# Junior Colloquium

#### Thursday, March 5, 2:30 in AB 102

Abstract: We are all very familiar with representing real numbers by infinite decimals, a representation which is very good at facilitating arithmetic and calculations. Similar representations can be done using any base. But there is another way of representing real numbers by sequences of positive integers which is not as well known: continued fractions. These are not tied to any choice of base, and give much more efficient ways of approximating real numbers by rational numbers. (However, they are not so good for doing arithmetic.) Continued fractions are usually regarded as part of number theory, but they arise naturally (and sometimes unexpectedly) throughout mathematics; thus every mathematician should be familiar with them. Until about 100 years ago they were an important part of the mathematics curriculum, but they have unfortunately mostly disappeared (to the best of my knowledge, we do not have any regularly offered courses which cover them). The goal of this talk is to partially fill this hole in our curriculum. I will describe the basic theory of continued fractions and discuss some of the beautiful and important aspects of the theory and its applications. The properties to be discussed range from purely algebraic ones to some deep probability theorems and applications in dynamical systems. Anyone who has had calculus and high-school algebra will be able to follow almost all of the talk. Some of the advanced results involve Lebesgue measure, but I will try to treat these fairly gently using the language of probability. Everyone is welcome to come; I will try to make the discussion interesting and stimulating for faculty and grad students as well as undergraduates. The talk will probably last somewhat more than 50 minutes, so we can thoroughly discuss the basics and also get to some of the deeper and more interesting topics. It will be no more than 75 minutes.

### Induction, plain and fancy

#### Thursday, December 4, 2:30 in LME 315

Abstract: Everyone knows the dull, prosaic induction technique: prove for 1, assume true for n, yada yada yada, it's always true. But there are
more interesting and subtle variations that you may never encounter.
Unless you come to this talk, where you will be dazzled by unusual variations on the inductive theme.

### Infinite matrices and applications

#### Thursday, November 20, 2:30 in LME 315

Abstract: In Linear Algebra, a matrix encodes a linear transformation between two finite dimensional vector spaces, like Rn. What happens if we deal with infinite dimensional vector spaces, like the space of polynomials or the space of all convergent sequences of real numbers? Then a linear transformation (also called operator) is given by an infinite matrix. In this talk I will give examples of classes of infinite matrices which encode certain linear transformations between infinite dimensional vector spaces. I will try to answer some questions like:

1. What operations with infinite matrices are permitted?
2. When does a matrix define a bounded operator?
3. Does an infinite matrix have eigenvalues and eigenvectors?
4. Can we calculate the norm of an infinite matrix?
5. I will also discuss a Theorem of Toeplitz involving infinite lower triangular matrices, which implies a discrete version of L'Hospital rule for sequences, called Stolz-Cesaro Lemma. I will give several examples of computations of limits using this lemma.

### If circles were squares...

#### Thursday, April 24, 2:30 in AB206

Abstract: Since a circle is defined solely in terms of distance, if we changed our definition of distance, the figure known as a circle would look different, as would other figures such as ellipses, parabolas and hyperbolas. We will define a reasonable distance so that circles are squares. What will ellipses look like? Will the obvious answer, rectangles, be correct? Or will the answer be, like all obvious answers to Pfaff's questions, wrong? Be sure to come and see the exciting answer to this and, as time permits, other questions.

### Dangers of the Infinite: Cantor and His Legacy

#### Tuesday, April 15, 2:30 in AB206

Abstract: Are there as many real numbers between 0 and 1 as there are between 0 and 3? Are there as many even integers as integers? As many integers as real numbers? The answers to some of these questions may surprise you. We will explore the answers to these questions, some similar questions, and the theory behind them, seeing along the way the different levels of infinity. Georg Cantor, considered to be the father of set theory, was the first mathematician to do real research dealing with the infinite. We will hear bits and pieces of his tragic tale and discuss some of some of his most surprising results and ideas, including the ever-elusive continuum hypothesis. Afterwards we examine the seven (sometimes eight) axioms which form the basic foundation for all of mathematics, and we will see how the logicians Kurt Gödel and Paul Cohen continued Cantor's work. We conclude with the answer to the question posed by the continuum hypothesis and mention some related ideas and results.

Prerequisite for understanding: A strong desire to learn about something so fascinating, it should be illegal. It would help to know some very basic set theory, but such knowledge is by no means necessary.

### Some comments on the Poincaré Conjecture

#### Monday, November 19, 5:30 in AB638

Abstract: Perelman recently settled the famous Poincaré conjecture and the geometrization conjecture. I'll explain what the conjecture says, give a little history, some examples, and the briefest outline of how Perelman approached and solved the problem.