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 in the second half. FAMUS is hosted by Jeff Rushall.

Come learn what mathematics is about and how the faculty ended up in our department at this popular event.

Note that talks are listed in reverse chronological order.

**Date:** April 17, 2015

**Speaker:** Dana C. Ernst (NAU)

**Guest:** Nellie Gopaul (NAU)

**Abstract:** The Prisoner’s Dilemma goes something like this. Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don’t have enough evidence to convict the pair on the principal charge. Simultaneously, the police offer each prisoner a bargain. If A and B both confess the crime, each of them serves 4 years in prison. If A confesses but B denies the crime, A will be set free whereas B will serve 5 years in prison (and vice versa). If A and B both deny the crime, both of them will serve 2 years in prison. In this talk, we will first discuss this classic game theoretic problem and then introduce an iterative version that consists of a round robin tournament of teams, where the winner is the team that spends the least amount of time in prison. [PDF of Flyer] [Slides]

**Date:** April 10, 2015

**Speaker:** Kevin Luna (undergraduate mathematics major at NAU)

**Guest:** Dr. Jim Swift (NAU)

**Abstract:** The Gradient Newton Galerkin Algorithm (GNGA) is a computational algorithm for finding numerical solutions to partial differential equations. The GNGA can be readily applied to the family of nonlinear elliptic PDEs of the form $\Delta u + su + f(u) = 0$ with Dirichlet boundary conditions on some piecewise smooth region that is a subset of $\mathbb{R}^n$, where $f(u)$ is the non-linearity of the PDE, and $s$ is a parameter. In this talk the GNGA will be discussed, and results from its application on the interval $[0, \pi]$ and the square region $[0, \pi] \times [0, \pi]$ will be examined when the non linearity is set to be $f(u) = u^3$. [PDF of Flyer]

**Date:** April 3, 2015

**Speaker:** Wilson Lough (undergraduate mathematics major at NAU)

**Abstract:** For any natural number $n$, finding all possible determinant values of $n\times n$ matrices with entries restricted to the set $\{1, -1\}$ is known as the determinant spectrum problem. For $n < 8$, the spectra consist of sets of consecutive integers in arithmetic progressions, but gaps appear in the spectra for $n\geq 8$. In this talk, I will present results involving the spectra of $n\times n$ matrices with entries restricted to the set $\{1, -1, i, -i\}$.

There is a “catch” to this talk: he/I/we are stuck on part of our “proof” that the $4\times 4$ spectrum for matrices with entries in $\{1, -1, i, -i\}$ is completely classified. We know what the spectrum is via brute force; but we are having a hard time verifying this algebraically. So, in lieu of a faculty interview, part of the talk will be an open Q&A, and hopefully someone in the audience can help us find a way to prove we are trying to prove. [PDF of Flyer]

**Date:** March 26, 2015

**Speaker:** Philip Doi (undergraduate mathematics major at NAU)

**Guests:** A subset of the 14 undergrad math majors who attended a conference in the past month (either SUnMaRC, which was at the University of Texas at El Paso or the CURM conference at BYU in Provo, Utah)

**Abstract:** In point-set topology, a Toronto Space is a topological space that is homeomorphic to every subspace of the same cardinality. The Toronto Problem asks if there is a non-discrete HausdorffToronto space of cardinality aleph-one. We will examine some of the literature known about Toronto spaces and will suppose the existence of non-discrete Hausdorff Toronto spaces, reviewing known results about these hypothetical spaces. Additionally, we will discuss the notion of logical independence for mathematical statements within the scope of axiomatic set theory; a notion that pertains to understanding the problem’s full nature. [PDF of Flyer]

**Date:** March 5, 2015

**Speaker:** Mel Theobald (undergraduate mathematics major at NAU)

**Guest:** Dr. Terry Blows (NAU)

**Abstract:** We consider $N$-dimensional, age-structured models of the normally 1-dimensional Beverton-Holt, Ricker, and Pennycuick population models. Our particular interest is in the impossibility of certain $p$-cycles in models of corresponding dimension. [PDF of Flyer]

