The following is a list of topics, with brief descriptions, about which members of our department would be willing to speak. Most topics can be adapted to audiences of students, at various levels, or teachers (the exceptions are noted). This list is by no means exhaustive, and if there is a topic in which you are interested and it does not appear on the list, please let us know and we will try to accommodate you.
[Contact Information] [York Math Department]
Title:
Puzzles, Surprises, and Famous Problems in Mathematics
Description:
A number of puzzles and problems in mathematics, such as the Königsberg Bridge Problem, The Birthday Problem, The Game of Rencontres, Lucas' Problem of the Married Couples, and The Jack-Queen Problem, will be discussed. No specific mathematical background is required.
Speaker:
Morton Abramson
Title:
Games and Mathematics
Description:
Students are usually highly motivated to learn how to win games. If the solution of a game requires some mathematics, then the students are indirectly motivated to learn that mathematics. Along the way they should learn something less tangible: a logical (or better a mathematical) way of approaching a difficult problem such as the solution of a game. Several examples of games and their solutions will be given. The level of mathematics required is appropriate for high school students from grades 9 to 13.
Speaker:
Richard Brown
Title:
Some Concepts of Topology.
Description:
If we bend or stretch a geometric figure, its shape may be changed quite considerably. For example, a square can be made into a circle or a triangle. But a figure-eight is "topologically" different and cannot be deformed into one of the above. A sphere is of a different kind again. Topology deals with properties of geometric entities which do not change under continuous transformations.
Speaker:
Richard Brown
Title:
Some Mathematical Monsters.
Description:
When we perform ordinary mathematical operations, even simple ones like counting or measuring, we may be aware of some mathematical monsters lurking near at hand. Teachers and texts usually avoid these monsters by skillful navigation, but we will confront several geometric monsters head on.

This talk is suitable only for mathematically adventuresome students, grades 9 - 12.

