Essay & Presentation Guidelines
Applied Math 206
Fall 2009
Description
The purpose of these essays is for you to study some topic beyond the syllabus in depth, and understand it well enough to be able to explain it in your own words. The essay should contain a significant amount of prose, which explains the motivation for the material, highlights the main ideas, and gives intuition for the mathematics. Of course, this prose should be supporting a mathematical development, with precise definitions and statements of results. You need not include proofs of all lemmas and theorems; selectively decide which to include or sketch according to how informative and important they are. Further originality beyond the exposition is of course welcome too. This may come in the form of examples, new proofs of some lemmas, or even some small new results. Be sure to clearly identify anything that is new! In order to ensure that you present things in your own words (and really understand the material), I recommend that you study the source material carefully, but then try to write your essay without looking at the source material (only consulting it when you get stuck).
Presentations
Each AM206 student will give a presentation on one of her/his essay topics during reading period. These are a chance for you to share something interesting you have learned with the rest of the class. Since the presentations are relatively short, you need to be even more selective about what to include; avoid the temptation to squeeze in too much material. If the audience leaves with one memorable idea or concept and an interest to learn more about the subject, you will have succeeded. Again due to the short time, you are encouraged to use slides/powerpoint (but again avoid the temptation to fill them with too much math). You may choose either your first or second essay topic for the presentation. The exact length and scheduling of the presentations will be done closer to the end of term.
Other Guidelines
- Essay 1 due date: Monday, November 2
- Essay 2 due date: Monday, December 7
- Presentations: Monday, December 7 and Wednesday, December 9.
- Length: 7-8 pages each (though anything in the 5-10 page range is acceptable)
- Target audience: your fellow AM206 students - i.e. a mathematically mature reader, but with no specific expertise in algebra beyond what we cover in AM106/206.
- Typesetting your essays in LaTeX is highly recommended. (For Windows, I recommend using the programs Miktex and Winedt.)
- All references you use should be listed at the end of the essay, and you should clearly distinguish material/exposition from the references from that which is original.
Topics
Below is a list of suggested topics with some suggested references, which will be updated periodically. You should feel free to suggest your own topic, e.g. one that relates to your own interests or research. If you do, you must discuss the topic and the references you plan to use with Salil for approval no later than 1 week before the essay is due.
- Classification of Finite Abelian Groups
- Gallian Chapter 11
- We will state but not prove this in lecture
- The Sylow Theorems
- Classification of Finite Simple Groups
- Symmetry Groups, with applications to chemistry or physics.
- Gallian Chapters 27-28 and the references therein.
- The articles by White mentioned in Chapter 2 and by Watson in Chater 27 may be good starting points for the applications.
- M. Tinkam. Group Theory and Quantum Mechanics. Dover, 2003.
- Applications of Group Theory to Counting
- Gallian Chapter 29 and the references therein
- Computer scientists may also be interested in the article Computational Polya Theory by Mark Jerrum.
- Representation Theory (of Finite Groups)
- Fourier Analysis on Finite Abelian Groups, with applications to random walks, computer science, and more.
- Barrington's Theorem: equivalence of log-depth circuits and constant-width branching programs
(via multiplication in S_5)
- already covered on Problem Set 3.
- Barrington, David A. Bounded-width polynomial-size branching programs recognize exactly those languages in ${\rm NC}\sp 1$. Journal of Computer & System Sciences. 38 (1989), no. 1, 150--164. 68Q15 (68Q25)
- Computational Group Theory
- Solving the Rubik's Cube.
- Integer Multiplication in Nearly Linear Time.
- Primality Testing
(and other number-theoretic algorithms)
- Galois Theory
- Algorithms for factoring polynomials over finite fields.
- J. von zur Gathen and D. Panario. Factoring polynomials over finite fields: A survey. J. Symb. Comput., 31(1/2):3–17, 2001.
- E. Kaltofen. Polynomial factorization: a success story. In J. Rafael Sendra, editor, ISSAC, pages 3–4. ACM, 2003.
- J. von zur Gathen. Who was who in polynomial factorization. In Barry M. Trager, editor, ISSAC, page 2. ACM, 2006.
- R. Lidl, H. Niederreiter. Introduction to finite fields and their applications.
- and the references therein
- Algebraic coding theory and list-decoding algorithms
- Cryptography from ideal lattices
- Grobner Bases (main tool for doing computations with ideals in multivariate polynomial rings)
- Understanding ruler-compass constructions using field extensions
- Gallian Ch. 23
- Chapter in Algebra by Michael Artin.
- Elliptic Curve Cryptography