Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach
نویسنده
چکیده
By “parallelogram geometry” we mean the elementary, “commutative”, geometry corresponding to vector addition, and by “trapezoid geometry” a certain “noncommutative deformation” of the former. This text presents an elementary approach via exercises using dynamical software, hopefully accessible to a wide mathematical audience, from undergraduate students and high school teachers to researchers, proceeding in three steps: (1) experimental geometry, (2) algebra (linear algebra and elementary group theory), and (3) axiomatic geometry. Introduction. Sometimes, fundamental research leads to elementary results that can easily be explained to a wide audience of non-specialists, and which already bear the germ of the much more sophisticated mathematics lying at the background. In the present work I will try to show that our joint paper with M. Kinyon [BeKi12] is an example of this situation, and that even in such classical domains as plane geometry of points and lines, something new can be said and, maybe, taught. In a nutshell, everything is contained in the following figure:
منابع مشابه
On Commutative Reduced Baer Rings
It is shown that a commutative reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.
متن کاملOn the commuting graph of non-commutative rings of order $p^nq$
Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...
متن کاملParticle Physics Models, Grand Unification and Gravity in Non-Commutative Geometry
We review the construction of particle physics models in the framework of noncommutative geometry. We first give simple examples, and then progress to outline the Connes-Lott construction of the standard Weinberg-Salam model and our construction of the SO(10) model. We then discuss the analogue of the Einstein-Hilbert action and gravitational matter couplings. Finally we speculate on some exper...
متن کاملTopological expansion of the Bethe ansatz, and non-commutative algebraic geometry
In this article, we define a non-commutative deformation of the ”symplectic invariants” (introduced in [13]) of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantit...
متن کاملSupersymmetric Quantum Theory and Non-Commutative Geometry
Classical differential geometry can be encoded in spectral data, such as Connes’ spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes’ non-commutative spin geometry encompassing noncommutative Riemannian, symplectic, complex-Hermitian and (Hyper-) Kähler geometry...
متن کامل