# 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*Χ*- that have broad applications in many fields. Come and find out what the^{2}measure*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**