk will aid in filling in a few gaps for those who are not familiar with the subject and be a bridge between quaternions and Dr. Hestenes work in Geometric Algebra and Calculus.
      (1135-51-590)


    • 5:15 p.m.
      Quaternions in Geometric Algebra and Physics.
      David Hestenes*, Arizona State University
      (1135-00-139) (David Hestenes was not able to attend conference)


 

  • Thursday January 11, 2018, 1:00 p.m.-3:50 p.m.
    AMS Special Session on Quaternions, II
    Room 18, Mezzanine Level, San Diego Convention Center
    Organizers:
    Terrence Blackman, Medgar Evers College, City University of New York
    Johannes Familton, Borough of Manhattan Community College, City University of New York
    jfamilton@bmcc.cuny.edu
    Chris McCarthy, Borough of Manhattan Community College, City University of New York


    • 1:00 p.m.
      Rodrigues, Euler, and Hamilton: The problem of Rotations.
      Johannes C. Familton, Department of Mathematics, BMCC, The City University of New York, NY 10007,.
      Lucio Prado*, Department of Mathematics, BMCC, The City University of New York, NY 10007,.

      Abstract. Rodrigues, Euler, and Hamilton: The problem of Rotations. Preliminary report. In 1840 Rodrigues wrote a paper about the laws of geometry that control the displacement of a solid system in space. This work was a precursor to Hamilton’s quaternions. Rodrigues’ work went unnoticed until 1846 when Cayley acknowledged Euler’s and Rodrigues’ priority describing orthogonal transformations in a letter to the Editors of the Philosophical Magazine. In this talk, we want discuss the history of Rodrigues’ work, and his 1840 paper. We will analyze its connection with Euler’s rotation theorem, and Hamilton’s quaternions.
      (1135-00-1243)


    • 1:30 p.m.
      Computational advantages and historical insights from viewing quaternion multiplication as geodesic vector addition on S2
      Bob Palais*, Utah Valley University

      Abstract. Computational advantages and historical insights from viewing quaternion multiplication as geodesic vector addition on S2. We will develop geodesic vector addition on the ordinary two-sphere, S2, a group operation isomorphic to unit quaternion multiplication. This description offers insight into algorithms that implement and interpolate rotations, the belt, plate, and tangle tricks from physics, and several constructions from Lie groups. We will use this geometric perspective to understand the algebraic structure of the product and the mathematical inevitability of its discovery in Rodrigues’ 1840 paper on SO(3) as a continuous group.
      (1135-22-415)


      2:30 p.m.
      Quaternionic manifolds.
      Justin Sawon*, University of North Carolina – Chapel Hill

      Abstract. Quaternionic manifolds. Complex manifolds can be described as open subsets of C^n, patched together with biholomorphic maps. But quaternionanalytic maps are too limited to define quaternionic manifolds in a similar way. Instead, one defines an almost quaternionic structure on the tangent bundle, and then imposes one of several possible integrability conditions. This yields “quaternionic manifolds” and “hypercomplex manifolds”. One can also add compatible Riemannian metrics, giving “quaternion-Kahler manifolds” and “hyperkahler manifolds”. In this talk we will describe these different constructions and then survey some of the main facts about the resulting manifolds.
      (1135-53-392)


    • 3:00 p.m.
      Hilbert-Blumenthal Quaternionic Surfaces.
      Joseph A. Quinn*, Universidad Nacional Autónoma de México
      Alberto Verjovsky Sola, Universidad Nacional Autónoma de México

      Abstract. Hilbert-Blumenthal Quaternionic Surfaces. Preliminary report. The group PSL2(O), where O is an order in Hamilton’s quaternions, acts discretely by Mobius transformations on hyperbolic 4- or 5-space (depending on the definition used for PSL2), giving rise to hyperbolic 4- or 5-manifolds. We introduce a generalization of this where O is instead an order in a definite quaternion algebra over a real quadratic number field, and the action now occurs on a product of two copies of hyperbolic 4- or 5-space via a Galois twist (analogous to the classical Hilbert-Blumenthal surfaces), giving rise to 8- or 10-dimensional manifolds. We present a fundamental domain for the cusp of such a manifold, which facilitates the study of its topology and dynamics. We also discuss analogous new fundamental domains for classical Hilbert-Blumenthal surfaces that were developed concurrently by the authors with the intention of generalizing to the quaternionic version, and which serve nicely to build intuition for the higher-dimensional setting.
      (1135-57-148)


    • 3:30 p.m.
      Spectral correspondences for Maass waveforms on quaternion groups.
      Terrence Richard Blackman*, Medgar Evers College, CUNY

      Abstract. Spectral correspondences for Maass waveforms on quaternion groups. We prove that in most cases the Jacquet-Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e., when a bijection is possible. The proof uses the Selberg trace formula.
      (1135-11-336)


 

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