Richard Evan Schwartz
Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton.
Thu, 02/24/2011 - 16:01 — SomNambulAuthors: Richard Evan Schwartz
Subjects: Dynamical Systems
AbstractIn this long paper we give a fairly complete analysis of outer billiards on
the Penrose kite. Our analysis reveals that this 2-dimensional non-compact
system has a 3-dimensional compactification, a certain polyhedron exchange map,
and that this compactification has a renormalization scheme. These two features
allow us to make some sharp statements concerning the distribution, large-scale
geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits.
For instance, one of our results is that the union of the unbounded orbits has
Hausdorff dimension 1.The Pentagram Integrals on Inscribed Polygons.
Tue, 04/27/2010 - 15:01 — SomNambulAuthors: Richard Evan Schwartz, Serge Tabachnikov
Subjects: Combinatorics
AbstractThe pentagram map is a natural iteration on projective equivalence classes of
(twisted) n-gons in the projective plane. It was recently proved ([OST]) that
the pentagram map is completely integrable, with the complete set of Poisson
commuting integrals given by the polynomials O1,...,O[n/2],On and
E1,...,E[n/2],En, previously constructed in [S3]. These polynomials are
somewhat reminiscent of the symmetric polynomials. It was observed in computer
experiments that if a polygon is inscribed into a conic then Oi=Ei for all i.
The goal of the paper is to prove this theorem.The 5 Electron Case of Thomson's Problem.
Fri, 01/22/2010 - 16:01 — SomNambulAuthors: Richard Evan Schwartz
Subjects: Metric Geometry
AbstractWe give a rigorous computer-assisted proof that the triangular bi-pyramid is
the unique configuration of 5 points on the 2-sphere that globally minimizes
the Coulomb (1/r) potential. We also prove the same result for the (1/r^2)
potential. The main mathematical contribution of the paper is a fairly
efficient energy estimate that works for any number of points and any power-law
potential.Elementary Surprises in Projective Geometry.
Tue, 10/13/2009 - 18:46 — SomNambulAuthors: Richard Evan Schwartz, Serge Tabachnikov
Subjects: Differential Geometry
AbstractWe discuss eight new(?) configuration theorems of classical projective
geometry in the spirit of the Pappus and Pascal theorems.Elementary Surprises in Projective Geometry.
Tue, 10/13/2009 - 18:46 — SomNambulAuthors: Richard Evan Schwartz, Serge Tabachnikov
Subjects: Differential Geometry
AbstractWe discuss eight new(?) configuration theorems of classical projective
geometry in the spirit of the Pappus and Pascal theorems.
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