[MD-sorular] Araþtýrma konularý (matematiðin günlük hayatta kullanýmýyla alakasý olmayan)

[Ali Ýlik]

Sık sık listeye matematiikle ilgili proje konusu soruluyordu.
Ve, konu olarak üniversite öğrencilerine (lise öğrencilerine verilmesi de
çok mantıklı değil)
"Matematiğin günlük hayattaki yeri ve önemi" -ne demekse- veriliyordu.

Şurada bir araştırma konuları listesi var. Tesadüfen buldum, unutmadan
listeye yolluyorum.
Faydalı olabilir. Bazıları iyi.

http://www.swarthmore.edu/NatSci/wstromq1/math97/PossibleProjectTopics1.htm

(Linkin ölme ihtimaline karşı listeyi aşağıya da alıyorum. İleride yine
birisi sorarsa bu mesaj referans olarak verilebilir.)

Pre-Math 97

April 17, 2006

*Possible Project Topics*



*1.  Perfect numbers.*  What are the even ones?  Are there any odd
ones?  Consult
work of Euclid, Euler, Peter Hagis.



*2.  Quadratic Sieve and Number Field Sieve.*  How to factor large numbers.



*3.  Recognizing primes.*  The largest known primes have millions of digits.
How do they know?



*4.  Unique factorization.*  Does it work in rings other than the integers?
Of quadratic extension fields  *Q*( ), which are Euclidian, and which have
unique factorization?  (Like all topics, must go beyond course material.)



*5.  Covering spaces and covering groups.*  See Kuga, *Galois' Dream.*  It's
topology, but it looks like Galois theory.  (Note:  This was covered
thoroughly in 104.)



*6.  Linear programming.*  Simplex methods (Gale's work?) and interior
methods (Khachian, Karmarkar)



*7.  "Normal" numbers.*  Are there any?  If so, name one.



*8.  P vs. NP.*  What does it mean?  It's a Millennium problem.



*9.  Navier-Stokes Equations.*  There's a Millennium problem about these,
too.  What is it about?



*10.  Ramsey numbers* and Ramsey theorems generally.  How many people have
to be in a room before there are four mutual friends or four strangers?



*11.  The Kepler conjecture.*  Hales has proved it.



*12.  The Universal Chord Theorem.*  Cities A and B are at the same
elevation, and connected by a 100-mile straight road.  Is there a 20-mile
stretch of the road that starts and ends at the same elevation?  What about
a 40-mile stretch?



*13.  The Ham Sandwich Theorem.*  Applications to fair division.



*14.  Equilibria in Game Theory.*  John Nash's papers won him a Nobel Prize.
This is a worthwhile topic even for two-person games.



*15.  Fundamental Theorem of Decision Analysis.*  To decide among lotteries,
you need a probability model and a utility function.



*16.  Combinatorial Games.*  Consult Berlekamp, Conway, Guy, *Winning
Ways,*or Conway,
*On Numbers and Games*, or a recent paper by Daniel Kaseorg.



*17.  Moment-Generating Functions and Characteristic Functions.*  These can
be used to prove the central limit theorem.



*18.  Generating Functions* as a way of counting things.  See Wilf, *
Generatingfunctionology*.



*19.  The Prime Number Theorem.*  See lead article in January, 2006, *American
Mathematical Monthly.*



*20.  Two-body orbits.*  It's just one differential equation; how can it be
hard?  (See Marsden and Ross, AMS Bulletin Nov. 2005)



*21.  Matrix multiplication.*  Can you multiply two 1000-by-1000 matrices
using less than a billion multiplications of numbers?



*22.  Error-Correcting Codes.*  See Hamming's book.



*23.  p-adic numbers.*



*24.  Quadratic forms and sums of squares.*  2-square identity, 4-square
identity – are there others?



*25.  Division rings over R*.  Quaternions, Wedderburn Theorem, etc.



*26.  Non-negative matrices and the Perron-Frobenius Theorem*

* *

*27.  Block designs / finite geometry*

* *

*28.  Continued fractions *and rational approximations to real numbers



*29.  Wallpaper groups and frieze groups*

* *

*30.  Transcendental numbers. * e, p, and the ab problem from Hilbert's
problems



*31.  Pseudo-random sequences* of 0's and 1's.  Polynomials over *Z*2.



*32.  Combinatorial matrix theory.*  Sign pattern matrices, Zeilberger's
proof, etc.



*33.  Linear preserver problems.*  What must a map Mn be if it preserves,
say, determinants?  Or eigenvalues, or rank?



*34.  Numerical range.*  Or, "field of values."  If  x*x = 1, what can  x*Ax
be?  Classical version and generalizations.



*35.  Group representations and applications.*  Chemists use them.  Why?



*36.  KNOTS, KNOTS, KNOTS.*

* *

*37.  Banach spaces or anything else in functional analysis. * Ask Mike
Lauzon about all this good harmonic stuff.

* *

*38.  Solving systems of polynomial equations.*  In linear algebra the
degree-1 version became routine.  With Gröbner bases, so does the degree-n
version.

* *

*PREVIOUS (as of Fall 2004) TOPICS:*

The Jordan Curve Theorem

Random Order: Sequences in Arbitrary Set Partitions

Two-Person Zero-Sum Games

Factorization Algorithms

A Proof of the Transcendentality of e

Penrose, Escher, Wang: Engrossing Periodicity Work

The Isoperimetric Problem

Knot Theory and Knot Invariance

Forced to Commute: Wedderburn's Theorem on Finite Division Rings

Ramsey Theory: Parties, Colors, and Tic-Tac-toe

Musings on Division Algebras, Particularly the Octonions

Euclidean Constructions

The Impossibility of Democratic OPOV Election Procedures

Giuga's Conjecture on Primality

Nonlinear Economic Dynamics

Modeling Epidemics - From Deterministic Models to Chain-Binomial Models

Fuzzy Logic and Probability

Decomposition Principle for Large-Scale Linear Systems

Math is Hard: An Introduction to Computational Complexity

An Introduction to Auction Theory

A Little Bit about Matrix Representations of Groups

The Jones Polynomial: A Knot Invariant

A Comparison of Error-Correcting Codes

The Axiom of Choice

Truth, Math, and Reconciliation: A Posthumous Nobel Prize (in Peace) for
Jean Charles de Borda?

Newton's Method, Chaotic Motion and Symbolic Dynamics

Godel's Incompleteness Theorem

Open and Closed Leontief Models and the Study of Positive Matrices

Financial Derivatives

Plane hyperbolic geometry

Option pricing models

Information theory

Projective geometry and cryptography

Geometry of curves in R3

Social choice theory

Boolean algebra

The Fundamental Theorem of Algebra

Game theory



(This is the end of the handout.  The actual topic list is infinite.)**

Ali
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