A View from the Top: Analysis, Combinatorics and Number Theory

Contents: Foreword. 1. The Cauchy-Schwarz inequality. 2. Projections in R[sup(3)]the elephant makes an appearance! 3. Projections in four dimensions. 4. Projections and Cubes. 5. Incidences and matrices. 6. Basics of grids over finite fields. 7. Besicovitch-Kakeya conjecture in two dimensions. 8. A gentle entry into higher dimensions. 9. Some basic counting, probability and a few twists. 10. A more involved taste of probability. 11. Oscillatory integrals and fun that lies beyond. 12. Integer points and a crash course on Fourier analysis. 13. Return of the Fourier transform. 14. It is time to say goodbye. Bibliography. This book is based on a capstone course that the author taught to upper division undergraduate students with the goal to explain and visualize the connections between different areas of mathematics and the way different subject matters flow from one another. In teaching his readers a variety of problem solving techniques as well, the author succeeds in enhancing the readers hands-on knowledge of mathematics and provides glimpses into the world of research and discovery. The connections between different techniques and areas of mathematics are emphasized throughout and constitute one of the most important lessons this book attempts to impart. This book is interesting and accessible to anyone with a basic knowledge of high school mathematics and a curiosity about research mathematics. The author is a professor at the University of Missouri and has maintained a keen interest in teaching at different levels since his undergraduate days at the University of Chicago. He has run numerous summer programs in mathematics for local high school students and undergraduate students at his university. The author gets much of his research inspiration from his teaching activities and looks forward to exploring this wonderful and rewarding symbiosis for years to come.…