February 21: Zach Gaslowitz
Title: Playing with Math
Abstract: We will discuss a few pencil-and-paper games that you can play with your friends, and we’ll see how we can use logic and mathematics to understand them on a deeper level. Does one player have an advantage over the other? Is there a ‘best’ way to play, and if so, how can we figure out what it is? You’ll come away with some sneaky moves to trick, and perhaps even amaze, you friends.
February 28: Gili Golan
Title: The Banach-Tarski Paradox
Abstract: The Banach-Tarski Paradox says that it is possible to cut a ball into 5 disjoint pieces and rearrange the pieces to get two balls of the same size. We would talk about the Axiom of Choice which implies the Banach-Tarski Paradox and discuss some Group Theory results which form the basis for the paradox.
March 14: Ryan Solava
Title: Is pi overrated?
Abstract: Yes, it is.
To celebrate this Pi Day, we will discuss the reasons that pi is not as great as everyone seems to think. I will propose an alternative to pi and examine its many benefits.
March 21: Hayden Jananthan
Title: Computability and the Church-Turing Thesis
Abstract: In the 1930s, Alonzo Church, Alan Turing, and Kurt Godel independently created their own definitions of what it meant for a function to be computable. Although extremely different, it was shown they all defined the same class of functions, and this observation lead to the Church-Turing Thesis:
“Every effectively calculable function (effectively decidable predicate) is general recursive.”
We will formally define what it means to be general recursive (now known as Partial Recursive), introduce another model of computation, and illustrate the Church-Turing Thesis for these two models.
March 28: José Gil-Férez
Title: The Hero and the Hydra: A Journey to Infinity and Beyond (and Back Again)
Abstract: The Hydra, with its many, many heads, poisonous breath, and so venomous blood that even its scent is lethal, is guarding the entrance to the Underworld. The hero needs to slain the monster, but with each head that is severed, the rage of the beast grows and many more heads spring up.
We realize that, for the task to be accomplished, all that is needed is to know the number of the beast. But its number is not part of this finite world of ours, and the search is going to lead us up the ordinal stairs, to the infinity and beyond. With good fortune on our side, we shall return, descending from the hinterlands of infinity in only a finite number of steps, just on time to defeat our foe.
From this journey, we will learn about our limitations, as finite beings, and how the multiple powers of our imagination would prove necessary to overcome them.
April 4: Blake Dunshee
Title: Baseball’s Pythagorean Theorem
Abstract: Baseball: America’s pastime and possibly its most polarizing sport. We will pretend for an hour that baseball is a slow, boring game and that we are co-owners of a Major League Baseball team. Quickly thegame becomes significantly more intriguing… at least from a mathematical perspective.
We will transform baseball into a board game. The players will each morph into their own set of intricately crafted, high maintenance die. We will turn the flight patterns of the ball and movements of players into spaces on a game board that our game piece travels along based on the results of our players’ die. This will allow us to advise our manager concerning strategic decisions like whether to bunt, steal, or swing away.
Ultimately, we want to know how well we can expect our team to perform. Our manager has told us his expectations of our offensive and defensive efficiency this season, but how will this translate into wins? We use probability density functions to derive a fabled formula known as baseball’s Pythagorean Theorem.
April 11: Matthieu Jacquemet
Abstract: This is going to be an IKEA talk. You will need: a wooden plank, plenty of nails, a hammer, a ruler, and plenty of string. On your plank, tap in nails to form a regular grid. Now take a piece of string, and form a polygon on your plank by fastening the two ends of your piece of string to the same nail. Question: what is the area of your polygon ?
We are going to see that this question can be answered by just counting nails. This will also give us the occasion to illustrate the typical process a mathematician (and more generally, a scientist) uses to study a question.
April 18: Derek Bruff
Title: Breaking the Vigenère: The End of Encryption in the 19th Century
Abstract: In a pre-digital age, messages meant to be kept secret were encrypted using pencil-and-paper ciphers. The pinnacle of 19th century encryption techniques was the Vigenère cipher, a polyalphabetic substitution cipher that had remained unbroken for over 250 years. (“Polyalphabetic substitution cipher.” Say that five times fast.) The Vigenère met its end, however, at the hands of a British mathematician who hated organ grinders and invented (but didn’t build) the first computer, who was challenged to break the Vigenère by a dentist from Bristol. In this talk on classical cryptography, we’ll see how the Vigenère works and how it was broken with a little bit of number theory.