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Joint Mathematics Meetings

• Baltimore Convention Center, Hilton Baltimore, and Baltimore Marriott Inner Harbor Hotel, Baltimore, MD
• January 16-19, 2019 (Wednesday – Saturday)
• Meeting #1145

Associate secretaries:

Steven H Weintraub, AMS shw2@lehigh.edu
Hortensia Soto, MAA hortensia.soto@unco.edu

AMS Special Session on Quaternions (link to the JMM2019 Website)

• Wednesday January 16, 2019, 8:00 a.m.-10:50 a.m.
  AMS Special Session on Quaternions, I
  Room 319, BCC
  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
  • 8:00 a.m.
    Involutions of groups of type $G_2$ over fields.
    John Hutchens*, Winston-Salem State University
    Nathaniel Schwartz, Washington College
    
    Abstract. Involutions of groups of type G_2 over fields. We define a generalized symmetric space to be the quotient G/H where G is an algebraic group and H is the fixed point group of an involution of G. Let C be an octonion algebra over a field k, then Aut(C) is a group of type G2 over k. We determine the Aut(C)-conjugacy classes of the k-involutions and their respective fixed point groups. It is shown that the classification of conjugacy classes of involutions of Aut(C) correspond to isomorphism classes of quaternion algebras for almost every field.
    (1145-20-93)
  • 8:30 a.m.
    Using quaternions to prove theorems in spherical geometry.
    Marshall A Whittlesey*, California State University San MarcosAbstract. Using quaternions to prove theorems in spherical geometry. The quaternions have long been known to be useful for describing rotations and reflections in 3 dimensional Euclidean space. We show how to use the quaternions as a tool to prove theorems in spherical geometry.
    (1145-51-232)
  • 9:00 a.m.
    Edge-minimal Graphs with Given Generalized Quaternion Automorphism Group.
    Lindsey-Kay Lauderdale*, Towson University
    Christina Graves, University of Texas at Tyler
    Stephen Graves, University of Texas at Tyler
    
    Abstract. Edge-minimal Graphs with Given Generalized Quaternion Automorphism Group. Preliminary report. For a finite group G, let e(G, m) denote the minimum number of edges among all graphs with m vertices and automorphism group isomorphic to G; if no such graphs exists, then consider e(G, m) to be undefined. This invariant is the subject of prior research by several authors, but its value is known only for two finite groups and a few other infinite families of finite groups. In this talk, we will consider the value of e(Q_{2^n}, m) for the generalized quaternion group, Q_{2^n}, where n ≥ 3. Specifically, if m ≥ 2^{n+1}, we determine the value of e(Q_{2^n}, m) ; the value of e(Q_{2^n}, m) is undefined provided m < 2^{n+1}. Additionally, we will discuss the sizes of connected edge-minimal graphs with quaternion symmetry and conclude with some open questions on the value of e(G, m) in general.
    (1145-05-474)

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    The following 3 presentations appear in the single video SawonQuinnBlackmanJMM2019 (below).
    

  • 9:30 a.m.
    Generalized twistor spaces of quaternionic manifolds.
    Justin Sawon*, University of North Carolina – Chapel Hill
    (1145-53-77)   (0:00 to 27:00 in video)
  • 10:00 a.m.
    Macfarlane hyperbolic 3-manifolds.
    Joseph A Quinn*, National Museum of Mathematics
    (1145-57-240)
  • 10:30 a.m.
    Spectral correspondences for Maass waveforms on quaternion groups.
    Terrence Richard Blackman*, Medgar Evers College, CUNY
  • (1145-11-2949)
• Abstracts of the three presentations appearing in the above video.
  
