**Pi Mu Epsilon Seminar**

**Ayla Gafni**

University of Mississippi

**Integer Partitions** (pdf)

An integer partition of n is a non-increasing sequence of positive integers whose sum is equal to n. For example there are 5 partitions of 4:

4, 3+1, 2+2, 2+1+1, 1+1+1+1

This simple definition is the seed of a deep mathematical theory that spans number theory, combinatorics, and analysis, dating back centuries. We will explore the patterns and relations that can be found among partitions and discuss some open problems that have mystified mathematicians for ￼generations.

**Jiranan Kerdboon and Casey Lochridge**

University of Mississippi

**Solving the 15-Puzzle** (pdf)

The 15-Puzzle is a sliding puzzle where the goal is to arrange the pieces in order from 1 to 15 with the empty space in the bottom right corner. In 1891, puzzle maker Sam Loyd offered $1000 to anyone who could solve the puzzle above with only pieces 14 and 15 switched. Nobody ever claimed the prize. We will discuss the reason that this challenge remains unsolved using modern algebra techniques and provide an algorithm to solve the puzzle given different arrangements of the pieces.

**Ken Ono**

University of Virginia and Emory University

**Prime Numbers** (pdf)

This informal discussion will be about one of the most important sequences of numbers {2, 3, 5, 7, 11, 13, 17, 19,….}, the set of prime numbers. Learn about classic works of Euclid and Gauss, and how their ideas have inspired the deepest works by contemporary mathematicians such as Ben Green, James Maynard, and Fields medalist Terry Tao. The deepest of these works is now so well understood that the proofs are accessible to advanced undergraduates. In fact, these results have been refined in various ways in the speaker’s REU summer programs.

**Thái Hoàng Lê**

Department of Mathematics, University of Mississippi

**Finding Hay in a Haystack** (pdf)

Existence results are ubiquitous in mathematics. Proving that something exists does not necessarily provide a method to find it. Stranger still, in many situations one can prove that a certain property is satisfied by many (or even most) objects, yet it is very difficult to find a single object satisfying the property. I will take you on a tour and show you examples of this nature in different areas of mathematics such as set theory, analysis, number theory and combinatorics.

**John Gardner**

Department of Economics, University of Mississippi

**Uncovering Hidden Variables** (pdf)

In latent-variable models, the distributions of observed random variables depend on underlying, unobserved variables. Because these models allow for the possibility that hidden forces shape the observable world, they have many applications in the social and physical sciences. I will discuss when, and how, the relationships between observed and unobserved random variables can be recovered from variation in observed variables. I will also introduce a computational technique for estimating latent-variable models from sample data.

**Sophia Fosdick**

Senior Mathematics Major, University of Mississippi

**A Career as an Actuary and the Path Towards Becoming One** (pdf) (Slides)

In this talk, I will describe the different areas of the actuarial field, comparing consulting and pure insurance, as well as discuss what a typical working day might look like. I will also provide information on the exams and other necessary steps towards attaining a career, including delving into internships, deadlines, and valuable skills for the profession.

**Thái Hoàng Lê**

University of Mississippi

**The Isoperimetric Problem** (pdf)

Among all closed, non-intersecting curves of the same length on the plane, which curve encloses the largest area? This question is known as the isoperimetric problem. While the answer, a circle, has been known since antiquity, this problem was only studied rigorously in the 19th century. I will talk about the history of this problem and its influence on modern mathematics.

**Sandra Spiroff**

University of Mississippi

**Automating the Hilbert-Kunz Multiplicity and F-Signature** (pdf)

We will discuss an automation of a calculation in Algebra arising from current research. The computation, which finds the volume of certain polytopes, is tedious and time-consuming to do by hand, but perfectly suited for computer programming. We begin by presenting the background of the automation and then see it in action. This work is part of a joint research project with Gabriel Johnson, who earned a BS in Computer Science through the Barksdale Honors College in 2018.

**Adam Jones**

University of Mississippi

**Seagulls and Solar Flares: Science, Data, and Data Science** (pdf)

Echoes, signals, and information—these things form our view of the world around us. In many ways the human experience is simultaneously a cacophony and symphony of data. What can this tell us about ourselves and the universe we live in? In this lighthearted talk, we will look at these and other topics related to both the science and philosophy of data.

