A TOPIC IN FRACTAL GEOMETRY

Fractal geometry is a vast subject that impinges on many areas of mathematics, science and social science. This project will investigate an area of fractal geometry beyond that covered in the Honours course. Possibilities include Julia sets and the Mandelbrot set, fractals in dynamical systems, classes of fractal constructions, random fractals, fractals in number theory, etc, etc. See some of the books on fractals for ideas.

FIXED POINT THEOREMS

There are many possible conditions on a set X and a function f : X → X that guarantee that f has a fixed point, i.e. there is an x ∈ X such that f(x) = x. For example, central theorems such as Brouwer’s fixed point theorem and Banach’s contraction mapping theorem give this conclusion in different contexts. Fixed point theorems have applications throughout mathematics, from matrix theory to differential equations. Write an account of some of these fixed point theorems and their applications. (Initial reference: DH Griffel ‘Applied Functional Analysis’ chapters 5,6.)

THE FINITE DIFFERENCE CALCULUS IN STATISTICS

A review of probabilistic and statistical problems in which the finite difference calculus arises. Areas covered could include the classical occupancy problem, sums of uniform variates and population size estimation. The essay could discuss both the probability models and the related inferential problems.

LATIN SQUARES: THEIR PROPERTIES AND APPLICATIONS

A latin square (of order n) is an n×n array filled with entries from a set of n symbols, with the property that each entry occurs once in each row and column. These objects were first introduced by Leonhard Euler in the eighteenth century, and have been much-studied since. Viewed algebraically, such a square corresponds to a generalization of a group, called a quasi-group; determining the number of n × n latin squares (for general n) is still an open problem in combinatorics. Latin squares also have applications in error-correcting codes and the design of experiments. This project explores these (and other) topics.

MATHEMATICS OF FINANCE

In recent years the financial industry has rapidly expanded. This is mainly due to the development of the Black-Scholes Equation which occurs in the theory of pricing financial options. The project will involve looking at some aspect of financial mathematical models, including computational techniques for solving the Black-Scholes equation.

THE AXIOM OF CHOICE

The axiom of choice (AC) is one of the most controversial mathematical axioms. Some mathematicians avoid using it, some use it without realizing, and others debate its truth. In this project you will study:

• The history of AC, how it came to be, and why it was/is so contentious;• Some equivalent conditions such as Hausdorff’s maximal chain condition, Zorn’s Lemma and the Teichm¨uller-Tukey Lemma;• Standard mathematical theorems that fail if AC is not assumed;• (potentially) undesirable consequences of AC;• Some theorems that only hold if AC is replaced by an alternative axiom that contradicts AC.

GRAPH HOMOMORPHISMS

This project is about graph homorphisms. A homomorphism is a map from a graph Γ to another graph Γ that preserves the edges of Γ. The study of graph homomorphisms is a relatively new area with connections to many other areas of mathematics as well as computer science and statistical physics. In this project you will study a classical result in the field: that every finite group is isomorphic to the group of automorphisms of a finite graph. Related results will also be investigated

THE MATHEMATICS OF CARL FRIEDRICH GAUSS

Carl Friedrich Gauss, the ”Prince of Mathematics,” exhibited his calculative powers when he corrected his father’s arithmetic before the age of three. He recovered from the subsequent smack on the head to become one of the greatest mathematicians of all time, making important contributions to areas of mathematics as diverse as algebra, statistics, geometry and many other besides. This project would look at the mathematical advances made by Gauss and how his work has helped shape modern mathematics.

EIGENVALUES

History of EIGENVALUES & Application in mathematics

Eigenvalues are an extremely important part of linear algebra and have many important applications throughout mathematics. This project would involve a study of the history of eigenvalues, methods for calculating them and the uses of eigenvalues.

METHODS TO COMPUTE DIGITS OF PI

The aim in this project is to give an overview over the known methods to compute (potentially many) decimal digits of the number Pi. A part of the project is to actually come up with programs computing digits of Pi, describe these programs and prove that they work.

CONSTRUCTIONS WITH COMPASS AND NO RULER

Geometric constructions with compass and ruler have been studied by mathematicians since ancient times. The task in this project is to prove that every point in the Euclidean plane that can be constructed from given points using compass and ruler can also be constructed by using only the compass.

SPACE-FILLING CURVES

A curve in the unit square is called space-filling if it covers the whole unit square; the figure below shows a space filling curve. The existence of space filling curves caused some disturbance in the 1920’s. In the 1920’s mathematicians were investigating how to tell the difference between Euclidean spaces of various dimensions topologically and examples of space filling curves made people wonder whether the problem might be harder than it appeared. Of course it’s easy to show that the line and the plane are not homeomorphic, for example removing a point disconnects the line. But it was not so clear why, say, the plane and three-dimensional space could not be homeomorphic. You will learn the basic definitions and facts about space filling curves. Then you will choose a more specialized topic (e.g. a special space-filling curve or a special property of space-filling curves) and discuss that topic in more detail. The main reference for the project is Hans Sagan, Space-Fillling Curves, New York, Springer-Verlag, 1994. Prerequisites: MT4004 is desirable but not essential.

