Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


16166.pdf (1008.14 KB)

ETD Abstract Container

Abstract Header

Discrete Approximations of Metric Measure Spaces with Controlled Geometry

Lopez, Marcos D
http://orcid.org/0000-0002-3977-4914
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529

Abstract Details

2015, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
The goal of this thesis is to study metric measure spaces equipped with a doubling measure and supporting a Poincare inequality, in terms of graph approximations. We show that a metric measure space is equipped with a doubling measure supporting a p-Poincare inequality if and only if it can be approximated in the pointed measured Gromov-Hausdor sense by a sequence of graphs equipped with doubling measures and supporting a discrete graph version of a p-Poincare inequality, with the doubling and Poincare constants uniformly bounded. We also show that given such a metric measure space, the upper gradient energy on the space is comparable to the weak limit of energies on the approximating graphs. This is in contrast to the approach taken by those who study geometric analysis on fractal spaces, where the energy on the fractal is a weak limit of energies on approximating graphs, but this limit energy need not correspond to energies that come from an upper gradient structure on the fractal space.
Nageswari Shanmugalingam, Ph.D. (Committee Chair)
James T. Gill, Ph.D. (Committee Member)
David Herron, Ph.D. (Committee Member)
Carl David Minda, Ph.D. (Committee Member)
Leonid Slavin, Ph.D. (Committee Member)
120 p.
Mathematics
Gromov Hausdorff convergence; doubling condition; Poincare inequality; analysis on metric measure spaces

Recommended Citations

Citations

  • Lopez, M. D. (2015). Discrete Approximations of Metric Measure Spaces with Controlled Geometry [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529

    APA Style (7th edition)

  • Lopez, Marcos. Discrete Approximations of Metric Measure Spaces with Controlled Geometry. 2015. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529.

    MLA Style (8th edition)

  • Lopez, Marcos. "Discrete Approximations of Metric Measure Spaces with Controlled Geometry." Doctoral dissertation, University of Cincinnati, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529

    Chicago Manual of Style (17th edition)

ucin1439305529
565
© 2015, all rights reserved.
This open access ETD is published by University of Cincinnati and OhioLINK.