**Xin Dang**

University of Mississippi

**Estimation of Pi by Statistical Methods** (pdf)

Pi is probably the most recognized mathematical constant in the world. In this talk, I will begin by discussing some interesting facts about Pi and reviewing the history of Pi approximation. Then, after a brief introduction of statistical concepts, I will present two statistical techniques for estimating Pi, called respectively the point and interval methods. To demonstrate these contrasting methods, I will show simulations generated through the statistical software R.

**David R. Bickerstaff**

Fellow of the Casualty Actuarial Society

￼Chairman of Bickerstaff, Whatley, Ryan & Burkhal

**Actuary 101: The Math of Pricing Unpredictable, Destructive Events** (pdf)

We know that unpredictable, financially-destructive events will occur… to someone… somewhere. At a fundamental level, insurance is a financial strategy by which a group of people can pool an individually small portion of their resources to redistribute them in the future to those in the group who become victims of costly misfortunate events. But how much should individuals in the group pay to ensure that this program is likely to be solvent? It is the work of an actuary to analyze real-world data and build mathematical models to attempt to address this type of question. I will discuss some basic principles and methods that guide actuaries in their work and suggest some curriculum directions that can lead mathematically-inclined students to employment in this field.

**Thái Hoàng Lê**

University of Mississippi

**The Banach-Tarski Paradox** (pdf)

In 1924 Banach and Tarski proved the following result: any solid ball in the 3- dimensional space can be cut into a finite number of pieces and these pieces can be reassembled to yield two identical copies of the original ball. Also, one can cut a solid ball and reassemble the (finitely many) pieces to obtain any other solid ball! In this talk, I will convince you (hopefully) that these paradoxes are not as strange as they sound, and I will show you some ideas behind them.

**Laura Sheppardson**

University of Mississippi

**When will I use this in real life?** (pdf)

What are you going to do with your math major? Math majors are in high demand in the job market. However, you’re unlikely to find a job ad for a mathematician. I’ll talk about opportunities beyond academia, what employers want, and how to get it. With this being the first meeting of the year, I’ll also share information on opportunities to watch for, including research, conferences, and competitions.

**Micah Milinovich**

University of Mississippi

**Solved and unsolved problems concerning prime numbers** (pdf)

An integer greater than one is said to be prime if its only positive divisors are one and itself. The distribution of the primes has fascinated mathematicians since antiquity. In this talk, I will outline some well known solved and unsolved problems concerning the prime numbers, trying to illustrate how modern number theorists think about them.

**Thái Hoàng Lê**

University of Mississippi

**Complete Order is Impossible** (pdf)

Any large enough system must contain a structured subsystem. This is the idea of Ramsey theory, a branch of combinatorics. For example, among any six people, we can find either three people who mutually know each other or three people who mutually do not know each other. In this talk, I will give a brief introduction to Ramsey theory, as well as its manifestations in other branches of mathematics such as number theory, analysis and logic.

**Rachna Prakash**

School of Accountancy, University of Mississippi

**Where Business Meets Mathematics** (pdf)

In this talk I will discuss various applications of math in business with a focus on how math is applied in the discipline of accountancy. Starting with a brief introduction on the historic links between accountancy and math, I will discuss how ubiquitous math is in different fields of business. I will end with an example of how math is used in accounting research with a brief discussion of my research on climate change.

**Xiaotu Ma**

St. Judes’ Hospital

**Genomics study of relapsed pediatric B-acute lymphoblastic leukemia** (pdf)

Genomics-based analysis is revolutionizing our understanding of cancers and holds the promise for understanding all diseases with a genetic-basis. In this talk, I will give a brief introduction on what we are looking for in a genomic study, from a perspective of variant detection and interpretation. I will then describe how we apply our techniques in a study of relapse ALL samples and stress the importance of detection of subtle signals from statistical perspectives.

