Undergraduate Colloquium

Spring 2015
Wednesdays 12:55 - 1:45
LCB 225

Pizza and discussion after each talk
Receive credit for attending
Past Colloquia

January 14    No Talk

January 21     Andrejs Treibergs
Inequalities of Analysis
Abstract: Inequalities take on increasing importance the more deeply a student studies analysis. In this talk I will begin describing the inequality comparing arithemtic, geometric and harmonic means of several numbers and present Cauchy’s elemetary proof. Power mean inequalities such as those of Holder and Minkowski will be discussed. Similar inequalities hold as well for integrals. All of them can be deduced from Jensen’s Inequality, which deserves to be better known. Some applications to geometry and physics problems will be given.

January 28     Tyler Johnson
Positively Polynomial
Abstract: Suppose I have some polynomial with non-negative integer coefficients. You are tasked with guessing the polynomial; I will tell you the value of the polynomial at any integer you choose. How many integers are required for you to be certain of the polynomial? What if the polynomial has multiple variables? The solution to this problem is surprisingly simple and satisfying; we will discuss a few other interesting variations of this problem as well.

February 4     Peter Alfeld
Hotel Infinity
Abstract: You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full. A new guest arrives and insists you give her a room. How do you accommodate her? The next day, a family with infinitely many members arrives, each of whom wants a private room. The next day infinitely many families, each with infinitely many members, arrive. Each family member insists on a private room. You can do it! Infinity is different.

February 11     Jennifer Kenkel
The King Chicken Theorems
Abstract: Consider a coop of chickens. In any pair of chickens, one pecks the other. However, there might not necessarily be a chicken who pecks every other bird. Instead, we call a "king chicken" one that, for every other chicken in the coop, either pecks it, or pecks a chicken who pecks it. By representing each chicken as a vertex and each pecking relationship with an edge, we can use graph theory to examine chicken politics. We will see every flock has a king, but this king is not necessarily unique, or even uncommon.

February 18     Evelyn Lamb
Mathematical Tie Knot Enumeration
Abstract: In 1999, the mathematicians Thomas Fink and Yong Mao enumerated all possible tie knots, or so they thought. In 2003, the Merovingian appeared in The Matrix Reloaded wearing a tie knot that didn’t appear on Fink and Mao’s list. What went wrong? Last year, four researchers developed a new way to generate and enumerate tie knots, and there are many more than we realized. Bring your own tie and learn a new knot.

So how many ways are there to tie a tie? Take the red pill and find out.

February 25     Tom Alberts
Branching Processes
Abstract: A branching process models the changes in a population level in which each individual in generation n produces some random number of individuals in generation n+1. They are a very simple but important part of probability theory and can be used to model reproduction within a bacteria colony, the spread of surnames in genealogy, or the propagation of neutron collisions in an atomic bomb. This talk will go over the basic models of branching processes and some interesting variants and then describe the beautiful mathematics behind one of the most important questions in the subject: what is the probability that the population ultimately goes extinct?

March 4     Thomas Goller
Theory of Everything
Abstract: Are you frustrated because pure mathematics courses like linear algebra, analysis, and modern algebra seem completely unrelated? Do you feel like you're starting from scratch every time you take a new math course? We will explore similarities among different fields of math by looking for structures that fit an abstract template called a "category". We will discuss this abstract template and touch lightly on examples from linear algebra, analysis, algebra, and more. Then we will see how a "functor" gives us the power to travel between two categories.

March 11     Kelly MacArthur
Magic or Markov Chain?
Abstract: We'll explore what a Markov Chain is and how some of its properties explain some seemingly surprising phenomenons within card games, random walks and number games. Then, we'll extend this thinking to consider how many shuffles are necessary to randomize a deck of cards, and a quick summary of the reasons behind this result.

March 18     No Talk - Spring Break

March 25     Vera Babenko
Deep Brain Stimulation for Parkinson disease
Abstract: Parkinson's disease is a degenerative disorder of the central nervous system that causes major movement dysfunction. An estimated 7 to 10 million people worldwide are living with Parkinson's disease. In this talk we will explore causes of this disease and mathematical models of one very popular treatment - Deep Brain Stimulation.

April 1     Drew Johnson
Busy Beavers and Big Numbers
Abstract: In this talk we will learn a little bit about a Turing machine, a theoretical model of computation. Along the way, we will have a contest, meet some furry creatures, encounter super-astronomical numbers, and see how the ability to express large numbers reflects the progress of civilization.

April 8     Adam Brown
An Invitation to Harmonic Analysis
Abstract: Fourier series are a central topic in the study of differential equations. However, it can be difficult to gain an intuition for these mysterious decompositions. We will explore how Fourier series naturally appear in representation theory, and how they can be used to solve differential equations. Generalizations of our techniques compose an extremely interesting field known as harmonic analysis. Some familiarity with calculus, complex numbers, and elementary linear algebra will be required.

April 15     Matt Cecil
Some Irrational Thoughts about π and $e$.
Abstract: $\pi$ and $e$ show up in just about every math course. $\pi$ even has its own day on the calendar! They are both irrational numbers and hence have a non-repeating decimal expansion. In this talk, I will discuss how you might find their digits by using approximations derived from power series. I will also prove that they are irrational. This talk should be accessible to anyone who has taken or is currently taking Calculus II.

April 22     Nelson H. F. Beebe
Pseudo-random numbers: mostly a line of code at a time
Abstract: Random numbers have an amazing range of application in both theory and practice. Approximately-random numbers generated on a computer are called pseudo-random. This talk discusses how one generates and tests such numbers, and shows how this study is related to importan mathematics and statistics - the Central-Limit Theorem and the Χ2 measure - that have broad applications in many fields. Come and find out what the Birthday Paradox, Diehard batteries, gorillas, Euclid, French soldiers, a Persian mathematician, Prussian cavalry, and Queen Mary have to do with random numbers.

April 29     No Talk - Reading Day
Papers are due for students enrolled in Math 3000-001 by 4:00