Narrow Results

Starting with the complexes, quaternions and the octonions we study five families of groups and homogeneous spaces each building geometrically on the previous families. For each family a particular theorem is studied from the perspective of the structure developed. The evolution of the spaces and the corresponding results form a sort of mathematical memoir of a career spanning more than 50 years.

We begin with a discussion on two apparently disconnected topics — one related to nonperturbative superpotential generated from wrapping an M2-brane around a supersymmetric three cycle embedded in a G₂-manifold evaluated by the path-integral inside a path-integral approach of Ref. 1, and the other centered around the compact Calabi–Yau CY₃(3, 243) expressed as a blow-up of a degree-24 Fermat hypersurface in WCP⁴[1, 1, 2, 8, 12]. For the former, we compare the results with the ones of Witten on heterotic worldsheet instantons.² The subtopics covered in the latter include an 𝒩=1 triality between Heterotic, M- and F-theories, evaluation of RP²-instanton superpotential, Picard–Fuchs equation for the mirror Landau–Ginzburg model corresponding to CY₃(3, 243), D = 11 supergravity corresponding to M-theory compactified on a "barely" G₂ manifold involving CY₃(3, 243) and a conjecture related to the action of antiholomorphic involution on period integrals. We then shown an indirect connection between the two topics by showing a connection between each one of the two and Witten's MQCD.³ As an aside, we show that in the limit of vanishing "ζ", a complex constant that appears in the Riemann surfaces relevant to defining the boundary conditions for the domain wall in MQCD, the infinite series of Ref. 4 used to represent a suitable embedding of a supersymmetric 3-cycle in a G₂-mannifold, can be summed.

A Dirac fermion is expressed by a four-component spinor, which is a combination of two quaternions and can be treated as an octonion. The octonion possesses the triality symmetry, which defines symmetry of fermion spinors and bosonic vector fields. The triality symmetry relates three sets of spinors and two sets of vectors, which are transformed among themselves via transformations G₂₃, G₁₂, G₁₃, G₁₂₃ and G₁₃₂. If the electromagnetic (EM) interaction is sensitive to the triality symmetry, i.e. EM probe selects one triality sector, EM signals from the five transformed world would not be detected and be treated as the dark matter. According to an astrophysical measurement, the ratio of the dark to ordinary matter in the universe as a whole is almost exactly 5. We expect quarks are insensitive to the triality, and triality will appear as three times larger flavor degrees of freedom in the lattice simulation.

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.

This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

In this paper, triality refers to the $S_3$-symmetry of the language of quasigroups, which is related to, but distinct from, the notion of triality as the $S_3$-symmetry of the Dynkin diagram $D_4$. The paper investigates a homogeneous method for rendering the linearization of quasigroups (over a commutative ring) naturally invariant under the action of the triality group, on the basis of an appropriate algebra generated by three invertible, non-commuting coefficient variables that is isomorphic to the group algebra of the free group on two generators. The algebra has a natural quotient given by setting the square of each generating variable to be $-1$. The quotient is an algebra of quaternions over the underlying ring, in a way reminiscent of how symmetric groups appear as quotients of braid groups on declaring the generators to be involutions. The corresponding quasigroups (which are described as quaternionic) are characterized by three equivalent pairs of quasigroup identities, permuted by the triality symmetry. The three pairs of identities are logically independent of each other. Totally symmetric quasigroups (such as Steiner triple systems) are quaternionic.

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and $p$-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work, we handle all (simply connected) outer forms of $D_4$.

We discuss basic properties (uniqueness and regularity) of viscosity solutions to fully nonlinear elliptic equations of the form F(x, D²u) = 0, which includes also linear elliptic equations of nondivergent form. In the linear case we consider equations with discontinuous coefficients.