Topics

What you should do
  1. Choose a topic. I'd like you to become the local expert on the topic. Tell me what topic you will work on so we can avoid collisions.
  2. Read background material on the topic; search the web. All of these topics have lots of references.
  3. Give me a 2 or 3 page outline of your work, with references.
  4. Prepare a 20-minute talk on your topic, and rehearse this talk with me.
  5. Give me a draft (5 to 10 pages) of your paper.
  6. Give the talk, and, following any necessary changes, submit your paper.

Randomness  Ben Tully
  1. General introduction to the subject (see S. B. Volchan, "What is a random sequence?" , Amer. Math Monthly 109 (2002) 46-63.) This talk will be used by several other students' presentations.
  2. Pseudo random number generators. See chapter 21 of [G]. See also http://www.nist.gov/dads/HTML/pseudorandomNumberGen.html
Addition chains  Steven Jaslar
  1. What is an addition chain, and why is it "useful"? Here [K] might be good, as well as the papers http://www.ccrwest.org/gordon/jalg.ps and http://www.daimi.au.dk/~mmj/crypt/addchain/addchain.pdf
  2. What is presently known, and what are the conjectures? This is in the papers cited above. See http://www.cpktec.com/demo.html.
  3. Students should understand something about the present knowledge of addition chains.
  4. An additional reference, somewhat technical but perhaps the first few sections may be accessible and useful.
Linear congruential generators as pseudo-random number generators  Jagan Pisharath
  1. Following the discussion of randomness.
  2. A definition is at http://www.nist.gov/dads/HTML/linearCongruentGen.html.
  3. Here is one discussion: http://random.mat.sbg.ac.at/~charly/server/node2.html and the books [G] and [K] also contain discussions.
  4. Students should understand the idea, and understand why certain LCG's are better than others.
Linear feedback shift registers as pseudo-random number generators  Justin Palumbo
NO NO NO: P vs NP as exemplified by a discussion of minesweeper.
  1. Following the discussion of randomness
  2. A definition is at http://www.nist.gov/dads/HTML/pseudorandomNumberGen.html
  3. The book [G] contains a discussion.
  4. Students should understand the idea, and be able to "generate" the bits of a LFSR given the initial fill.
GCD via Euclidean algorithm  Robert Burton
  1. Discuss the history of the Euclidean algorithm.
  2. Exhibit the algorithm with suitable examples. One reference is http://www.cut-the-knot.org/blue/Euclid.shtml.
  3. Show how this applies so that students in the class can find by hand the multiplicative inverse of N mod P where P is, say, a prime with #(P)=6 or 7.
Efficiency of GCD  Nathan Wilson
  1. How many steps on average does the GCD algorithm take?
  2. What is the "worst case"? (The Fibonacci numbers)
Knapsack or subset sum cryptosystems  Gregory Ryslik
  1. What is it? (Another "hard" math problem underlying a cryptosystem)
  2. How can it be broken? Sources include [G] below and many other references.
Factoring
  1. Description of history of the problem, comparison with primality (see, e.g., http://www.cse.iitk.ac.in/news/primality.html).
  2. Current state of the art. [C & P] is a good reference.
Fast(er) multiplication  Dale Gold
  1. Aim: show that there are faster ways to multiply than #(n)2.
  2. Discuss Karatsuba multiplication. A general introduction is available here: http://mathworld.wolfram.com/KaratsubaMultiplication.html. Other references are http://www.math.ubc.ca/~cass/programs/fast-multiplication.dvi and http://people.cs.uchicago.edu/~kaharris/cspp55001/karatsuba-ofman.pdf.
  3. A possible continuation of this is Fourier transform multiplication. Useful background for understanding this is some linear algebra.
NvJ
As mentioned in the diary, this is sort of a converse to the von Neumann problem. Here is a proper statement, I hope:
  You are given a fair coin and a number p with 0<p<1. You must come up with a procedure to simulate a coin which comes up heads with probability p and tails with probability q=1-p. The procedure must terminate almost surely, that is, with probability 1. No further information about p is given: it can be any number between 0 and 1. You also need to analyze the "running time" of your procedure: the expectation for the number of flips of the fair coin needed to contribute one simulated p/q flip.
I think that I know a way to do this problem. As pointed out in class by several students, if p is rational (a quotient of two integers) there seem to be several ways to do the problem. But, as I remarked, I'd like to know how to do it for p=1/sqrt(2), which doesn't seem too rational to me.
So I don't know how hard this problem is, or how involved a writeup it would need (would it be comparable to other topics, for example?). I don't know.
Solved by Mr. Jaslar! More to come.

More broadcasting  Charles Siegal
On Monday, October 13, I will begin an analysis of the broadcasting model and how it could break for binary trees. I think that this analysis could be extended. Here are some suggestions:

  1. Instead of binary trees, how about trees whose branching rule is multiplication by B (a positive integer) at each stage (binary is the case B=2). I think it should be possible to duplicate my reasoning in this case. Computer simulation with some data supporting your theoretical deductions would be useful.
  2. After doing the preceding case, how about trying trees that alternate branching between binary (B=2) and ternary (B=3)? What will the probabilities turn out to be? I don't know, and this could be a bit interesting. Computer simulation with some data supporting your theoretical deductions would be useful.
  3. What happens if we alternate between 2 and 3 at random? An analysis of this may be quite difficult. So here definitely computer simulation would be useful.
More Ramsey Theory  Lisa Facchini
I will show that if a complete graph is large enough (!) and if the edges are two-colored, then there must always be some "monochromatic" small complete graph inside it. (These terms will all be explained later.) This project would attempt to duplicate the reasoning given in class for graphs which are 3-colored, then 4-colored, etc. Or another variation: how about 3-coloring a graph, and seeing how large it must be before sme small complete 2-colored graph must be inside it? A student who takes up this project will need to read ahead in my notes, and talk with me about them.

The Four Color Theorem
Read about the history and controversy and report on them. See http://matholymp.com/ARTICLES/4color.pdf for an initial source. The books by Thomas L. Saaty and Paul C. Kainen (The four-color problem : assaults and conquest, QA612.19.S2 1986) and Oystein Ore (The four-color problem, QA3.P8 v.27) are currently in the library. Look at Kempe's original paper (available on the web, I believe, through a Rutgers browser). Read Professor Zeilberger's opinion and interview him: perhaps invite him to come to class.

Scale-free networks
Read Barabási's book, and describe scale-free networks. Discuss one application of the networks. See also the text Six Degrees: The Science of a Connected Age by Duncan Watts, also, by the same author, Small Worlds.


Bibliography

As mentioned above, a search of the web for any of these topics will provide many refences and some of these will be useful. Here are some texts which are good:

Maintained by greenfie@math.rutgers.edu and last modified 9/30/2003.