**Date:** February 27, 2015

**Speaker:** Taryn Laird (NAU)

**Abstract:** It’s nice to change things up now and then, and with almost half of the regular FAMUS attendees being out of town for SUnMaRC, I thought this might be a good chance to dispense with the usual talk and faculty interview and, instead, show a movie! The movie is the well-known cartoon “Donald Duck in Mathmagic Land,” and FAMUS is being hosted by graduate student Taryn Laird. [PDF of Flyer]

**Date:** February 20, 2015

**Speakers:** Jordan Hunt and Zach Parker (undergraduate mathematics majors at NAU)

**Guest:** Matt Fahy (NAU)

**Abstract:** An amicable pair $(m,n)$ is a pair of two positive integers with the property that the proper divisors of $m$ sum to $n$ and the proper divisors of $n$ sum to $m$. The smallest amicable pair is $(220, 284)$. While there are many interesting questions one can ask about amicability (Do finitely many or infinitely many amicable pairs exist? Does there exist an amicable pair with $m$ and $n$ having opposite parity? Do there exist “amicable triples”? etc.), the focus of this talk is the search for amicable pairs in other algebraic structures that possess unique factorization, most notably the ring of Eisenstein integers. [PDF of Flyer]

**Date:** February 13, 2015

**Speaker:** Dana C. Ernst (NAU)

**Guest:** Dr. Roy St. Laurent (NAU)

**Abstract:** In the 1930s, the famous Hungarian mathematician, Paul Erdős, thought that for any infinite $\pm1$-sequence, it would always be possible to find a finite subsequence consisting of every $n$th term up to some point summing to a number larger than any you choose. But Erdős could not prove his conjecture, which he referred to as the “discrepancy problem.” About a year ago, Lisitsa and Konev published a proof that is a significant step towards proving Erdős’ problem. Their computer-assisted proof resulted in headlines such as “Wikipedia-size maths proof too big for humans to check” because their proof is as large as the entire content of Wikipedia, making it unlikely that it will ever be checked by a human being. In this episode of FAMUS, we will tinker with Erdős’ puzzle and attempt to wrap our heads around its difficulty. [PDF of Flyer] [Slides]

**Date:** February 6, 2015

**Speaker:** Jeff Rushall (NAU)

**Guest:** Mike Falk (NAU)

**Abstract:** For those of you who are unfamiliar, The Futility Closet is a website containing all sorts of interesting, bizarre and shocking facts in various categories, one being science and mathematics. I’ve compiled several dozen of the more amazing math facts and will sharing several of them in this talk: geometry, number theory, magic squares, and so on, including some rather mysterious open problems. Our fearless leader, Mike Falk, is serving as the faculty guest. [PDF of Flyer]

**Date:** January 30, 2015

**Speaker and Guest:** Ellie Kennedy (NAU)

**Abstract:** The talk this week is titled “A Report on the 2015 Nebraska Women’s Conference”, and is given by Ellie Kennedy, who is also the faculty guest. This conference, which targets undergraduate women who study mathematics, was held just last weekend. Ellie took several of our majors to the conference; among the student attendees, two presented a talk and two presented a poster, and they all have lots of pictures to show and stories to tell. [PDF of Flyer]

**Date:** January 23, 2015

**Speaker:** Jeff Rushall (NAU)

**Guest:** None this week

**Abstract:** The talk this week is titled “The Biggest Number in the Universe” and is given by Jeff Rushall. Believe it or not, some people HAVE crowned “the world’s biggest number,” and although not everyone agrees with their definition of “biggest” or with their choice of this “biggest number,” it all makes for a fun presentation. This biggest number, Graham’s Number, is the answer to a question in Ramsey Theory. Jeff will present background info and motivate the existence of this number, with can only be described using Knuth up-arrow notation. There will be no faculty guest this week. [PDF of Flyer]