  • Presentation 1 (from 0:00 to 27:00 in video): Abstract (Sawon): Preliminary report. Quaternionic manifolds are equipped with families of complex structures (in some cases, local and/or almost complex structures). The twistor construction is a convenient way of packaging these different complex structures into a single complex manifold, known as the twistor space. Quaternionic manifolds also admit large families of generalized complex structures, in the sense of Hitchin, and one would like to package these all in a single generalized complex manifold. Together with my student Rebecca Glover, we constructed such generalized twistor spaces for hyperkahler manifolds. In this talk, we describe this construction and its extension to another class of quaternionic manifolds: quaternion-Kahler. From 0:00 to 27:00 in video.
  • Presentation 2 (from 29:00 to 54:00 in video): Abstract (Quinn). Macfarlane hyperbolic 3-manifolds. We identify and study a class of hyperbolic 3-manifolds whose (generalized) quaternion algebras admit a geometric interpretation analogous to Hamilton’s classical model for Euclidean rotations. We call these Macfarlane manifolds, as
    they incorporate a modern generalization of Alexander Macfarlane’s classical ideas about hyperbolic quaternions. We characterize these manifolds arithmetically, and show that infinitely many commensurability classes of them arise in diverse topological and arithmetic settings. We then use this perspective to introduce a new method for computing their Dirichlet domains. We also give similar results for a class of hyperbolic surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds. From 29:00 to 54:00 in video.
  • Presentation 3 (from 54:00 to 1:35:50 in video): Abstract (Blackman). 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. From 54:00 to 1:35:50 in video.

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• Wednesday January 16, 2019, 2:15 p.m.-6:05 p.m.
  AMS Special Session on Quaternions, II
  Room 319, BCC
  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

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  • 2:15 p.m.
    Maxwell, Clifford, and Hestenes.
    Johannes Familton*, BMCC City University of New York
    Richard Friedberg, Columbia University (emeritus)
    
    Abstract. Since Maxwell first wrote his equations they have been rewritten in many different ways. Maxwell himself first wrote them as components (1862-1865) and then in quaternions (1873). In this talk we will look at the 4D Clifford algebra formalism and how it has been used in Hestene’s version of Geometric Algebra to derive Maxwell’s equation.
    (1145-78-531)

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    2:45 p.m.
    Quaternions in Geometric Algebra and Physics.
    David Hestenes*, Department of Physics, Arizona State University
    
    Abstract.  Geometric Algebra and Calculus has emerged as a unified mathematical language for the whole of physics —- a language that simplifies formulation and solution of all fundamental equations while providing new insights into the geometric structure of physics [1]. In this talk I discuss how quaternions fit into Geometric Algebra with emphasis on rotational dynamics, spinors and Hopf fibrations in electrodynamics. [1] Geometric Calculus website: http://geocalc.clas.asu.edu Most papers and books on GC can be accessed or traced from here and links to other websites, especially one at Cambridge University.
    (1145-15-150)

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  • The following 2 presentations appear in the single video “Palais PengelleyJMM2019” (below).
  • 3:45 p.m.
    Computational advantages and historical insights from viewing quaternionic interpolation of three-dimensional rotations as geodesic vector interpolation on $S^2$
    Bob Palais*, Utah Valley University
    (1145-22-2929)   0:00 to 25:00 (in video)
  • 4:15 p.m.
    How can symmetries of a rectangle, tethered up to homotopy, provide a physical model for the quaternion group? Generalizations?
    William A. Bogley, Oregon State University
    David Pengelley*, Oregon State University
    (1145-55-2199)   30:30 to 54:00 (in video)

• Abstracts of the two presentations appearing in the above video.

Presentation 1 (from 0:00 to 25:00 in video): Abstract (Palais): Computational advantages and historical insights from viewing quaternionic interpolation of three-dimensional rotations as geodesic vector interpolation on S^2. We compare three methods for interpolating two three-dimensional rotations: Directly in SO(3) matrix form; Using the Euler transform from S^3 to SO(3); Using the analog of ordinary vector interpolation for directed geodesic arcs on S^2. We also compare the spherical triangles used by Euler and Rodrigues to locate an axis for general and composed rotations, and note an interesting consequence of the spherical triangles of Harriot and Girard. From 0:00 to 25:00 in video.

