Abstract: Kaushiki Dunusinghe

Mentor: Dr. Cheng

My summer research will be in the eld of harmonic analysis. Har- monic analysis is an area of mathematics which studies the representation of signals or functions as the superposition of basic waves. Basic waves are called harmonics", and hence the name harmonic analysis".

The mathematical study of harmonic analysis is concerned with examining mathematical objects by decomposing them into many simple building blocks. In doing this, the problem of studying a complex mathematical object is reduced to into studying many elementary components which are easier to understand. For instance, if we have a function (a musical sound, for example) that can be decomposed into a sum of pure harmonics cosine and sine, then we can study these relatively simple curves to get a better understanding of our original functions behavior. The combination of innovative numerical tools for signal processing and rich analytical tools derived from harmonic analysis have been applied in areas such as acoustics, optics, electrical engineering, signal image processing, and data compression.

In particular, my research will focus on solving partial dierential equations (PDEs) using tools from harmonic analysis, such as Fourier series and Fourier transformations. A PDE is an equation that involves more than one independent variable x; y; : : :, a dependent variable u, and partial derivatives of the dependent variable u.These equations arise in many physical applications of Harmonic analysis. As a part of my research project, I will also be learning how to use LATEX, a high-quality typesetting system especially designed for the production of technical and scientific documentation. This program allows authors to concentrate on the content of the work without worrying about the appearance of their document. LATEX will allow me to document all my research work in a presentable manner and I will be able to use it in my future works in the eld of mathematics as well.