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MATH 5411 3.00AF (Fall 2016)

*Analysis for Teachers
*

Problem Sets

Due Dec. 12

Solutions.
### Math Project

You should find a topic that interests you, and run it by me. I can help
turn a broad and fuzzy topic into something precise for you to focus on, or
I can simply suggest references. Basically you should learn something
mathematical that you didn't know before (and which relates in some way
to analysis), and then share it with the class.
The presentation should be 20 minutes, and should focus on what the topic is,
and why you think it is useful or interesting. You can leave details out of
the presentation that you put into the project. I want your projects to
convince me that you really did learn some mathematics, at more than just
a superficial level. Reports should be roughly 8-10 pages.

Some ideas for topics (you don't have to pick one of these, and some would
need paring down before they're suitable topics anyway)

- The Gibbs phenomenon in Fourier Analysis
- Wavelets
- Bezout curves in computer graphics
- Splines in computer graphics
- Quadrature methods in integration
- Frechet derivatives
- Convexity and optimization
- Calculus of variations
- Frenet frames and the geometry of curves
- Mandelbrot set (in more detail than we did)
- Arithmetic-Geometric means
- Brownian motion and stochastic calculus
- Gaussian curvature of surfaces
- Minimal surfaces and soap films
- Bernoulli numbers (eg proving the formulas I just stated in class)
- Pade approximation of reals by rationals
- p-adic numbers and fractals
- Harmonic functions and Laplace's equation
- PDE's and characteristics
- Infinite series for pi
- Ramanujan and applying analysis to count partitions
- The gamma function
- Bessel functions
- Buffon needle problem
- Hyperbolic sines and cosines
- Heine Borel theorem
- Bolzano Weierstrass theorem
- Bernstein's proof of the Weierstrass approximation theorem
- Ascoli-Arzela theorem
- Hilbert space
- Cauchy's theorem (in complex analysis)
- Contour integration
- Simulated annealing
- Banach Tarski paradox
- Space filling curves
- Fixed point theorems

### Pedagogy Project

Your first project will involve choosing a topic related to analysis, and
indicating how it could be incorporated into your teaching practice. This
could be as a classroom topic, an enrichment activity, or something related.
You'll give a short (10-15 minute) presentation in class (tentatively on
October 24) giving us the general idea, and should hand in a report (5-10
pages) about it, tentatively by October 31.
The topic could draw on something from class, like fractals, chaos,
iteration, inequalities, etc.
Since calculus and functions are mostly part of analysis,
it could also be a calculus topic that goes beyond the standard material
in the curriculum.

I'd like you all to try to e-mail me a possible topic, before class on
September 26, so that I can confirm that your proposal is on the right
track. If you need help finding a suitable topic, we can discuss it by
e-mail or by phone. Or if necessary, before class on September 26.
By the end of class that day, I'd like everyone to have their topic
approved, so you can start work.

Due in class, Monday Sept. 26.
Due in class, Monday Sept. 19.

Solutions.
Sample R program:
Sierpinski.R