Figure 1: A space filling curve.

ALGEBRAIC NUMBERS AND CONVERGENCE

Numerical computer experiments “show” that the powers xn of x = 3+√8 approache integer values as n tends to infinity. The number 3+√8 is an example of a so-called algebraic Pisot number. In this project you will study questions related to the convergence or divergence of, forexample, powers of certain algebraic numbers. The project involves a nice mixture of methods from both algebra (rings and fields) and analysis. Since this topic is related to ergodic theory, which is one of the active research areas at this school, this project is well suited for a student aspiring to do post graduate work in analysis. For a good introduction to the subject see R Salem, Algebraic Numbers and Fourier analysis, Heath, 1963. This project is also suitable for an enlarged MMath project. Prerequisites: A basic knowledge of group theory.

ORDINALS AND CARDINALS IN NAIVE SET THEORY

The number of elements in a (typically infinite) set A is called the cardinality of the set A and denoted by card A. It is well known that a set is finite, countable or uncountable; however, uncountable sets may have different sizes (cardinality), e.g. card R < card {A | A ⊆ R}. Cardinal numbers may thus be viewed as a generalization of the concept of a finite, countable or uncountable set. The first purpose of this project is to make precise the notion of the cardinality, card A, of a set A. The second purpose is to write an exposition of some aspects of cardinal numbers.

SHARKOVSKY’S THEOREM

Consider a continuous map f of the unit interval [0, 1] into itself. A point x is called periodic if there is some n such that fn(x) = x. Call the minimal such n the period of x. A natural question about the dynamical system f is: which of the natural numbers appears as the period of some periodic point? Sharkovsky proved the remarkable fact that if there is a periodic point of period three, then there are periodic points of all other periods. In this project you will study this theorem, along with some of its generalizations.

MATHS AND MUSIC

Musical instruments exhibit mathematics and physics in action through the normal modes of oscillation and harmonics they support. Some notes sound pleasing when played together, while others clash. The arrangement of different pitch notes into musical scales has been done in many cultures and can be studied mathematically. The Equal Tempered scale we use has 12 intervals in an octave, but others use 19 or 24 intervals.

See the following web pages and their links for more details:http://en.wikipedia.org/wiki/Music and mathematics

http://en.wikipedia.org/wiki/Physics of musichttp://mathforum.org/library/drmath/sets/select/dm music math.html

Shortest Path: Dijkstra’s Algorithm

1. List some real-life examples of the problem.

2. Formulate the problem in graph theoretical terms. Before you do this you mighthave to introduce some new terminology that is not covered in class.

Note thatSection 13.1 of Grimaldi [2] has the directed graph version whereas Section 5.5 of Liu [5] has the undirected graph version of the algorithm.

3. State the algorithm.

4. Give two examples showing how the algorithm works. One of these should involve a graph with |V | _ 6 and |E| _ 12 and the other one a graph with |V | _ 10 and |E| _ 15. You might want to use an undirected graph in one of these examples. In each of these examples illustrate every stage of the algorithm with a different picture. (This means for every change of i in Grimaldi’s notation and for every change of P in Liu’s.) Explain what is happening between the pictures at one point in the first example and at two points in the second.

5. You do not have to prove anything or explain why the algorithm works. You do not have to estimate the complexity of the algorithm.

References:

1. Athanasios, Papagelis. “Dijkstra Algorithm.” Accessed 29 March 2007:

http://students.ceid.upatras.gr/˜papagel/project/kef5 7 1.htm

2. Grimaldi, Ralph P. Discrete and Combinatorial Mathematics, 5th ed.

Pearson, 2004:

Section 13.1.

3. Ikeda, Kenji. “Dijkstra’s Algorithm.” Accessed 29 March 2007:

http://www-b2.is.tokushima-u.ac.jp/˜ikeda/suuri/dijkstra/Dijkstra.shtml

4. Laffra, Carla. “Dijkstra’s Shortest Path Algorithm”, Accessed 29 March

2007:

http://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/

DijkstraApplet.html.

5. Liu, C. L. Elements of Discrete Mathematics, 2nd ed. McGraw-Hill, 1985:

Section 5.5.

Minimal Spanning Tree: Kruskal’s and Prim’s Algorithms

1. List some real-life examples of the problem.

2. Formulate the problem in graph theoretical terms. Before you do this you might

have to introduce some new terminology that is not covered in class.

3. State the algorithms.

4. Give two examples in which you will apply both of the algorithms. One of these

should involve a graph with |V | _ 6 and |E| _ 12 and the other one a graph with

|V | _ 10 and |E| _ 15. In each of these examples illustrate every step of the

algorithm with a different picture. (This means for every change of i in Kruskal’s

algorithm and for every change of T in Prim’s algorithm in the notation of Section

13.2 of Grimaldi [3].) Explain what is happening between the pictures at one point

in each of the first pair of examples and at two points in each of the second pair.

