The “Friday Afternoon Mathematics Undergraduate Seminar” (FAMUS) is a weekly event consisting of announcements, talks, and faculty interviews. FAMUS takes place most Fridays at 3:00-4:00pm in Room 164 of the Adel Mathematics Building. Typically the first half of FAMUS consists of a talk on a mathematical topic while an interview of a faculty member takes place in the second half. FAMUS is hosted by Jeff Rushall.

Come join us for some entertaining talks! Refreshments always served.

Note that talks are listed in reverse chronological order.

**Date:** May 4, 2018

**Speaker:** Matthew Daunt (NAU)

**Abstract:** Inspired by the board game, Othello is a game in which a 4 by 8 board of disks are each chosen randomly to be either white or black (0 or 1). The color of each disk is an independent Bernoulli random variable with probability of white being .5. The score in the game is determined by the number of lines connected from the top to the bottom of the board. These lines can either be vertical, diagonal, or skew diagonal. I generalized this game to a board of size r by c, allowing multiple colors, and unequal probability of the colors. My research focuses on understanding patterns in the general structure of the probability distribution, including the mean and variance of the score for each version of the game.

The faculty guest is Todd Wolford. [PDF of Flyer]

**Date:** April 20, 2018

**Speaker:** Mike Rozinski and Jordan Wright (NAU)

**Abstract:** The Elo rating system is most widely known for its use in tournament chess. This system produces numerical ratings which represent the measure of a player’s skill set. The system, named after its creator Arpad Elo, is utilized for many purposes, including the pairing of players in a tournament and predicting the outcome of a two-player game. We will outline all relevant characteristics of Elo’s rating system and examine key features of the algorithm it uses. We will then deduce potential issues by cross-referencing current multi-player rating systems that use rudimentary paired comparisons, and construct a fairness criteria that addresses these issues. With this fairness criteria, we will assemble a rating system that extends Elo’s two-player algorithm to multi-players and discuss its success in accurately predicting a players ability.

The faculty guest is John Hagood. [PDF of Flyer]

**Date:** April 6, 2018

**Speaker:** Justin Sima and Riley Waechter (NAU)

**Abstract:** A graph is a bunch of dots/vertices and lines/edges. A perfect ternary tree looks like the family tree that would be drawn if:

- Adam and Eve had 3 kids,
- who each had 3 kids,
- who then each had 3 kids,
- who in turn then each had 3 kids,
- etc. …

A prime vertex labeling of the dots/vertices is a game: label each of the $n$ dots in a graph with one of the integers $1, 2, 3, \ldots, n$, so that any pair of vertices connected by an edge are given integer labels that share no common factors aside from 1. This game/problem has been around for about 30 years, and Riley and Justin think they have an idea that solves the puzzle. Their progress along these lines will be the focus of these two short talks. Does their idea work? Come and see. And then…

Instead of a faculty guest, any in the audience who wish to stick around and help with the construction of a life-size Menger Sponge will be rewarded with all of the **pizza and icream they can eat**. [PDF of Flyer]

**Date:** March 30, 2018

**Speaker:** Kaitlyn Lee and Mason Sargent (NAU)

**Abstract:** A Hadamard matrix is a square matrix with entries of 1 or -1, that has the biggest possible determinant among all such matrices. Although first created as the solution to a pure math problem, Hadamard matrices now have zillions of interesting and important applications. A quaternary complex Hadamard matrix is a square matrix with entries of $1, -1, i$, or $-i$, that again has maximal determinant. It turns out that these types of Hadamard matrices are fundamental components of quantum computing machines, which makes them both interesting and important. Mason and Kaitlyn have spend the past 6 months trying to modify techniques for building Hadamard matrices to instead build complex Hadamard matrices. Their progress along these lines will be the focus of these two short talks.

The faculty guest this week is Bianca Luedeker. [PDF of Flyer]

**Date:** March 16, 2018

**Speaker:** Zach Parker (University of Vermont)

**Abstract:** Solutions to the Pythagorean Theorem, called Pythagorean triples, have been known since antiquity. There are many different ways to find such triples. We explore one strategy toward the solution, utilizing more sophisticated methods and ultimately hoping to somehow generalize these techniques to a harder problem, namely Fermat’s Last Theorem This talk serves as a brief introduction into the world of algebraic number theory, a strange crossroads of many important areas of mathematics including complex analysis, ring and field theory, and Galois theory. [PDF of Flyer]

**Date:** March 9, 2018

**Speaker:** Charlie Katerba (Montana State University)

**Abstract:** This is actually a 2-part talk. [PDF of Flyer]

Part I: Catalan’s Constant is a funny number that pops up all over the place in mathematics and the sciences: in number theory, in computational mathematics, in topology,… The list goes on and on. Charlie will explain what it is, how it is computed, and why it is interesting.

