Following in the spirit of Greg Knese's excellent talk from last quarter, I show how
some concepts in complex analysis (three lines theorem and entire
functions) were used to prove a (sharp) basic inequality in classical
Fourier analysis (Hausdorff-Young). I also show how a result of mine from 1974 in operator theory leads to that same inequality, as well as some new (at the time) inequalities on some noncommutative groups including the two dimensional ax+b group. This motivates an introduction to abstract harmonic analysis, which I survey in anticipation of future talks I am planning to give on some 21st century applications of this noncommutative theory to engineering (e.g. robotics, wavelet transforms).

Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle
materials by the injection of a pressurized viscous fluid. In this talk I provide examples of natural HF and situations in which HF are used in industrial problems. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO2 sequestration and the enhancement of fracture networks to capture geothermal energy.
I describe the governing equations in 1-2D as well as 2-3D models of HF, which involve a coupled system of degenerate nonlinear integro-partial differential equations as well as a free boundary. I demonstrate, via re-scaling the 1-2D model, how the active physical processes manifest themselves in the HF model and show how a balance between the dominant physical processes leads to special solutions. I discuss the challenges for efficient and robust numerical modeling of the 2-3D HF problem and some techniques recently developed to resolve these problems: including robust iterative techniques to solve the extremely stiff coupled equations and a novel Implicit Level Set Algorithm (ILSA) to resolve the free boundary problem. The efficacy of these techniques is demonstrated with numerical results.
Relevant papers can be found at: http://www.math.ubc.ca/~peirce