ESP Biography

The number 41 is a sum of two squares (25+16). Can you write 37 as a sum of two squares? How about 43 or 47? To a mathematician, the next obvious step is to find the pattern and make a conjecture. Only once we know what is true is it possible to prove it. We will illustrate how mathematical research is done by finding an answer to the question of which numbers are a sum of two primes and then sketching a proof using the arithmetic of the Gaussian integers.

Suppose I have a rectangular bar of chocolate made up of nine squares. I want to separate it into the nine pieces by cutting it with a knife. What is the minimum number of cuts needed to do so? Once you've solved this, can you use the idea to solve other interesting problems? Through answering this question and similar ones, we will explore how mathematicians think about problem solving.

The number 41 is a sum of two squares (25+16). Can you write 37 as a sum of two squares? How about 43 or 47? To a mathematician, the next obvious step is to find the pattern and make a conjecture. Only once we know what is true is it possible to prove it. We will illustrate how mathematical research is done by finding an answer to the question of which numbers are a sum of two squares and then proving it using techniques called the geometry of numbers. This question has a pretty answer which leads to many fruitful generalizations, but the goal of this class is to illustrate the process of mathematics.

On a gameshow, there are three doors. Behind two are goats, and behind one is a wonderful surprise. You pick a door. The game show host opens a different door, revealing a goat. She asks if you want to change your mind and pick the third door in hopes of getting the wonderful surprise. Should you? We will discuss this tricky question and several other results in probability and statistics which seem paradoxical. The problem is not with mathematics, but with our understanding of concepts like independence and expectation and our lack of appreciation of confounding variables. Come learn, so that you will not be fooled by the data.

The number 41 is a sum of two squares (25+16). Can you write 37 as a sum of two squares? How about 43 or 47? To a mathematician, the next obvious step is to find the pattern and make a conjecture. Only once we know what is true is it possible to prove it. We will illustrate how mathematical research is done by finding an answer to the question of which numbers are a sum of two primes and then proving it using the arithmetic of the Gaussian integers.

Why is Shakespeare so strange? Why is Chaucer even stranger? And what's up with Beowulf? Have you ever wondered how Latin evolved into Spanish, French, Italian, and so many other languages? In this class, students will get a taste of \textbf{Historical Linguistics}, the study of \textit{how language changes over time}. By comparing words from several related languages, students will be able to reconstruct what the words had been in the shared \emph{proto-language} in addition to the regular sound changes that led to the modern variants. In addition to building general problem solving skills, language reconstruction is like solving a puzzle--fun and rewarding!

Can every problem be solved? We will first look at a very limited model of computation, finite state automata, and see how to prove that some problems are beyond their power to solve. Finite state automata are very simple, so this is not surprising. What is more surprising is that we will see that all computers run into the same sort of problem.

Which positive integers are the sum of two squares? For example, 5 = 1+4, but 3 cannot be written as the sum of two squares. We will investigate and form a conjecture, and then prove it using techniques developed by Hermann Minkowski. These techniques now go by the name the geometry of numbers, and are an elegant application of geometric ideas to algebraic number theory.