25 research outputs found
The combinatorial Mandelbrot set as the quotient of the space of geolaminations
We interpret the combinatorial Mandelbrot set in terms of \it{quadratic
laminations} (equivalence relations on the unit circle invariant under
). To each lamination we associate a particular {\em geolamination}
(the collection of points of the circle and edges of convex
hulls of -equivalence classes) so that the closure of the set of all of
them is a compact metric space with the Hausdorff metric. Two such
geolaminations are said to be {\em minor equivalent} if their {\em minors}
(images of their longest chords) intersect. We show that the corresponding
quotient space of this topological space is homeomorphic to the boundary of the
combinatorial Mandelbrot set. To each equivalence class of these geolaminations
we associate a unique lamination and its topological polynomial so that this
interpretation can be viewed as a way to endow the space of all quadratic
topological polynomials with a suitable topology.Comment: 28 pages; in the new version a few typos are corrected; to appear in
Contemporary Mathematic
Topological regluing of rational functions
Regluing is a topological operation that helps to construct topological
models for rational functions on the boundaries of certain hyperbolic
components. It also has a holomorphic interpretation, with the flavor of
infinite dimensional Thurston--Teichm\"uller theory. We will discuss a
topological theory of regluing, and trace a direction in which a holomorphic
theory can develop.Comment: 38 page
Schubert calculus and Gelfand-Zetlin polytopes
We describe a new approach to the Schubert calculus on complete flag
varieties using the volume polynomial associated with Gelfand-Zetlin polytopes.
This approach allows us to compute the intersection products of Schubert cycles
by intersecting faces of a polytope.Comment: 33 pages, 4 figures, introduction rewritten, Section 4 restructured,
typos correcte