**Welcoming meeting** (pdf)

We welcome you to the first Pi Mu Epsilon meeting of the 2016-17 school year! At this meeting, we will discuss the goals of Pi Mu Epsilon, future events, and how to join. To finish up, we will have graduate students from the math department answer questions about graduate school and how to apply to graduate programs.

**Maksym Derevyagin**

University of Mississippi

**Let’s Get Rational!** (pdf)

I am going to discuss continued fractions, which are expressions used to represent real numbers, and Diophantine approximations. The latter deals with problems such as whether a given real number is rational or irrational; if the number is irrational then the problem is to determine how well it can be approached by rational numbers.

**Gerard Buskes**

University of Mississippi

**From A (luminum Carbonate via kitchen cloth) to Z (ebra)** (pdf)

We will discuss 1) a mathematical object that connects the Musee du Conservatoire des Arts et Metiers in Paris with Oxford, Mississippi, 2) a mathematical object found in Dutch kitchens, and 3) an animal from Africa.

**Samuel Lisi**

University of Mississippi

**The Hairy Ball Theorem** (pdf)

Imagine a soccer ball entirely covered in hair. The Hairy Ball Theorem says that you can’t comb the hair flat everywhere — there will always have to be a tuft of hair sticking up somewhere. This famous theorem of topology was first stated by Poincaré (late 1800s) and proved by Brouwer in 1912. A different way of saying this is that if you consider the Earth and look at the windspeed at each point, there is at least one place where the wind is not blowing. This will lead us into an exploration of some basic topological properties of surfaces and their relationship to dynamical systems.

**James Reid**

University of Mississippi

**Mathematics and the Movies** (pdf)

We will view and discuss several short clips from movies that present mathematics in a correct, incorrect, and in a humorous fashion.

NOVA documentary “The Proof” (pdf)

**Jo Howard**

Special Guest (pdf)

**Jo Howard**

**Relating Mathematical Studies to Life Applications** (pdf)

Jo Howard graduated from the University of Mississippi with degrees in mathematics in 1967. She has since worked on applying mathematics to a number of fields including: designing trajectories for Boeing that put a man on the moon in 1969, programming the space simulator in Houston, Texas, and data management systems in the oil and gas industries.

**William Staton**

University of Mississippi

**Powerful Numbers** (pdf)

We will call a positive integer “powerful” if for every prime p, if p divides n, then p^2 divides n. That is, every prime in the factorization of n appears to at least the second power. We will discuss some algebraic properties of the set of powerful numbers to emphasize an important property of primes. Then we will show that there are infinitely many pairs of consecutive powerful numbers, such as 8 and 9.

**Eleanor Anthony**

University of Mississippi

**PME Induction and Student Talk** (pdf)

Come and join us as we welcome new members to the Pi Mu Epsilon Honor Society and elect officers for 2015. Eleanor Anthony will give a short talk on her undergraduate mathematics research.

**Samuel Lisi**

University of Mississippi

**Morse Theory on Surfaces** (pdf)

Have you ever wondered how many holes your doughnut has? We will see how calculus can tell us something about the topology of a surface.

**Joshua Adam Gray**

University of Mississippi

**Metadata, Social Networks, and Paul Revere** (pdf)

Dr. Kieran Healy, a sociologist at Duke University, wrote an article last summer discussing how “metadata” from colonial America could have been used to identify Paul Revere as a key figure of the American Revolution before it began. Dr. Gray will discuss the graph theory underlying some of these ideas and how it pertains to modern social networks.

**Sam Watson**

Massachusetts Institute of Technology

**Schramm-Loewner Evolutions** (pdf)

Dr. Watson will explain in non-technical terms why the study of “conformally invariant two-dimensional statistical physics models” is compelling and beautiful.

**Amanda Hall**

University of Mississippi

**The Tutte Polynomial Formula for the Class of Twisted Wheel Graphs** (pdf)

Amanda will talk about her SMBHC Senior Thesis this Thursday. Come to here about her research and the research process.

**James Reid**

University of Mississippi

**A Beautiful Mind, Equilibria, and Traffic** (pdf)

We will explore the life and works of John Nash, the subject of the popular movie *A Beautiful Mind*. In particular, we will discuss the concept of Nash-Equilibria and its implications to the flow of vehicular traffic.

