Math 50780: SUMR Honors topics

Spring `14


Weekly schedule

Instructor: Jeffrey Diller (click for contact info, general policies, etc.)

Official Time and place: MWF 2-2:50 PM in Hayes-Healy 125

Textbook: Analysis: An Introduction by Richard Beals. Possibly other sources later in the course

What we'll cover: this course will serve as an introduction to Lebesgue measure/integration and to Fourier Analysis. In terms of the textbook, my plan is to cover chapters 10-14 and whatever else time allows me to achieve. One of my goals is to help prepare you for an introductory graduate course in real analysis at Notre Dame or elsewhere.

The Lebesgue integral is a more comprehensive alternative to the Riemann integral. The main difference is that instead of partitioning the domain of a function as one does with Riemann sums, one partitions its range. The effect is that one can integrate many more functions than before, and more importantly, one gets much better convergence theorems for integrals involving sequences of functions.

Fourier series are almost as old as Calculus, having been studied by e.g. Euler. They were taken up (a little later) by Fourier himself to solve physical problems concerning heat. They have been and remain a central topic in analysis. The basic idea is that any decent periodic function should be in some sense expressable as an infinite sum of sines and cosines with arbitrarily high frequencies. Making rigorous sense of this idea is one of the main things that motivated Lebesgue integration to begin with. The applications to e.g. partial differential equations, number theory, geometry, signal processing, electromagnetism, quantum mechanics etc, etc are legion.

How you will be evaluated:

Homework: assigned and collected on Fridays, worth 50% of your final grade. I highly encourage you to work together on homework assignments, but I expect you to write up solutions yourself. No copying allowed! Occasionally I give out extra credit problems. On these, I expect you to work alone. I plan to grade homework myself, ideally at the rate of 3 or 4 problems a week. If I'm not satisfied with your solution to a given problem I might ask you to redo your solution and turn it back to me along with your earlier attempt.

Midterm Exam: Monday 3/17 in class, worth 20% of your final grade. Possibly with a take home portion

Final Exam: Monday 5/5 from 4:15-6:15, location TBA. comprehensive and worth 30% of your final grade. Possibly with a take home portion.