Artículos con la etiqueta ‘Geometric Topology (math.GT)’

Generalizations of the Kolmogorov-Barzdin embedding estimates

Por • 11 feb, 2014 • Category: Ciencia y tecnología

We consider several ways to measure the `geometric complexity’ of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness’, based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.

GLC actors, artificial chemical connectomes, topological issues and knots

Por • 25 dic, 2013 • Category: Educacion

Based on graphic lambda calculus, we propose a program for a new model of asynchronous distributed computing, inspired from Hewitt Actor Model, as well as several investigation paths, concerning how one may graft lambda calculus and knot diagrammatics.

On Hilbert’s fourth problem

Por • 13 dic, 2013 • Category: Educacion

Hilbert’s fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired from an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert’s problem, especially those that arose from Busemann’s work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry.

The Euler characteristic of an even-dimensional graph

Por • 21 jul, 2013 • Category: Crítica

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so be rewritten as an average 1-E[K(x,f)]/2 over a collection two dimensional “sectional graph curvatures” K(x,f) through x. Since scalar curvature, the average of all these two dimensional curvatures through a point, is the integrand of the Hilbert action, the integer 2-2 X(G) becomes an integral-geometrically defined Hilbert action functional.

Updates on Hirzebruch’s 1954 Problem List

Por • 23 may, 2013 • Category: Educacion

We present updates to the problems on Hirzebruch’s 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about contact structures, and about (complementary pairs of) foliations.

Ahlfors circle maps: historical ramblings

Por • 24 nov, 2012 • Category: Ciencia y tecnología

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer’s translation of “Kreisabbildung”), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein’s ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler’s planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann’s Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\”uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem.

Graphic lambda calculus and knot diagrams

Por • 12 nov, 2012 • Category: Educacion

In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.