June Lester - Curriculum Vitae - Home

Mathematical Publications

Note: some of the papers listed below appear in more than one section. There are 39 distinct papers. Follow the links to see papers on

 

Metric Vector Spaces

A metric vector space is a vector space which has a (usually indefinite) scalar product. I first became fascinated with these spaces as a beginning master's student. Geometrically interesting in their own right (as Euclidean n-space or Minkowski spacetime, for example), they are also invaluable as coordinate spaces: it's quite extraordinary just how many classical geometries can be coordinatized by n-tuples subject to some indefinite scalar product. And looking at these geometries through their coordinate spaces often makes obvious the isomorphisms between different models of the same geometry, or even between different geometries: the same coordinate space implies the same or related geometries.

On Null-Cone Preserving Mappings. (with M. A. McKiernan) Math. Proc. Camb. Phil. Soc. 81 (1977) 455 - 462

Cone Preserving Mappings for Quadratic Cones over Arbitrary Fields. Canad. J. Math. 29 (1977) 1247 - 1253

Some Properties of Non-Positive Definite Real Metric Vector Spaces. Utilitas Math. 12 (1977) 327 - 333

A Characterization of Non-Euclidean, Non-Minkowskian Inner Product Space Isometries. Utilitas Math. 16 (1979) 101 - 109

Transformations of n-Space Which Preserve a Fixed Square-Distance. Canad. J. Math. 31 (1979) 392 - 395

Conformal Spaces. J. Geom. 14 (1980) 108 - 117

Transformations Preserving Null Line Sections of a Domain: the Arbitrary Signature Case. Resultate Math. 9 (1986) 107 - 118

A Metric Vector Space Proof of Miquel's Theorem. C. R. Math. Rep. Acad. Sci. Canada 9 (1987) 59 - 62

The Octahedron Theorem in Minkowski Three-Space: A Metric Vector Space Proof of Miquel's Theorem in the Laguerre Plane. J. Geom. 30 (1987) 196 - 202
Back to the top of this page

 

Geometric Characterization Problems

The basic characterization problem: determine those transformations of a geometric space which preserve some particular feature or measurement of the space, without recourse to regularity assumptions (linearity, continuity, ... ) or even without assuming bijectivity. The classical example is the Beckman-Quarles theorem: functions from the Euclidean plane to itself preserving pairs of points a distance 1 apart must be Euclidean motions (the functions need not be assumed bijective, or even single valued).

There are characterization theorems for many other spaces and spacetimes: for a survey, see a book chapter I wrote:

Distance Preserving Transformations. Chapter 16 of Handbook of Incidence Geometry, p. 921 - 944, ed. F. Buekenhout, Elsevier Science B. V., 1995.

On Null-Cone Preserving Mappings. (with M. A. McKiernan) Math. Proc. Camb. Phil. Soc. 81 (1977) 455 - 462

Cone Preserving Mappings for Quadratic Cones over Arbitrary Fields. Canad. J. Math. 29 (1977) 1247 - 1253

A Characterization of Non-Euclidean, Non-Minkowskian Inner Product Space Isometries. Utilitas Math. 16 (1979) 101 - 109

Transformations of n-Space Which Preserve a Fixed Square-Distance. Canad. J. Math. 31 (1979) 392 - 395

On Distance Preserving Transformations of Lines in Euclidean Three-Space. Aequationes Math. 28 (1985) 69 - 72

Euclidean Plane Point Transformations Preserving Unit Area or Unit Perimeter. Archiv Math. (Basel) 45 (1985) 561 - 564

Transformations Preserving Null Line Sections of a Domain: the Arbitrary Signature Case. Resultate Math. 9 (1986) 107 - 118

A Characterization of Motions as Bijections Preserving Circumradius or Inradius 1. Monatsh. Math. 101 (1986) 151 - 158

Martin's Theorem for Euclidean n-Space and a Generalization to the Perimeter Case. J. Geom. 27 (1986) 29 - 35

Orthogonal Spheres. C. R. Math. Rep. Acad. Sci. Canada. 8 (1986) 231 - 235

On Line Mappings Which Preserve Unit Triangles. Utilitas Math. 31 (1987) 81 - 84

Angle-preserving Transformations of Spheres. Aequationes Math. 32 (1987) 52 - 57

A Beckman-Quarles-Type Theorem for Coxeter's Inversive Distance. Canad. Math. Bull. 34 (1991) 492 - 498

Many of my characterization theorems are also discussed in two books by Walter Benz:

Geometrische Transformationen unter besonderer Berücksichtigung der Lorentztransformationen. B.I. Wissenschaftsverlag, Mannheim 1992

Real Geometries. B.I. Wissenschaftsverlag, Mannheim 1994

Back to the top of this page

 

Spacetime Geometry

Both of the previous topics come together here: the simplest spacetime, Minkowski spacetime is an example of a metric vector space, while Alexandrov's theorem characterizes its transformations (Lorentz transformations) by the fact that they preserve pairs of points connected by light signals. Another example: Zeeman's theorem characterizes the causality-preserving transformations of Minkowski spacetime, for example. I've done generalizations of these and other theorems for several other spacetimes.