Part II: How I got here is a short tale of how one person traveled from high school, to studying abroad, to college and a math degree, to graduate school and a Ph.D., and beyond. Several pictures might be shown, and several stories (most not inappropriate) might be told.

**Date:** March 2, 2018

**Speaker:** Marcela Gutierrez and Viola McCarty (NAU undergrads)

**Abstract:** Marcela and Viola have been working on a generalization of this problem: Can one find and organize every single set of integer solutions to the Pythagorean Theorem? You know some of these solutions: $(3,4,5)$, $(5,12,13)$, etc. It turns out that there is a clever way to depict ALL such solutions to $a^2 + b^2 = c^2$ in a pretty “tree,” a fact that has been known since 1970. Marcela and Viola have been trying to depict ALL integer solutions to an equation that looks a lot like the Pythagorean Theorem, namely: $a^2 + b^2 + c^2 = d^2$. There are lots of solutions to this equation; they include $(1,2,2,3)$, $(2,3,6,7)$, etc. In fact, Marcela and Viola have found a way to organize and display all such solutions in a beautiful tree graph. Their result is very, very pretty.

The faculty guest this week is Sarah Watson. [PDF of Flyer]

**Date:** February 23, 2018

**Speaker:** Dylan King (former student at the University of Nebraska Omaha)

**Abstract:** A Hilbert matrix is a nice, square matrix whose entries are a nice pattern of nice fractions of the form 1/n, where n is any positive integer. Originally developed as a tool in approximation theory, Hilbert matrices are now interesting all by themselves, especially because their inverses have a rather bizarre and surprising structure. I will explain how Hilbert matrices and their inverses are constructed, and then state some open questions about these inverses that are going to make excellent undergraduate research projects.

The faculty guest this week is new department member Ye Chen. [PDF of Flyer]

**Date:** February 16, 2018

**Abstract:** This week’s FAMUS will be kind of like a beauty pageant: several faculty will be on display, pitching possible projects, and students will decide whether they like a particular project and will then communicate their interest to that faculty member. On the other hand, this won’t be like a beauty pageant, because in the end the faculty decide who does/doesn’t work with them. The faculty who will be hawking their wares, so to speak, include:

```
- Dana Ernst
- Nandor Sieben
- Roy St. Laurent
- John Neuberger
- Jeff Rushall
```

Come see what projects faculty have in mind for next year. [PDF of Flyer]

**Date:** February 9, 2018

**Speaker:** Dylan King (former student at the University of Nebraska Omaha)

**Abstract:** Cops and Robbers is traditionally a pursuit game played on a finite, connected graph; $K$ cops and a single robber are placed on the vertices of a graph and take turns moving to adjacent vertices. If a cop occupies the same vertex as the robber then the cops win, but if the robber is always able to avoid that situation then he wins. Much of the literature is focused on determining the cop number of a graph, which is the minimum number of cops needed to always capture the robber. My research focused on the infinite analog to the traditional Cops and Robbers game.

The guest to be interviewed is also Dylan King. [PDF of Flyer]

**Date:** February 1, 2018

**Speaker:** Roy St. Laurent (NAU)

**Abstract:** The Envelope Problem is another probability-based paradox - sort of - in the theme of both the Sleeping Beauty Problem and the Monty Hall Problem, the topics of our last two FAMII. Here is a description of The Envelope Problem:

You are given two indistinguishable envelopes, each containing money, one contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?

Dr. St. Laurent will present the history of the problem and explain why resolving it is so controversial.

The faculty guest this week is David Deville. [PDF of Flyer]

**Date:** January 26, 2018

**Speaker:** Jeff Rushall (NAU)

**Abstract:** The Monty Hall Problem is a simple probability problem - somewhat similar to the Sleeping Beauty Problem - and every bit as controversial. The controversy surrounds a Parade Magazine column on this problem written by the infamous Marilyn Vos Savant in the 1990s. In this column, she offered a solution that was rejected by many mathematicians (but she was actually correct!). Anyways, I’ll explain the problem, show the correct solution, explain why many math geeks were mistaken, and discuss some generalizations.

The faculty guest this week is Ian Williams. [PDF of Flyer]

**Date:** January 19, 2018

**Panelists:** Dana Ernst, Angie Hodge, Roy St. Laurent and Steve Wilson (all NAU)

**Moderator:** Jeff Rushall (NAU)

**Abstract:** “The Sleeping Beauty Problem” is the latest controversy to attract my attention. It addresses a very simple question: is the probability that a fair coin flip ends up heads REALLY 1/2? It turns out that this is not necessarily always the case. I’ll spend about 15 minutes explaining the problem/controversy/paradox, and then I will moderate a debate that features, at a minimum, faculty members Dana Ernst, Angie Hodge, Roy St. Laurent and Steve Wilson. Other faculty may appear, and several faculty who cannot attend have offered their opinions, which I will share during my presentation. [PDF of Flyer]