Speaker:
Richard Brown
Title:
Dealing with Statistics in Secondary School.
Description:
A discussion of some elementary and useful ideas in statistics, which unify various aspects of mathematics studied in the curriculum. Some instructive examples are presented, together with a bibliography of helpful references.
Speaker:
Gene Denzel (Willing to speak on any other topics of interest in statistics or probability.)
Title:
Topics in the History of Mathematics.
Description:
Episodes from the history of mathematics including such topics as: the origins of the number system, Greek mathematics, what the Egyptians and Babylonians knew, the rise of mathematical notation, mathematics and astronomy, non-Euclidean geometry.
Speakers:
Israel Kleiner; Trueman MacHenry; Pat Rogers; Abe Shenitzer (We are willing to speak, to students or teachers, on many other topics in the history of mathematics.)
Title:
The Relativity of Mathematics.
Description:
Is 1 + 1 = 2? Is the sum of the angles of a triangle 180 degrees? Does the equation x(squared) + 1 = 0 have exactly two solutions? Is ab = ba? Although the answer to all these questions seems to be an unequivocal "yes", the questions are, in fact, meaningless as they stand. The purpose of this talk will be to explain the issues underlying these questions.
Speaker:
Israel Kleiner
Title:
Fermat's Last Theorem: a Theorem at Last (for teachers)
Description:
Fermat stated the theorem, namely that the equation x^n+y^n=z^n has no solutions for positive integers x, y, and z if the exponent n is greater than 2, in 1637. Andrew Wiles of Princeton University proved it (with the assistance of R. Taylor) in 1994. I will give an historical sketch of the 350-year quest for the solution of what was arguably the most famous mathematical problem.
Speaker:
Israel Kleiner
Title:
Hypercomplex Numbers.
Description:
A brief historical survey of the various number systems (integers, rationals, reals, complex numbers) will be given, and extensions to "hypercomplex" number systems discussed. [Familiarity with complex numbers will be helpful.]
Speaker:
Israel Kleiner
Title:
Number Theory.
Description:
Topics and problems in number theory (e.g. modular arithmetic, diophantine equations, properties of prime numbers, perfect and amicable numbers), suitable for students at various levels, will be discussed.
Speaker:
Israel Kleiner
Title:
Some Discoveries (Inventions) that Shook the Mathematical World. (For Teachers)
Description:
A number of topics, such as the irrationality of ˆ2, noneuclidean geometries, noncommutative algebras, logical paradoxes, and unprovable propositions will be discussed with a view to showing their great impact on subsequent developments in mathematics.
Speaker:
Israel Kleiner
Title:
Why Should a Teacher of Mathematics Know Some History of Mathematics? (For Teachers)
Description:
The relevance of the history of mathematics for the teaching of mathematics will be discussed.
Speaker:
Israel Kleiner
Title:
Symmetry.
Description:
There are infinitely many ways of covering a wall with a repeated pattern, but these patterns all fall into 17 types. We discuss ways of "measuring" the symmetry of design, illustrating with examples from nature and art.
Speaker:
Trueman MacHenry
Title:
Geometric Algebra.
Description:
The Greeks did not use algebraic notation to solve problems in algebra, still they were able to solve algebraic problems. We shall give some examples of their method and discuss some famous problems that grew out of them.
Speaker:
Trueman MacHenry
Title:
Maxima and Minima - How Calculus Can Help and Mislead.
Description:
By means of examples we show how calculus helps to solve some maximization and minimization problems, but how the uncritical application of some so-called "rules" can lead to incorrect results.
Speaker:
Martin Muldoon
Title:
On Colouring Maps.
Descriptions:
How many colours are needed to colour any map in such a way that no two adjacent countries will have the same colour? This seemingly innocuous question stumped amateur and professional mathematicians for over a hundred years. It was finally answered, with the help of a computer, in 1976. But the nature of the solution raises other questions, in particular that of what we mean by a "proof".
Speaker:
Martin Muldoon
Title:
Peace, War and Mathematics.
Descriptions:
In its application to science and as one of the finest examples of human creativity, mathematics has surely been of great benefit to mankind. But many mathematicians, from Archimedes at the defence of Syracuse to the Manhattan Project during World War II, have used their talents to help produce weapons of destruction. Sometimes, work done for peaceful or neutral ends has had military applications; undeniably, some work done for military purposes has had beneficial results. What are the social responsibilities of mathematicians, scientists and citizens, given the potential for total destruction inherent in today's weapons?
Speaker:
Martin Muldoon
Title:
Mathematics in Careers.
Description:
We present some examples of what may be unfamiliar applications and other aspects of mathematical thinking with a view of explaining the relevance of a mathematical education to a variety of occupations.
Speaker:
Martin Muldoon
Title:
Chaos, Fractals and Dynamical Systems
Description:
In this talk, I give a nontechnical introduction to chaos and some related concepts. Chaos will be demonstrated in the iteration of certain quadratic functions. Implications of chaos theory for the predictive value of certain mathematical models (such as those used in weather forecasting) will be discussed. Some examples of fractal sets will be described and their relation to chaos will be discussed.
Speaker:
Martin Muldoon
Title:
Fractions, Equations and .
Description:
How continued fractions can be used to solve some equations and to get good fractional approximations to and other numbers.
Speaker:
Alfred Pietrowski
Title:
Actuarial Mathematics.
Description:
There will be an outline of the actuarial profession and a discussion of the type of mathematical background which is required.
Speaker:
David Promislow
Title:
The Tower of Hanoi.
Description:
The ancient Brahman priests were to move a pile of 64 discs of decreasing size one at a time from one stand to another, after which the world would end. The rules of the game permit the use of one other stand, but at no time may a larger disc be placed on a smaller disc. The task of estimating how long this would take at the rate of one move per second reveals interesting insights into the nature of mathematical discovery.
Speaker:
Pat Rogers
Title:
Infinity.
Description:
Paradoxes and arithmetic with infinite sets and numbers.
Speakers:
Pat Rogers; Trueman MacHenry; Israel Kleiner
Title:
Examples of Meaningful Presentation of Mathematical Material. (For Teachers)
Description:
I am sometimes asked by teachers to show how high school topics in mathematics can be made more meaningful. Here are some of the topics I discuss.
• Why do we compute with signed numbers the way we do?
• Why do we add or multiply fractions the way we do?
• Why is 2^0 = 1 ?
• Is the connection between the roots of a quadratic polynomial and its coefficients restricted to quadratic equations?
• What is the importance, ifany, of the fact that if a is a root of a polynomial P(x), then x - a is a factor of that polynomial?
• What is the point of factorization?
• Why are polynomials important?
• Why are trigonometric functions important?
• Why are complex numbers important?
• Why are conic sections important?
Speaker:
Abe Shenitzer
Title:
Transformations.
Descriptions:
The three main issues to be discussed are:
• The use of transformations for the solution of problems.
• The role of transformations in Felix Klein's definition of "Geometries".
• Is the teaching of transformations in school the answer to all our problems?
Speaker:
Abe Shenitzer
Title:
Mirrors and Polyhedra.
Description:
When 2 or 3 mirrors are arranged at some particular angles, the reflections of points and line segments in these mirrors will give us pictures of regular and semi-regular polyhedra. We will show how to construct all Platonic solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron, as well as some of the semi-regular solids.
Speaker:
Asia Weiss
Title:
All the Way to the International Mathematical Olympiad (IMO).
Description:
Canadian students have been competing in the International Mathematical Olympiad since 1981. In July 1995 they competed on home ground when IMO-95 took place at York University in Toronto. We review the history of the IMO and the Canadian performance in it. The methods used to select and train prospective students for this competition and others, such as the Canadian Mathematical Olympiad, are discussed. We examine a few accessible questions from past exams and try with the participation of the audience to solve them.
Speaker:
Joan Wick Pelletier

## Contact Information:

Israel Kleiner
for High School Liaison Committee
Department of Mathematics and Statistics
York University