Park City Mathematics Institute
Research Course-Harmonic Analysis
Project Abstract
Research Course-Harmonic Analysis
Authors: Brian Hopkins, Stephen Badgley, Westley Knight, Katherine Maschmeyer, Melvin Peralta, Mark Walth
This teacher group participated in the Undergraduate Summer School course "Introduction to Harmonic Analysis" led by Ricardo Sáenz of the University of Colima (Mexico). The group attended the 1pm lecture and then spent an hour debriefing and working on problem sets. The first class laid out the plan for the three weeks: background and properties of harmonic functions, Hilbert transforms, the Dirichlet principle, and heat flow on fractals. Growing from Newton's work on heat diffusion, harmonic functions satisfy a second order partial differential equation and a given boundary condition. Until fractals, domains (in arbitrary dimensional real space) were the ball (spherical harmonics) and the upper half-space (Poisson kernel). There were many tools for creating "nicer" functions and various "relaxed" definitions (e.g., weak derivatives, weak continuity), two approaches to handling these difficult functions. In addition to the stated prerequisites multivariable calculus, linear algebra, and some real analysis, we used topics from complex analysis, topology, and measure theory. Historically, the material spanned from Newton to Hardy & Littlewood and Calderón & Zygmund to the recent work on fractals.
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