Presentation 2 (from 30:30 to 54:00 in video): Abstract (Pengelley): How can symmetries of a rectangle, tethered up to homotopy, provide a physical model for the quaternion group? generalizations? Preliminary report. The 8-element quaternion group can be represented by rectangle symmetries up to tethered homotopy. But tethered how? Via a strip, or strings, which may realize different groups? And can this be generalized, e.g., to a tethered tetrahedron, icosahedron, or other objects? What groups arise? From 30:30 to 54:00 in video.

• 4:45 p.m.
  Visualization and separation of chromatic information in natural and medical images based on a quaternion algebra framework.
  Nektarios A Valous*, National Center for Tumor Diseases (NCT)
  Rodrigo Rojas Moraleda, National Center for Tumor Diseases (NCT)
  Neha Pandey, Interdisciplinary Center for Scientific Computing (IWR)
  Dirk Jaeger, National Center for Tumor Diseases (NCT)
  Inka Zoernig, National Center for Tumor Diseases (NCT)
  Niels Halama, National Center for Tumor Diseases (NCT)
  
  Video not available.
  
  Nektarios A Valous
  Abstract. Visualization and separation of chromatic information in natural and medical images based on a quaternion algebra framework. Colors can be represented by vectors constructed by a linear combination of three primaries; in the perceptually nonuniform RGB color space the basic element i is chosen to describe red, j green, and k blue. Consequently, color pixels can be encoded by a linear combination of the three basis vectors in a hypercomplex algebra framework, e.g. quaternions. This encoding provides the opportunity to process color images in a geometric way, hence the quaternionic representation of color allows image analysis to be performed in a coherent manner. By conveniently rewriting the quaternionic representation of natural and medical images with simple algebraic operations, it is feasible to decompose an image into different spectral representations that can visualize and separate the contextual chromatic information. This pixel-based approach is computationally efficient thus taking advantage of parallel architectures in computing systems. The benefits of the proposed approach for medical images can be translated in two components: i) as a means for optimized manual assessment by clinicians (color visualization), and ii) as a key step of digitally separating chromatic regions of interest for further quantification in automated processing pipelines (color separation).(1145-68-30)

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  5:15 p.m.
  Split-quaternion representation of a functional hierarchy of a biologic system.
  Garri Davydyan*, Russian Medical Academy of Postgraduate Education
  
  Abstract. Preliminary report. Previously it was proposed that three regulatory patterns (negative feedback, positive feedback and reciprocal links) determine a functional cor of biologic systems. As a math structure each pattern is represented by a second order matrix over R, M(2,R). Evolution of biologic systems occurs through the formation of more complex, organized in hierarchy, steady functional structures. It is assumed that R, C, H entries on M(2,*) module represent a sequence of hierarchical levels obtained by a functional splitting of characters during biologic development.
  (1145-16-250)

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  5:45 p.m.
  Quaternion-valued Solutions to the Korteweg-deVries Equation.
  John Cobb, College of Charleston
  Alex Kasman*, College of Charleston
  Albert Serna, College of Charleston
  Monique Sparkman, College of Charleston
  
  Abstract. Preliminary report. Soliton equations are nonlinear PDEs whose solutions have surprising particle-like behavior and which can be solved exactly using algebro-geometric methods. Because there has been recent interest in non-commutative generalizations, we have undertaken a study of quaternion-valued solutions to the KdV equation as a student research project. The solutions produced by our methods include rational solutions, periodic solutions and breather soliton solutions. Among the theorems we prove about them is one in which the surprising effect of non-commutativity on the phase shift (essentially, the “bounce” when two solitons collide) is made apparent. The talk will end with a list of open questions that we hope the experts at this session will be able to help us to answer
  (1145-35-61)

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  End of Session.

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