232 research outputs found
Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
We introduce some algebraic geometric models in cosmology related to the
"boundaries" of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers
between aeons. We suggest to model the kinematics of Big Bang using the
algebraic geometric (or analytic) blow up of a point . This creates a
boundary which consists of the projective space of tangent directions to
and possibly of the light cone of . We argue that time on the boundary
undergoes the Wick rotation and becomes purely imaginary. The Mixmaster
(Bianchi IX) model of the early history of the universe is neatly explained in
this picture by postulating that the reverse Wick rotation follows a hyperbolic
geodesic connecting imaginary time axis to the real one. Penrose's idea to see
the Big Bang as a sign of crossover from "the end of previous aeon" of the
expanding and cooling Universe to the "beginning of the next aeon" is
interpreted as an identification of a natural boundary of Minkowski space at
infinity with the Big Bang boundary
Unimodular Hausdorff and Minkowski Dimensions
This work introduces two new notions of dimension, namely the unimodular
Minkowski and Hausdorff dimensions, which are inspired from the classical
analogous notions. These dimensions are defined for unimodular discrete spaces,
introduced in this work, which provide a common generalization to stationary
point processes under their Palm version and unimodular random rooted graphs.
The use of unimodularity in the definitions of dimension is novel. Also, a
toolbox of results is presented for the analysis of these dimensions. In
particular, analogues of Billingsley's lemma and Frostman's lemma are
presented. These last lemmas are instrumental in deriving upper bounds on
dimensions, whereas lower bounds are obtained from specific coverings. The
notions of unimodular Hausdorff size, which is a discrete analogue of the
Hausdorff measure, and unimodular dimension function are also introduced. This
toolbox allows one to connect the unimodular dimensions to other notions such
as volume growth rate, discrete dimension and scaling limits. It is also used
to analyze the dimensions of a set of examples pertaining to point processes,
branching processes, random graphs, random walks, and self-similar discrete
random spaces. Further results of independent interest are also presented, like
a version of the max-flow min-cut theorem for unimodular one-ended trees and a
weak form of pointwise ergodic theorems for all unimodular discrete spaces.Comment: 89 pages, 1 figure. This version of the paper is a merging of the
previous version with arXiv:1808.02551. Earlier versions of this paper were
titled `On the Dimension of Unimodular Discrete Spaces, Part I: Definitions
and Basic Properties
Limit theory of combinatorial optimization for random geometric graphs
In the random geometric graph , vertices are placed randomly in
Euclidean -space and edges are added between any pair of vertices distant at
most from each other. We establish strong laws of large numbers (LLNs)
for a large class of graph parameters, evaluated for in the
thermodynamic limit with const., and also in the dense limit with , . Examples include domination number,
independence number, clique-covering number, eternal domination number and
triangle packing number. The general theory is based on certain subadditivity
and superadditivity properties, and also yields LLNs for other functionals such
as the minimum weight for the travelling salesman, spanning tree, matching,
bipartite matching and bipartite travelling salesman problems, for a general
class of weight functions with at most polynomial growth of order
, under thermodynamic scaling of the distance parameter.Comment: 64 page
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