Topics in Analysis

One of the most prevalent and interesting modules in my second year was the study of Differential Equations. During this module, we considered Partial Differential Equations; in particular, the method of Separation of Variables and the use of Fourier Series. Even though the methods introduced were complete, no statement of convergence of such series was given. Now, during my final year, I have started to study such series in a Fourier Analysis course and matters of convergence are now being considered, which is satisfying this need for rigour that I desired two years ago. Instinctively to further investigate such topics, I then began to study the text ``An introduction to Harmonic Analysis'' by Y.Katznelson. Even though this was a challenging read, the methods introduced were elegant, well explained and diverse, further expanding my knowledge on the subject, whilst still intriguing me further. The ideas introduced in this book continue to fascinate me to a level beyond the scope of the Fourier Analysis course. The structure and supervision provided to me during my masters project has given me the confidence and motivation to pursue these concepts further. With the guidance of the University of Birmingham, I know I am ready to explore this dynamic area of mathematics, which has interested me throughout the entirety of my undergraduate degree. I will deliver an excellent thesis, which will develop the rich theory needed to help model many essential natural phenomena.