Academic Year 2018  2019
Spring 2019
Cool Mathematics
Abstract: I will in fact talk about some unsolved, or recently solved, problems in Mathematics. The main purpose of the meeting, however, will be to organize the Undergraduate Colloquium for those interested in taking it for credit (1 hour credit/no credit). For many participants this will be the first class in which they have to write a technical report. This is a complicated yet gratifying task. I will discuss some of the issues involved and also give a first introduction to the use of LaTeX.
Area of Lattice Point Polygons
Abstract: A wonderful theorem about a planar polygon whose vertices have interger coordinates was discopvered by Georg Pick in 1899. The area is given by the number of integer lattice points inside the polygon plus half the number of lattice points on the boundary minus one. An inductive proof will be given based on combining the formulas of two smaller polygons glued along a common boundary. The impossibility of a similar formula in higher dimensions will also be discussed.
The Mathematics of Rainbows
Are rainbows bridges to heaven? Or bows to shoot arrows of lightning? Is there really a pot of gold at the rainbow's end? "Where does the rainbow end, in your soul or on the horizon?" (Pablo Neruda).
In this talk we will try to answer these questions (and many more) on the naure of rainbows by using many resources of geometrical optics.
Gems from the Secondary Classroom
Abstact: The modern secondary (grades 712) mathematics classroom embraces the practice of utilizing multiple representations of a single concept to challenge and deepen student understanding, as well as foster creativity. Too often, humans confound their conceptual understanding with a particular convention or procedure. Challenging students' apparent understanding via unfamiliar representations exposes latent misunderstandings among excelling students, while offering cleanslate access points to struggling students. In this talk, we explore a few new approaches to old ideas from your life as a secondary math student. Come prepared for engagement and surprise!
A Fractional Introduction to Fractals
Abstract: How long is the coast of Great Britain? This question, although at first seemingly simple, has a peculiar answer when approached theoretically from different length scales due to the fractaltype nature of coastlines. In this talk we will introduce the notion of fractals, their unique mathematical properties, and their prevalence in nature, as well as view examples of such strange and fascinating objects.
Numbers in other bases
Abstract: When we represent a number say 31.235 we usually do so in base 10 and our representation refers to the fact that this number is 3 times 10 plus 1 times 1 plus 2 times 1 tenth plus 3 times one onehundredth. One can represent numbers in bases of natural numbers and even in bases for numbers greater than one that are not necessarily natural numbers. This talk will delve into this interesting world talking about how to define such expansions and issues of uniqueness.
Hotel Infinity Revisited
Abstract: Consider a hotel with an infinite number of rooms and no vacancies. If a guest arrives, is it possible to find a room for them? If an infinite number of infinite families arrive, what then? This problem is not just an interesting exercise, as it has deep connections to the ideas of analysis. In this talk we will explore these connections, as well as ways to expand on the idea of the infinite hotel.
Instability in Aesthetics : the fluid dynamics of painting
Abstract: Mathematicians studying fluid motion and painters have more in common than you think. Unstable flows create some of the most visually interesting images such as the iconic dripping fluid threads of Jackson Pollock. Similarly, a well known Mexican muralist, David Alfaro Siqueiros invented an accidental painting technique to create new textures. Why do these patterns occur? What is the effect of the paint's density? We will explore the experimental setup of pouring layers of paint of different densities on a horizontal surface and investigate how the RayleighTaylor instability drives pattern formation from this dual viscous layer.
No Talk  Spring Break
AWM; Applying for Math Grad School
Abstract: Maybe you are curious about what grad school is, maybe you're considering to apply someday, or maybe you've looked into it and it seems too overwhelming. If you've been in any of these situations, this workshop is for you! We will provide information about the path to grad school. We will talk about what is needed and what to expect during the application process. We will also have Q&A with professor that have been involved in the admissions process.
