September 17: Michael Montgomery
Abstract: Groups are found throughout mathematics to capture symmetries of the world around us. There is a famous group that you have interacted with if you have ever braided a friend’s hair. We will explore this incredible group and see what it can tell us about other mathematical objects.
October 1: Brian Luczak
Title: Continuous, Differentiable, and Everything in Between
Abstract: Many calculus students work with the concepts of continuity and differentiability on a regular basis.
However, a natural question is whether these are the only useful ideas in discussing the behavior of a function.
Could we define a new class of functions that are somehow ‘smoother’ than continuous, and yet not quite differentiable?
To explore this idea, we explain the concept of Holder continuity and examine many strange and fascinating functions of Weierstrass and Hardy from the late 19th century.
October 8: Hayden Jananthan
Title: Transcending the Rational
Abstract: The mathematical journey students take typically introduces them to numbers of higher and higher complexity, starting with the natural numbers (0,1,2,…), adding the negative integers (-1,-2,-3,…), then non-integer rational numbers (1/2,22/7,-25/3…), and finally irrational numbers (e, pi, sqrt(2),…). But not all irrational numbers appear to have been made equal; when considering the square root of 2, it seems like a simpler beast than that of pi. One way of distinguishing between them is that the former is algebraic while the latter isn’t (it is transcendental). We will see that in a strong sense, ‘almost all’ real numbers are transcendental. In spite of that, it is remarkably hard to actually prove that a given real number (like e or pi) is transcendental, so we will examine some of the major results concerning transcendental numbers.
October 15: Professor Glenn Webb
Title: An Environmental Model of Honey Bee Colony Collapse Due to Pesticide Contamination
Abstract: A model of honey bee colony collapse is developed, based on the contamination of forager bees in environmental regions contaminated with pesticides. An important feature of the model is the daily homing capacity each day of foragers bees. The model consists of difference equations describing the daily homing of uncontaminated and contaminated forager bees, with an increased homing failure of contaminated bees. The model quantifies colony collapse in terms of the fraction of contaminated bees subject to this increased homing failure. If the fraction is sufficiently high, then the hive falls below a viability threshold population size that leads to rapid disintegration. If the fraction is sufficiently low, then the hive can rise above the viability threshold and attain a stable population level.
October 29: Professor Jose Gil-Ferez
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 severed head, 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 finitely many jumps, 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.
November 5: Andy Jarnevic
Title: Classical Game Theory
Abstract: Classical game theory explores what happens when rational actors make simultaneous decisions. In this talk we will explore the primary tools and concepts of classical game theory, and apply them to topics such as games of rock-paper-scissors, whether to watch football or go to the opera, and whether or not to snitch on your friend if you find yourself in prison.
November 12: Dr. Lauren Ruth and Professor Alex Cameron
Title: Intro to Quantum Computing: The Deutsch-Jozsa Algorithm
Abstract: Want to learn a quantum algorithm? After introducing qubits and quantum gates and comparing them to classical bits and classical logic gates, we will describe the Deutsch-Jozsa algorithm. This example of a quantum algorithm illustrates how a quantum circuit can outperform a classical circuit.
November 19: Bogdan Chornomaz
Title: Probabilistic methods in combinatorics
Abstract: Typically, probability theory deals with things which are uncertain on the very ground level. Of course, to struggle with uncertainty is harder, but what can we do? How much simpler would things be if only our objects and actions were clearly determined. Or would they?
Surprisingly, it turns out that if our objects are by no means indeterminate, but still complicated enough, it might be convenient to study them as if they were probabilistic. Think of it as saying “whatever” at some point in an argument, but in a strictly mathematical way. In this talk, we will consider some simple examples where these methods can be fruitfully applied.