JOJohnson
Johnson
*This is an algorithmically generated author record based on publicly available information.
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Johnson has a total of 3 co-authors. Here are the top co-authors by number of shared publications:
- Alexey G. Gorinov
Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia
1 shared publication - George F. McNulty
Department of Mathematics, University of South Carolina, Columbia, USA
1 shared publication - Ralph Howard
Department of Mathematics, University of South Carolina, Columbia, USA
1 shared publication
Latest Publications
- 1 June 2020
- Published by Association for Mathematical Research in Arnold Mathematical Journal
- Vol. 6 (2) , 213-229
Maps That Must Be Affine or Conjugate Affine: A Problem of Vladimir Arnold
A k-flat in a vector space is a k-dimensional affine subspace. Our basic result is that an injection \(T:{{\mathbb {C}}}^n\rightarrow {{\mathbb {C}}}^n\) that for some \(k\in \{1,2,\ldots ,n-1\}\), T maps all k-flats to flats of \({{\mathbb {C}}}^n\) and is either continuous at a point or Lebesgue measurable, is either an affine map or a conjugate-affine map. An analogous result is proven for injections of the complex projective spaces. In the case of continuity at a point, this is generalized in several directions, the main one being that the complex numbers can be replaced by a finite-dimensional division algebra over an Archimedean ordered field. We also prove injective versions of the Fundamental Theorems of affine and projective geometry and give a counter-example to the surjective version of the latter. This extends work of A. G. Gorinov on a problem of V. I. Arnold.- 1 June 2020
- Published by Association for Mathematical Research in Arnold Mathematical Journal
- Vol. 6 (2) , 213-229
Maps That Must Be Affine or Conjugate Affine: A Problem of Vladimir Arnold
A k-flat in a vector space is a k-dimensional affine subspace. Our basic result is that an injection \(T:{{\mathbb {C}}}^n\rightarrow {{\mathbb {C}}}^n\) that for some \(k\in \{1,2,\ldots ,n-1\}\), T maps all k-flats to flats of \({{\mathbb {C}}}^n\) and is either continuous at a point or Lebesgue measurable, is either an affine map or a conjugate-affine map. An analogous result is proven for injections of the complex projective spaces. In the case of continuity at a point, this is generalized in several directions, the main one being that the complex numbers can be replaced by a finite-dimensional division algebra over an Archimedean ordered field. We also prove injective versions of the Fundamental Theorems of affine and projective geometry and give a counter-example to the surjective version of the latter. This extends work of A. G. Gorinov on a problem of V. I. Arnold.