5. You do not have to prove anything or explain why the algorithms work. You donot have to estimate the complexity of the algorithms.

References:

1. Athanasios, Papagelis. “Kruskal Algorithm.” Accessed 29 March 2007:http://students.ceid.upatras.gr/˜papagel/project/kruskal.htm

2. Athanasios, Papagelis. “Prim Algorithm.” Accessed 29 March 2007:http://students.ceid.upatras.gr/˜papagel/project/prim.htm

3. Grimaldi, Ralph P. Discrete and Combinatorial Mathematics, 5th ed. Pearson, 2004: Section 13.2.

4. Ikeda, Kenji. “Kruskal’s Algorithm.” Accessed 29 March 2007:http://www-b2.is.tokushima-u.ac.jp/˜ikeda/suuri/kruskal/Kruskal.shtml

5. Ikeda, Kenji. “Prim’s Algorithm.” Accessed 29 March 2007:http://www-b2.is.tokushima-u.ac.jp/˜ikeda/suuri/dijkstra/Prim.shtml

6. Liu, C. L. Elements of Discrete Mathematics, 2nd ed. McGraw-Hill, 1985: Section 6.7.

Maximum Flow: Ford-Fulkerson Algorithm

1. List some real-life examples of the problem.

2. Formulate the problem in graph theoretical terms. Before you do this you might have to introduce some new terminology that is not covered in class.

3. State the algorithm. You do not have to use the Edmonds-Karp algorithm to de- termine an f-augmenting path. You might prefer to use the algorithm as given in Section 6.8 of Liu [4]. Note that you will not need all the material presented in Section 13.3 of Grimaldi [2] in this context.

4. Give two examples showing how the algorithm works. One of these should involve a graph with |V | _ 6 and |E| _ 12 and the other one a graph with |V | _ 10 and |E| _ 15. In each of these examples illustrate every stage of the algorithm with a different picture. (This means every time the flow changes.) Explain what is happening between the pictures at one point in each of the examples.

5. You do not have to prove anything or explain why the algorithm works. In particular, you do not have to state or prove the max-flow min-cut theorem. You do not have to estimate the complexity of the algorithm.

References:

1. Chalidabhongse, Thanarat Horprasert. “Network Flow.” Accessed 29 March 2007:

http://www.cs.pitt.edu/˜kirk/cs1501/animations/Network.html

2. Grimaldi, Ralph P. Discrete and Combinatorial Mathematics, 5th ed. Pearson, 2004: Section 13.3.

3. Ikeda, Kenji. “Ford-Fulkerson’s Algorithm.” Accessed 29 March 2007:http://www-b2.is.tokushima-u.ac.jp/˜ikeda/suuri/maxflow/

Maxflow.shtml

4. Liu, C. L. Elements of Discrete Mathematics, 2nd ed. McGraw-Hill, 1985: Section 6.8.

5. Vatter, Vince. ”Graphs, Flows, and the Ford-Fulkerson Algorithm.” Accessed 29 March 2007:

http://www-groups.mcs.st-andrews.ac.uk/˜vince/teaching/summer04/flow.pdf

Complete Matching: Hall’s Theorem

1. List some real-life examples of the problem.

2. Formulate the problem in graph theoretical terms. Before you do this you might have to introduce some new terminology that is not covered in class.

3. State the theorem and present two different proofs that do not use the max-flow min-cut theorem. Note that the proof in Section 13.4 of Grimaldi [1] uses the max- flow min-cut theorem. You may want to use the other references below. Also note that you will not need all the material presented in Section 13.4 of Grimaldi in this context.

4. At least one of your proofs should suggest an algorithm. Demonstrate how this algorithm works on a graph with |V | _ 12 and |E| _ 18.

5. Give an example of a problem that is not stated directly in graph theoretical terms and can be solved using the theorem. (Suggestion: An 8×8 chessboard has exactly three pieces on each row and each column. Show that some of these pieces can be removed in such a way that there is exactly one piece left on each row and on each column.)

References:

1. Grimaldi, Ralph P. Discrete and Combinatorial Mathematics, 5th ed. Pearson, 2004: Section 13.4.

2. van Lint, J. H. and R. M. Wilson. A Course in Combinatorics, Cambridge, 1992: Chapter 5.

3. Liu, C. L. Introduction to Combinatorial Mathematics, 2nd ed. McGraw-Hill, 1968: Section 11.2.

4. McKernan, James.“Matching and Hall’s Theorem.” Accessed 29 March 2007:

http://www.math.ucsb.edu/˜mckernan/Teaching/04-05/Spring/137B/l 18.pdf

5. Simon, Istvan. “Bipartite Matching.” Accessed 29 March 2007:http://www.mcs.csuhayward.edu/˜simon/handouts/4245/hall.html