USRA Summer Seminar
Suggested Topics
The following gives a list of possible
topics for your summer project.
You are also free to come up with your own choice as well.
You should fix yourself on a topic, and let me know what it is,
as soon as possible.
To avoid duplications, I will
post
your choices on this site, as soon as they are made.
Project topics will be assigned on a first-come, first serve basis:
I expect that some of the more popular topics will be chosen
first, so it is in your interest to make a fast decision!
Primality testing:
Given a large integer N, (of a hundered or so digits, say) how can
one determine that it is prime? The naive approach, trial division, requires
about N or the square root of N operations, which is
very expensive. Remarkably, there are algorithms that decide whether N
is prime in a power of log(N) operations, which is
much better. Give a presentation of some of the
best known algorithms: the algorithm of Miller and Rabin, and the
more recent algorithm of Agrawal et. al. which is the first prvably
polynomial-time algorthm for primality testing.
References:
Miller, Gary L. Riemann's hypothesis and tests for primality.
Working papers presented at
the ACM-SIGACT Symposium on the Theory of Computing (Albuquerque, N.M.,
1975).
J. Comput. System Sci. 13 (1976), no. 3, 300--317.
Carl Pomerance, Recent developments in primality testing, Mathematical
Intelligencer 3, no. 3 (1981) 97-105.
Manindra Agrawal et. al. Primes is in P. Can be downloaded at
http://www.cse.iitk.ac.in/users/manindra/primality.ps.
Lenstra's factorization algorithm based on elliptic
curves over finite fields.
Reference:
Lenstra, H. W., Jr.
Factoring integers with elliptic curves. Ann. of Math. (2) 126
(1987), no. 3, 649--673.
Elliptic curve cryptosystems.
Reference.
Koblitz, Neal.
A course in number theory and cryptography. Second edition. Graduate
Texts in Mathematics, 114. Springer-Verlag, New York, 1994.
Shor's factorization algorithm based on "quantum computers". Recently
a probabilistic algorithm was discovered for factoring integers in
polynomial time on a quantum computer, a computing device which uses
elementary particles to store information and exploits the peculiar
properties of these particles (as described by quantum mechanics)
to achieve a fast factorization algorithm. This discovery has caused alot of
excitement among number theorists, theoretical computer scientists, and
physicists, and represents a remarkable confluence of the three subjects.
This would be a good topic for someone who already has some
background in physics (and the rudiments of quantum mechanics in particular).
Reference:
Peter Shor,
Polynomial-Time Algorithms for Prime Factorization
and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing
Volume 26, Number 5
pp. 1484-1509.
Dirichlet's Theorem on primes in arithmetic progressions.
Reference.
Serre, J.-P.
A course in arithmetic. Translated from the French. Graduate Texts in
Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.
The prime number theorem.
Reference.
Davenport, Harold.
Multiplicative number theory. Second edition. Revised by Hugh L.
Montgomery. Graduate Texts in Mathematics, 74.
Springer-Verlag, New York-Berlin, 1980.
Continued fractions and ergodic theory.
Reference.
Khintchine, A. Ya. Continued fractions.
Translated by Peter Wynn. P. Noordhoff, Ltd.,
Euler's calculation of zeta(2k) and its developments.
Reference
Dunham, William. Journey through genius.
The great theorems of mathematics. Penguin
Books, New York, 1991.
Koblitz, Neal. p-adic numbers, p-adic analysis, and
zeta-functions. Second edition. Graduate
Texts in Mathematics, 58. Springer-Verlag, New York-Berlin, 1984.
Iwasawa, Kenkichi. Lectures on p-adic L-functions. Annals of Mathematics
Studies, No. 74. Princeton University Press, Princeton, N.J.; University
of Tokyo Press, Tokyo, 1972.
The irrationality of zeta(3).
Reference.
van der Poorten, Alfred. A proof that Euler missed... Apery's
proof of the irrationality
of zeta(3).
An informal report. Math. Intelligencer 1
(1978/79), no. 4, 195--203.
Groningen 1963.
The theory of partitions.
Reference.
Andrews, George E. The theory
of partitions. Reprint of the 1976 original. Cambridge
Mathematical Library. Cambridge
University Press, Cambridge, 1998.
Schoof's algorithm for counting the number of points on an elliptic
curve over a finite field.
Reference.
Schoof, Rene. Elliptic curves over finite fields
and the computation of square roots mod
p. Math. Comp. 44 (1985), no. 170, 483--494.
Computing the zeroes of the Riemann zeta-function.
Reference.
Turing, Alan. M.
Some calculations of the Riemann zeta-function. Proc. London Math. Soc.
(3) 3, (1953). 99--117.
Odlyzko, Andrew M. Analytic computations in number theory. Mathematics
of Computation 1943--1993: a half-century of computational
mathematics (Vancouver, BC, 1993),
451--463, Proc. Sympos. Appl. Math., 48,
Amer. Math. Soc., Providence, RI, 1994.
p-adic numbers.
References.
Koblitz, Neal p-adic numbers, p-adic analysis, and
zeta-functions. Second edition. Graduate
Texts in Mathematics, 58. Springer-Verlag, New York-Berlin, 1984.
Gouvea, Fernando Q. p-adic numbers. An introduction. Second
edition. Universitext.
Springer-Verlag, Berlin, 1997.
Cubic and biquadratic reciprocity.
References.
Ireland, Kenneth; Rosen, Michael. A classical introduction to
modern number theory. Second edition. Graduate Texts in Mathematics, 84.
Springer-Verlag, New York, 1990.