Math 477, Topics in Real Analysis
Spring 2007, 8:30–9:30 MWF, 243 Olin-Rice
office hours: MWF: 3:30-5:00 pm, T 1:00-2:30 & 4:00-5:00 pm, Th:
2:00-5:00 pm, make an appointment, or stop by and see if I'm in
``If it be then your Pleasure, ye Lovers of Study, come always; be
not restrained through any Fear, nor retarded by too much Modesty, what
you may do by your Right, you shall make me do willingly, nay gladly and
joyfully. Ask your Questions, make your Enquiries, bid and command; you
shall neither find me averse nor refractory to your Commands, but officious
and obedient. If you meet with any Obstacles or Difficulties, or are retarded
with any Doubts while you are walking in the cumbersome Road of this Study
of Mathematics, I beg you to impart them, and I shall endeavour to remove
every Hindrance out of your Way to the best of my Knowledge and Ability.''
Isaac Barrow, March 14, 1664
Description
This seminar will follow the historical development of the Lebesgue integral,
including the insights into properties of the real numbers that were uncovered
in the second half of the 19th century. This course revolves around several
big questions that were tackled in the late 19th century:
- Is every continuous function of a real variable representable by its
Fourier series? Is there a trigonometric series that is not a Fourier series?
Can a function have more than one representation as a trigonometric series?
- If f is continuous on [0,1], what can be said about its differentiability?
What if we put additional restrictions on f such as monotonicity?
- What do we mean by a "small" subsets of [0,1]? Specifically, nineteenth
century mathematicians worked with sets of type n (first species sets), nowhere
dense sets, and sets of outer measure 0. How are these types of sets related?
How did the understanding of these types of sets develop?
- If a function f is differentiable at every point on the interval
[a,b], is its derivative necessarily Riemann integrable? If not, what definition
of integrability will ensure that the derivative is integrable?
- Given a 2-dimensional set E, when can the integral of f(x,y)
over E be evaluated by calculating iterated integrals?
- When can a series be integrated term-by-term?
Web Site
All assignments for this course and reminders of what is coming due will be
available on a Moodle site that will be set up before classes begin in January.
Grades
- An in-class presentations with a written report. The written report will
be due one week after the in-class presentation. 15%
- Homework due each Monday by 5pm. Late assignments will be penalized 10%
if less than 1 day late, 5% for each additional day late. 40%
- Final paper based on one of the big questions listed above. This paper should
be written for a Math 377 student who wants to know about the most important
results in real analysis in the late 19th or early 20th century. What were
the issues, what difficulties were encountered, and how were they resolved?
It should be a significant paper that clearly explains and developes the ideas.
I expect a typed paper in the range of 15-25 pages. Include sketches of key
proofs and references. Outline is due March 30; first draft is due April 15;
final version is due May 5. 25%
- Take-home mid-term exam. 20%
Texts
A Radical Approach to Lebesgue's Theory of Integration by David Bressoud,
manuscript available at the Macalester bookstore.
Additional Resources
- Companion encyclopedia of the history and philosophy of the mathematical
sciences
- Dictionary of Scientific Biography, in the reference section
of the Library, very extensive biographies
- Mathematical thought from ancient to modern times by Morris
Kline, extensive history of mathematics
- Men of mathematics, by Bell, selected mathematical biographies
- The History of Modern Mathematics, vol 1\/, by Rowe and McCleary,
includes article on Cantor's continuum hypothesis
- From the calculus to set theory, by Grattan-Guinness, essays
on aspects of development of analysis
- Measure and the integral, by Lebesgue, Lebesgue's measure theory
in his own words
- Great moments in mathematics (after 1650), by Eves, includes
articles on development of set theory
- Classics of mathematics, by Calinger, includes translations
of articles by Weierstrass, Cantor, Peano, Poincar\'e, Lebesgue and biographies
of Cantor, Lebesgue, Peano, Poincar\'e, Riemann, and Weierstrass
- A Source book in classical analysis, by Birkhoff, includes translations
of articles by Riemann, Heine, Jordan, Weierstrass, Poincar\'e, Dirichlet,
Lipschitz, Osgood, Peano, du Bois-Reymond, and Volterra
Links
St. Andrews
History of Mathematics Site