Board Games as Random Walks on Graphs
Abstract: A directed graph is a set of vertices and directed edges (arrows pointing from one vertex to another). We can assign a probability to each edge and imagine randomly moving from vertex to vertex along these edges (a 'random walk'). We will describe this mathematically and show how it applies to the analysis of board games. We'll show how this can improve your Monopoly strategy by determining the most likelytobelandedupon properties. We'll also discuss how to answer the question I ask myself whenever I play my threeyear old in Hi Ho CherryO: ``how much longer can this game possibly go on?" A little familiarity with basic probability and linear algebra is all that is necessary to understand this discussion.
Isolation as a means of epidemic control
Abstract: There are few options in the face of an unforeseen epidemic outbreak; isolation is one of them. When implemented in full and without delay, isolation is very effective. Flawless implementation, however, is seldom feasible in practice. I will present in this talk a simple epidemic model called "SIQ" with an isolation protocol, focusing on the consequences of delays and incomplete identification of infected hosts. The continuum limit of this model is a system of Delay Differential Equations, the analysis of which reveals clearly the dependence of epidemic evolution on model parameters such as disease reproductive number, probability and speed of identification of infected hosts, recovery rates and duration of immunity. Our model offers estimates on critical response capabilities needed to curb outbreaks, and predictions of endemic states when containment fails.
The Sperner Simplex
Abstract: Frank Stenger's talk is about the Sperner simplex, and how it is connected with the Brower and other fixed point theorems. He then also describes his combinatorial method of computing this simplex, and he presents explicit two dimensional examples. Students are encouraged to review properties of determinants for this talk.
Fall 2018
Cool Mathematics
Abstract: I will in fact talk about some unsolved, or recently solved, problems in Mathematics. The main purpose of the meeting, however, will be to organize the Undergraduate Colloquium for those interested in taking it for credit (1 hour credit/no credit). For many participants this will be the first class in which they have to write a technical report. This is a complicated yet gratifying task. I will discuss some of the issues involved and also give a first introduction to the use of LaTeX.
The King Chicken Theorem
Abstract: Consider a flock 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 flock, either pecks that chicken, 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.
Foraging in an Uncertain World
Abstract: Suppose you are an insect, foraging for sustenance across patches of plants. When your random search lands you on a plant, you must make some decisions. How long should you visit this specific plant? And when you leave, should you stay close or travel to a new patch that is less depleted? We will use the Marginal Value Theorem to find your optimal strategy in a world where you have total information. Then we will introduce uncertainty in the number and types of plants to see how this changes your strategy. We will also consider the plants' perspective, to see how their neighbors help or hurt them based on your foraging behavior.
The Porisms of Steiner and Poncelet
Abstract: Draw two circles, one inside the other. Starting anywhere
between the two circles we can draw a chain of circles tangent to both. In
some cases the chain will close up on itself. Steiner's remarkable porism
states that whether the chain closes does not depend on where we start. We
will prove this during the talk, and discuss another such result due to
Poncelet if time permits.
How Do Bacteria Talk? Understanding Bacterial Quorum Sensing by Mathematical Modeling
Abstract: Quorum sensing is a bacterial communication mechanism that uses signalreceptor binding to regulate gene expression based on cell density, resulting in group behaviors such as biofilm formation and bioluminescence. In certain bacterial species such as Vibrio harveyi, several parallel quorum sensing signaling pathways drive a single phosphorylationdephosphorylation cycle, which in turn regulates quorum sensing target genes. Through mathematical modeling, we investigate how signal integration is done at single cell levels and how bacteria use quorum sensing to measure its social and physical environment.
On the Labouchere Betting System
Fractal Patterns, Game Theory and Bali's Rice Terraces
Abstract: Bali's famous rice terraces, when seen from above, look like colorful mosaics because some farmers plant synchronously, while others plant at different times. The resulting fractal patterns are rare for humanmade systems and lead to optimal harvests without global planning. We'll describe a model of this system, introducing some notions from game theory, to help understand this peculiar example.
