Hoppis, Jared Edward2019-11-132019-11-132019-12-01http://hdl.handle.net/2097/40238Arne Beurling first studied extremal length, namely the reciprocal of 2−Modulus, in the plane, and then later studied it jointly with Lars Ahlfors. Beurling was interested in extremal length because he wanted a conformal invariant to study harmonic measure. One of the differences between R² and R[superscript N] with N ≥ 3, is that there are far fewer conformal maps in the latter case. This naturally suggests defining a larger class of functions that distort N-Modulus by a bounded amount. This gives rise to the notion of quasiconformal mappings, see [1]. There have been many recent developments in the discrete theory of p-Modulus, and a natural question is “Can the discrete theory tell us anything about the continuous theory?” There are two ways to try and answer this question. The first is to approximate a domain with a mesh of points and study if discrete p-Modulus of families of walks on the mesh converges to continuous p-Modulus on the domain. This line of inquiry has been pursued in the recent literature [10, 11, 21, 25, 12]. The second way to answer the question is to try to come up with a dictionary of results by developing a way to pair up results for the discrete theory and the continuous theory. This is where this thesis is developed. In [6], with Nathan Albin, Pietro Poggi-Corradini, and Nageswari Shanmugalingam, we establish a relationship between ∞-Modulus of a family of paths connecting two points in general metric spaces and the “essential” shortest path metric between two points. This result is inspired by a similar relationship in the discrete setting established in [5]. In [4] N. Albin, Jason Clemens, Nethali Fernando, and P. Poggi-Corradini show that p-Modulus, 1 ≤ p < ∞, can be related to other metrics. Using the work of Aikawa and Ohtsuka, we show that a similar modulus metric can be defined, with some slight modification, in R2, for 2 < p < ∞. Note that, in the continuous setting, with N-dimensional Lebesgue measure, we cannot hope to get a metric for 1 ≤ p ≤ N because for these values the p-Modulus of the family of curves connecting two distinct points is zero. We are currently working to adapt the argument to dimension N ≥ 3 and in metric measure spaces (X,d,µ) where µ is a Borel regular measure. en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/ModulusReciprocalityFulkersonDualityAikawaOhtsukaDissertation