**Organizational Meeting:** Induction of New Members and Officer Elections (pdf)

**Caroline Turnage-Butterbaugh**

University of Mississippi

**Gaps Between Primes** (pdf)

Caroline will talk about some recent exciting results in number theory, starting with Yitang Zhang’s theorem on prime gaps, the Polymath project that grew from it, and the Maynard-Tao result that improved upon this work. She will also talk about a conjecture of Erdös and Turán that she proved recently as joint work with William Banks and Tristan Freiberg from the University of Missouri.

**Sooyeon Lee, Catherine Putnam, Stephan Roberts, Christopher Schwanke, Caroline Turnage-Butterbaugh**

University of Mississippi

**Graduate Studies in Mathematics** (pdf)

Are you considering going to graduate school in mathematics? Do you wonder what graduate school in mathematics is like? Do you have questions about applying to graduate schools? If you’ve answered yes to any of these questions, come ask a panel of five of the math department’s graduate students anything you’d like to ask about graduate school!

**Sandra Spiroff, Jonathan Broom, Travis Dirle, Adam Flaherty, and Thomas Naugle**

University of Mississippi

**“There’s ca$h in math”-an overview of the SMILE program at LSU** (pdf)

Come hear about the summer SMILE program at Louisiana State University and how you can earn a stipend while learning mathematics, for example, Jordan decomposition and the Z-transform, or associating graphs to commutative rings. An overview of the program will be given by Dr. Spiroff, followed by the students briefly describing their summer projects.

**Sam Cole**

University of Mississippi

**“The Hardest Logic Puzzle Ever” and Logical Connectives** (pdf)

In 1986 logician George Boolos published “The Hardest Logic Puzzle Ever”, a variation on the classic “Knights and Knaves” puzzle. Boolos’ puzzle is as follows: “Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.” In order solve this puzzle the logical connectives encountered in mathematics are needed to construct questions. The properties of these connectives and the solution to many variants of this problem will be presented.

**Przemo Kranz**

University of Mississippi

**Octonions-or how many square roots of -1 are there?** (pdf)

As complex numbers are constructed out of real numbers, so quaternions are constructed out of complex numbers. Quaternions, in turn, are the building blocks of a still more “complex” set of objectsoctonions. Octonions form an eight dimensional normed division algebra, a basis of which contains seven elements whose squares are all -1.

**Erwin Miña-Diaz**

University of Mississippi

**Inversion in Planar Geometry** (pdf)

Inversion, or reflection about circles, is a transformation that generalizes the idea of reflection (mirror image) about a line. Inversion has several beautiful properties. For instance, lines and circles are transformed into lines and circles as well, and angles between curves are preserved under inversion. In this talk, we shall introduce this concept, discuss some of its most basic properties, and apply it to solve some problems in planar geometry.

**Brad Cole**

University of Mississippi

**Drunk Cops and Robbers: How To Catch A Thief… Slowly** (pdf)

The game of Cops and Robbers can be played out, in a mathematical sense, on a graph—i.e., on a set of vertices and edges, like a square, star, hexagon, or other figure. In the game, the players designated as “cop” and “robber” take turns, respectively, moving from vertex to vertex along graph edges. Should the players ever occupy the same vertex at the same time, the robber is captured and the cop wins. The theory behind these graph pursuit-evasion games has been well studied. In 2011, a version of this game was investigated where the robber is moving randomly; a “drunk” robber. This talk will explore the reversed role: the cop is now “drunk” while the robber moves perfectly. The expected capture time will be demonstrated for the simplest 1-cop win graphs (paths), 2-cop win graphs (cycles) and, time permitting the k-d-regular tree.

If this talk were a movie, then the plot would be: How would the movie Taken (or Taken 2, the sequel) work out if Liam Neeson’s character had been drinking constantly?