In another, disjoint spacetime direction, my favourite paper Does Matter Matter? uses some ideas from classical inversive geometry to construct a spacetime model in which the location of infinity is relative (i.e. observer-dependent). Under some very natural assumptions about proper time, the model predicts cosmological redshifts with unexpectedly realistic properties. It also predicts an age for the universe of about 25 billion years (comfortably more than the stars in it, unlike the situation in more classical theories).

The Beckman-Quarles Theorem in Minkowski Space for a Spacelike Square-Distance. Arch. Math. (Basel) 37 (1981) 561 - 568 [summary in C. R. Math. Rep. Acad. Sci. Canada 3 (1981) 59 - 61]

Alexandrov-type Transformations on Einstein's Cylinder Universe. C. R. Math.Rep. Acad. Sci. Canada 4 (1982) 175 - 178

Transformations of Robertson-Walker Spacetimes Preserving Separation Zero. Aequationes Math. 25 (1982) 216 - 232

A Physical Characterization of Conformal Transformations of Minkowski Spacetime. Ann. Discrete Math. 18 (1983) 567 - 574

Conformal Minkowski Spacetime I: Relative Infinity and Proper Time. Il Nuovo Cimento 72B (1982) 261 - 272

Conformal Minkowski Spacetime II: A Cosmological Model. Il Nuovo Cimento 73B (1983) 139 - 149

Separation Preserving Transformations of de Sitter Spacetime. Abh. Math. Sem. Univ. Hamburg 53 (1983) 217 - 224

The Causal Automorphisms of de Sitter and Einstein Cylinder Spacetimes. J. Math. Phys. 25 (1984) 113 - 116

Relative Infinity in Projective de Sitter Spacetime and its Relation to Proper Time. Ann. Discrete Math. 37 (1988) 257 - 264

Zeeman's Lemma on Robertson-Walker Spacetimes. J. Math. Phys. 30 (1989) 1296 - 1300

The Effect of a Relative Infinity on Cosmological Redshifts. Astrophysics and Space Science 207 (1993) 231 - 248

Does Matter Matter? Physics Essays 11 (1998) 481 - 491
Back to the top of this page

 

Complex Triangle and Polygon Geometry

This topic began as a minor recreational problem and expanded into a major research project. Basically what I've done is to develop a rather productive complex cross ratio formalism for triangle geometry. First, I use a single complex number, called shape, to describe Euclidean triangles and prove theorems about similar triangles. Second, I use another complex number to coordinatize the plane relative to a given triangle and to prove theorems about it. Third, I relate the two: the coordinate of any special point of a triangle is a corresponding function of its shape. This function can be used to discover and prove theorems about special points by reducing the proofs to complex algebra.

Triangles I: Shape. Aequationes Math. 52 (1996) 30 - 54

Triangles II: Complex Triangle Coordinates. Aequationes Math.52 (1996) 215 - 245

Triangles III: Complex Triangle Functions. Aequationes Math. 53 (1997) 4 - 35

A generalization of Napoleon's Theorem to n-gons. C. R. Math. Soc. Canada 16 (1994) 253 - 257

My triangles work has been extended to other planes: see for example Shapes of Polygons, R. Artzy, J. Geom. 50 (1994) 11 - 15 and Shape-Regular Polygons in Finite Planes, R. Artzy and G. Kiss, J. Geom. 57 (1996) 20 - 26.

The discover of the circle that has come to be known as the Lester circle was one of many theorems in Triangles III. For details of this theorem and a bibliography of papers it has inspired, please see the Lester Circle website.
Back to the top of this page

 

 Miscellaneous Topics

This includes some pre-Ph.D work on functional equations and various other work.

A Canonical Form for a System of Quadratic Functional Equations. Colloq. Math. 35 (1976) 105 - 108

The Solution of a System of Quadratic Functional Equations. Ann. Polon. Math. 37 (1980) 113 - 117

Points of Difference: Relative Infinity in the Euclidean Plane. J. Geom. 46 (1993) 92 - 118

Worlds of Difference: Relative Infinity in the Hyperbolic Plane. Mitt. Math. Ges. Hamburg 13 (1993) 93 - 117

Orthochronous Subgroups of O(p, q). Linear and Mulitlinear Algebra 36 (1993) 111 - 113
Back to the top of this page