Abstract: A group of prisoners is given an opportunity to play a game for their freedom. Each prisoner has a hat, either white or black (both equally likely), simultaneously placed on their heads. Absolutely no communication between prisoners is possible after this point. Prisoners cannot see their own hat, but they can see the hats of the others. The warden then rings a bell, and the prisoners must all simultaneously guess a color (white or black) or pass. They all win their freedom when at least one prisoner guesses the color of their own hat without any incorrect guesses being made. If an incorrect guess is made (or if everyone passes), they are all marched back to prison.
The prisoners may work together to devise a strategy before the game begins. What is an optimal strategy for the prisoners to secure their freedom? (Hint: if 45,000 prisoners fill RiceEccles stadium, there is a strategy that guarantees their freedom 99.996948242% percent of the time.)
The BehrensFisher Problem
Abstract: "Which of the two populations is bigger on average?" This is an often asked question in statistics. An often provided answer is by way of a two sample ttest assuming the populations are approximately normally distributed, with about the same degree of variability. However, what if the populations have different degrees of variability? It is not clear what the appropriate test should be! This is the classic BehrensFisher problem.
We will discuss some solutions to particular avatars of this general problem.
Me, You and Math
Abstract: How does learning mathematics in classrooms differ from the way mathematicians do mathematics (or does it)? Who does mathematics and why? What does the educational research say about the value of discussing mathematics? How can we create more inclusive classrooms and dispel the genius myth that lingers within the mathematics community? Does the structure of math classes impact student success? In this talk, we'll explore these questions and possibly challenge our own ideas about our individual roles within our mathematics community.
No talk, Thanksgiving Break.
The Mathematics of Rainbows
Abstract: Are rainbows bridges to heaven? Or bows to shoot arrows of lightning? Is there really a pot of gold at the rainbow’s end? “Where does the rainbow end, in your soul or on the horizon?” (Pablo Neruda)
In this talk we will try to answer these questions (and many more) on the nature of rainbows by using many resources of mathematical physics.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.
Academic Year 2017  2018
Spring 2018
Cool Mathematics
Abstract: I will in fact talk about some unsolved, or recently solved, problems in Mathematics. The main purpose of the meeting, however, will be to organize the Undergraduate Colloquium for those interested in taking it for credit (1 hour credit/no credit). For many participants this will be the first class in which they have to write a technical report. This is a complicated yet gratifying task. I will discuss some of the issues involved and also give a first introduction to the use of LaTeX.
What's the deal with being a high school teacher?
Abstract: Are you thinking about maybe teaching middle or high school math someday? Maybe as a career or even as a bridge to graduate school or some other career? Perhaps you’ve thought about it as a profession after you retire from something more lucrative? There is a shortage of math teachers in Utah (and across the country); teaching is incredibly rewarding, but also challenging because there is a lot to know to be successful in a classroom. This colloquium will focus on three questions: 1) What is mathematical knowledge for teaching? 2) What’s in the “Common Core”? and, 3) What are the paths into teaching secondary math?
Tim Jones, Ph.D.
The Joy of Teaching Kids Math
Abstract: My career has taken me from particle physics at CERN to defense work at Lockheed Martin to start ups in Silicon Valley. I have worked with tremendous people on some amazing projects – but nothing has been more rewarding to me than the past decade I have spent teaching math to secondary kids (7^{ }– 12). When we teach math at our school, we teach kids that math is hard – it is not builtin and requires a significant effort on the part of the student to learn math. We teach kids how to do math – that it is not guesswork. Math, to us, is a language – a language that is used to communicate ideas and to communicate logical reasoning. Not surprisingly, our algebra standards are therefore quite high. We do not use gimmicks to teach math – we do not need to. Students get the same intrinsic rewards as scientists, engineers, and mathematicians when they can do math – when they can solve problems. When students are taught at a level that is appropriate to them as an individual, they can learn math efficiently. All students – independent of socioeconomic background can be successful – when given a chance. Teaching math requires teachers that have a drive to change the world – a willingness to invest in kids – a willingness to do the right thing. Math teachers, however, must be smart in math. Not x + 3 = 5 smart, but Fourier Series smart.
