Mentor: Professor Leslie Cheng
This summer I am researching harmonic analysis with an emphasis on oscillatory integral operators with Professor Leslie Cheng as my advisor. My research is primarily to teach myself the background that I will need for writing my M.A. thesis on harmonic analysis.
I will start by reading selected chapters from A Panorama of Harmonic Analysis by Steven G. Krantz. The book begins with a quick overview of measure theory and functional analysis, subjects usually only taught in graduate level courses. Then Krantz dedicates several chapters to developing the ideas of Fourier analysis, a powerful tool in harmonic analysis, which is the analysis of a function by decomposing it into elementary sine and cosine curves. Fourier analysis is used in applied differential equations, mathematical physics, engineering, and pure mathematics. The remaining chapters discuss integral operators, which is what my research will involve, and wavelets.
After reading the necessary chapters from A Panorama of Harmonic Analysis, I will read a thesis of a Bryn Mawr alum, Kiki Kemp, whose thesis is entitled Lp Estimates for Oscillatory Integral Operators. My M.A. thesis will either expand on that of Kemp’s thesis, or will be on a related subject.
If there is time after learning the background of harmonic analysis, I will work on a book project on harmonic analysis, which is an introductory book that Professor Cheng and Professor Rhonda Hughes are working on to make harmonic analysis and wavelets more accessible to undergraduate students
This summer research opportunity will not only prepare me for writing my thesis, but it will also give me an idea of what an independent research experience in pure math is like, which will prepare me for graduate school.