**Christopher Schwanke, Daniel Chausse, Stephan Roberts, Michael Azlin, Caroline Turnage-Butterbaugh, Lisa Ewell, and Lynsey Cargile**

University of Mississippi

**Hilbert’s Hotel: A Play** (pdf)

Hilbert’s Hotel is very different than other hotels, for it has infinitely many rooms! Come see the very strange things that can occur when dealing with an infinite number of objects. Things like the sign in front of the hotel, which reads: “No Vacancy – Rooms Available!”

** Sandra Spiroff, Jonathan Broom, Ben Moore, and Lindsay Pittman**

University of Mississippi

**“I Know What You Did Last Summer” – an overview of the SMILE (Summer Math Integrated Learning Environment) program at LSU** (pdf)

Are you interested in earning a summer stipend while learning some new and interesting mathematics? Then this meeting is for YOU. We will be highlighting the SMILE program at Louisiana State University and how you can apply for summer 2013. In particular, three Ole Miss students will give a brief discussion of their projects and experiences in Baton Rouge.

**William Staton**

University of Mississippi

**Gaussian Integers and Rational Tangents of Rational Angles** (pdf)

**Caroline Turnage-Butterbaugh, Chris Schwanke, Kai Yu**

University of Mississippi

**Panel of Graduate Students to Answer Questions about Graduate School in Mathematics**

**John Dever**

University of Mississippi

**Infinitesimals, Differentials, and Derivatives in Normed Vector Spaces** (pdf)

By establishing properties of a certain class of functions, the infinitesimals, between normed vector spaces, the concepts of the differential and derivative are easily generalized to normed vector spaces. The relationship between differentials and directional derivatives will be discussed and properties such as the chain rule will be established.

**John Estes**

University of Mississippi

**Paths, Independence, and Fibonacci**

**Sandra Spiroff, Daniel Chausse, Tanya Dewland, W. Chapman Smith**

University of Mississippi

**I Know What You Did Last Summer** (pdf)

Come hear about the summer SMILE program at Louisiana State University and how you can earn a stipend while learning mathematics, for example, geometric constructions or the spread of HIV. An overview of the program will be given by Dr. Spiroff, followed by the students briefly describing their projects.

**David Benko**

University of South Alabama

**eBay and the art of bidding**

Bidding on eBay is easy – no math is involved, right? We will disprove the claim by demonstrating a scientific way of bidding. We will also offer some ideas that should be implemented at eBay to make it a better auction marketplace. Finally, we propose a fair system which is somewhere between socialism and capitalism. Whether you’re a buyer or seller or president of eBay, you must see this talk!

**Zhu Cao**

University of Mississippi

**Elementary Theory of Partitions**

**William Staton**

University of Mississippi

**A 15-minute Introduction to Extremal Graph Theory**

**Vlad Timofte**

University of Mississippi

**Repeating Atomic Configurations You May Not Be Unique!**

**Xin Dang**

University of Mississippi

**PageRank — secret of Google’ success**

**Erwin Mina-Diaz**

University of Mississippi

**Kinematics Method in Planar Geometry** (pdf)

From the study of Physics and Calculus, undergraduate students are quite familiar with Kinematics,”the theory of vectors and velocities”. In this talk we shall discuss how the concept of velocity and its properties can be used to solve some purely mathematical problems in planar geometry.

**Ryan Daileda**

Trinity University

**Factorization Beyond the Integers** (pdf)

We all know that any integer can be factored uniquely into primes. What if we look at other sets which have the same axiomatic properties as the integers? Does an analogous property hold? It turns out that the answer is ‘no,’ and we will explore how badly this can fail.

**James Reid**

University of Mississippi

**Durer’s Melencolia and Spherical Polyhedra** (pdf)

Albrecht Durer (1471-1528) is regarded by some as the preeminent artist of the Renaissance in Northern Europe. Many different interpretations of the symbols included in his engraving the “Melencolia” have been given. There have been numerous interpretations of the shape of the stone block given on the left side of the figure. We investigate whether this block can be the model of a spherical polyhedron.

**William Staton**

University of Mississippi

**Generating Functions and Integer Partitions**

**Sandra Spiroff**

University of Mississippi

**Unique Factorization and a Connection to Gambling**

**John Conlon**

Department of Economics, University of Mississippi

**Fool Asset Price Bubbles with Rational Economic Agents**