If you would like to talk to Tim about trying out teaching, you can contact him at tjones@apamail.org
Sequences and Differential Equation
Abstract: You learned arithmetic sequences and geometric sequence in an elementary algebra class. In this talk, I will talk about another type of sequence which is little bit more complex than arithmetic/geometric sequences. We will find its nth term by solving a corresponding differential equation. That means we will use differentiation, integration, taylor expansion, linear differential equation etc, to understand the sequence.
Cupid's Identity:
Valentine's Day = pi Day  One Month = e Day + One Week
Abstract: On this day of love, can we quell the recent antagonism between pi and e, as documented in an infamous recent debate at Williams College? Let's meditate away the irrational differences (and sums and products) between these transcendental numbers and instead toast the innumerable occasions on which they work together in harmony.
Rebecca Hardenbrook
A Knotvice’s Guide to Untangling Knot Theory
Abstract: Consider a single piece of string. Now, knot it using a variety of twists and turns, finishing by combining the two ends together so that the string is one continuous loop. Is it possible to untangle this into the trivial knot? Although it may seem impossible to tell for the most complicated of knots, much work has been done over the past century to determine such an answer. This work has led to the development of a new field in mathematics: knot theory. In this talk, we will explore the history and basics of knot theory and, hopefully, won’t find ourselves too tangled up along the way.
Spot the Math!
Abstract: Spot It! is a pattern recognition game. There is a deck of 55 cards. Each card shows eight different symbols. Any two cards have exactly one symbol in common. I will explain the rules of the basic version of the game during the talk, then I will answer the following two questions:
1) How can we generate such a deck?
2) In particular, what's the total number of symbols in the whole deck?
Both questions turn out to be quite difficult. However, they become simpler if we use a little bit of geometry. The only prerequisite for this lecture is to know that the equation of a line is y=a+bx.
Yuri Tschinkel, New York University
Rational points
Abstract: The Gauss circle problem asks about the number of vectors with integer coefficients in a circle of growing radius. More generally, one can consider this problem in spaces of higher dimension, and impose polynomial conditions of the vectors. In many cases, this problem can be solved, applying a range of tools from harmonic analysis and analytic number theory. I will discuss a general conjecture concerning the distribution of lattice points satisfying polynomial conditions, in spheres of growing radius, as well as some recent results establishing this conjecture.
Nicholas Cahill
Bias and Threat: Understanding Sexism in STEM
Abstract: One of the largest ongoing projects in American education has been the attempt to understand the role sexism plays in gaps in performance and achievement between men and women in the sciences. While it may at first have seemed like a simple attitude problem, the gap has proven to be a complicated challenge, with many subtle and less subtle factors at play in the minds of educators and students, and in the broader environment where learning takes place. As the tenacity of this problem has become clear, it is more important now than ever for mathematicians to be understand the threat! This talk will be a brief overview of some of the most important concepts we use to understand what sexism is and how it operates in the sciences.
The Infinite Monkey Theorem
Abstract: Would you be surprised to learn that, given an infinite amount of time, a monkey sitting at a typewriter hitting keys at random will almost surely type the complete works of William Shakespeare (or any other given string of letters)? Maybe not, since after all a lot can happen in an infinite amount of time. But how long would you have to wait? Surprisingly, low long you'd have to wait depends on more than just the length of the string. We'll discuss and prove these facts. No monkeys or advanced mathematics required, just some basic probability (which I will review).
Anna Nelson
On the rheology of cats: are cats fluid?
