- Project reports are due December 12. I've also posted the third problem set, which is due at the same time. You can either give me a hardcopy early (on Dec 5), or e-mail me your solutions. If your solutions are handwritten, you can e-mail me a scan.
- Possible project topics.
- The course meets Mondays 6-9 in S525 Ross (changed from VC 114)
- There isn't a textbook - we'll pull out readings from multiple sources instead.
- We'll spend as much time as possible exploring topics. Either with pencil and paper or via computers. In particular, you should ensure that you have access to some programming language that you're familiar with. It doesn't matter so much which language though. Mathematica or MatLab or Maple are fine, but can be expensive. SageMath does many of the same things, but is a free download. C++ or Python are often bundled with computers, and if not, can usually be downloaded free. I'm a probabilist, so I like R, which can also be downloaded free. To do that, go to https://www.r-project.org, select the CRAN mirror at the University of Toronto, and download a version that will work on your system.
- Another thing I'd encourage (but don't insist on) is that you explore TeX or LaTeX. Again, these can usually be downloaded free. I use TeXShop on a Mac, but there are other implementations out there. TeX can be used to produce lovely mathematical documents (or presentations, using Beamer). Mathematicians typically express mathematics via e-mail using informal versions of TeX. So knowing a little is really useful. There is a learning curve, but once you're used to it, you'll never go back to equation editors.

- Course outline
- Problem sets
- Sample R program. This now includes two routines, one of which plots all at once, and one which plots one point at a time.

- Sep 12: fractals
- Sep 19: fractals. Also
- 1-d Brownian motion (graph) generated in R
- 2-d Brownian motion (range) generated in R
- Sierpinski carpet (9MB file), generated in R as an iterated function system with 20,000 steps.

- Sep 26: chaos. Also
- Some R code to display iterations and bifurcation: perioddbl.R
- An example of the output: Iteration with c = -1.82
- bifurcation diagram (11.5MB file)

- Oct 3: Complex numbers. Also
- Oct 17: Mobius transformations and hyperbolic geometry. Links:
- Pseudosphere (if you Google
*pseudosphere*you'll find other nice images) - Escher's hyperbolic tilings (plus a bunch of his Euclidean tilings and hyperbolic versions of them).
- The tiling referred to at the end of the notes is hyp019 on the following page
- If you like these images, there are more Escher or math-art images at the official Escher site, the American math society, or the Bridges conference site. Note that the latter will be in Waterloo in 2017.

- Pseudosphere (if you Google
- Oct 24: Pedagogy project presentations
- Oct 31: Julia and Mandelbrot sets, the Millennium problems
- Julia set for Newton's method applied to z^3-1 (from the Wikipedia entry for Julia sets)
- Julia set for a quadratic polynomial
- Movie of how the Julia set for z^2 + c changes, as c moves in a circle of radius .7885 about the origin.
- Mandelbrot set (see Wikipedia entry for more)
- Handout on the Millennium prize problems.

- Nov 7: Analytic number theory:
Bernoulli numbers and the
Riemann hypothesis
- Related Wikipedia entries: Riemann zeta function, Prime counting function

- Nov 14: Inequalities (revised version of the notes handed out in class).
- Nov 21: Approximations and splines, LaTeX and Beamer tutorial
- Notes on polynomial approximation and Fourier series
- LaTeX introduction
- LaTeX samples (downloads): Nov14 notes, calculus exam,
- Beamer sample: TeX file (will download), with pictures removed. Plus the typeset pdf file (with pictures)

- Nov 28: Project presentations
- Dec 5: Project presentations

- We're going to start out by looking at Fractals, including examples,
Hausdorff dimension, and the Chaos game. If you want to do some background
reading, some nice books are:
*Fractals everywhere*by Michael Barnsley*Chaos, fractals, and dynamics*by Robert Devaney*Fractals for the classroom*by Peitgen et al.

- We'll spend some time learning about algebra and mappings over
the complex numbers. There are lots of good references for this, but my
favourite is
*Complex Analysis*by Lars Ahlfors

- We'll look at iteration and chaos, over both the real and complex numbers.
Some references are the Devaney book, mentioned above, or
*Chaos and fractals, the mathematics behind the computer graphics*by Robert Devaney and Linda Keen.*Chaos - making a new science*by James Gleick.

- We'll explore some topics in analytic number theory, including the
Riemann zeta function. And maybe try to understand some of the Millenium
problems. Some references:
*Explorations in Calculus*by Robert M. Young, MAA 1992*A radical approach to Real Analysis*by David Bressoud, MAA 2007

- Inequalities are a very classical topic, and you'll find treatments in
most analysis texts. A readable book, about nothing but inequalities, is
*The Cauchy-Schwarz master class*by J.M. Steele, Cambridge 2004

- We'll study some approximation theory, Fourier analysis, and splines
(perhaps including applications to computer graphics). Bressoud's book
(referenced above) has some material on Fourier analysis. For more details see
*Invitation to classical analysis*by Peter Duren, AMS 2012.

- At various times we'll talk about other aspects of Calculus, including trying to understand why the calculus we teach today looks so different from the way it looked when Newton and Leibiniz first invented it. Bressoud's book (referenced above) discusses this at length, explaining how our notions of rigour and proof have evolved. Basically, old-style intuitive arguments started running into problems with examples in which it wasn't clear what was true and what wasn't. This forced people to rethink their ideas of what words like function and continuity meant. The way you teach calculus is set up to fit with that 19th century rigorous approach. It requires enormous hindsight before this approach seems natural, so it is not actually surprising that our students struggle with the level of abstraction involved. The inventors of 17th century calculus didn't think that way at all, and would probably have struggled too. Knowing this can inform our teaching.