Abstract: In this talk we will determine whether the claim that cats are liquid is solid! Using principles of rheology and fluid mechanics, we will study the flow and deformation of cats in different time scales. No prior knowledge of fluid mechanics is required!
The scallop theorem
Abstract: Bacteria and other small swimming organisms can't coast. Once they stop moving their flagella, they come to a complete stop instantaneously. They live in a world where inertia doesn't matter. We, on the other hand, live and swim in an inertia dominated world. The scallop theorem is a beautiful and simple geometric argument explaining why if inertia does not matter, you can't swim with a single hinge or via reciprocal motion. While the scallop theorem tells us what does not work, it does not explain the rich swimming behavior observed in nature. To answer this question, applied mathematicians use a combination of model, analysis and simulation starting from the famous NavierStokes equations.Stein's Paradox and Shrinkage
Abstract: Suppose we want to compute the batting average of Adam Eaton. Then, our natural estimator would be Hits/At Bats. Now suppose, we want to compute a batting average for each of the top 10 MLB players. Our natural estimator is no longer the best choice. In fact, the moment you are interested in 3 or more players, you are better off not using the natural estimator.
Charles Stein pointed out this paradox much to the horror of the statistical community in 1955. Today, a host of shrinkage estimators owe their origins to his result.
Fall 2017
No Talk  First Week of Classes
Cool Mathematics
Abstract: I will in fact talk about some unsolved, or recently solved, problems in Mathematics. The main purpose of the meeting, however, will be to organize the Undergraduate Colloquium for those interested in taking it for credit (1 hour credit/no credit). For many participants this will be the first class in which they have to write a technical report. This is a complicated yet gratifying task. I will discuss some of the issues involved and also give a first introduction to the use of LaTeX.
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.
Panel of Faculty, Postdocs, and Graduate Students
Applying for and attending graduate school
Abstract: This weeks Undergraduate Colloquium will be aimed at helping undergraduates answer the following questions:

 Should I apply to graduate school?
 How do I apply to graduate school?
 What will it be like when I'm in graduate school?
Pseudorandom numbers: {mostly} a line {of code} at a time
Abstract: Random numbers have an amazing range of application in both theory and practice. Approximatelyrandom numbers generated on a computer are called pseudorandom. This talk discusses how one generates and tests such numbers, and shows how this study is related to important mathematics and statistics  the CentralLimit 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.
The Angel Problem and other Games of No Chance
Abstract: I'll discuss John Conway's "Angel Problem," a simple sounding game for two players (the angel and the devil). The game was introduced in 1982, and the problem of finding a winning strategy remained unsolved until 2005. We'll talk about several variants of the game, and how these can be solved. Finally, we'll discuss one or two other games that fall under the umbrella of combinatorial game theory.
Casey Johnson (Math Department Alumnus)
Student Opportunities at Department of Defense (with a little bit of math)
Abstract: We discuss at a high level a variety of securityrelated positions within the U.S. Government that may be of interest to mathematicians. This will include professional opportunities, as well as internships available to undergraduates. As time permits, we will turn our attention to several related algebra problems and show how they can be combined to construct a primitive computer.
Complex and Tropical Nullstellensatze
Abstract: If a system of linear equations has no common solution, then by eliminating variables, one can obtain 1 = 0 as a linear combination of the given equations. Hilbert’s Nullstellensatz or “zero set theorem” says that the same thing happens with polynomial equations provided that one allows for complex solutions (and polynomial combinations). In the world of tropical numbers we consider polynomial relations and then there is a result of the same nature. We’ll talk about this and also about why the Nullstellensatz is the foundation of Algebraic Geometry.
Big Data Employer Panel
Location  LCB Loft (4th floor)
Abstract: Want to know what it's really like to work in a job relating to math, data, and statistics? Do you have questions about what it takes to land a position in "the real world" or what classes you should focus on now for your future career success? Hear from former students and get your questions answered! Food and networking to follow the panel.
What can you do with a slide rule?
I'll describe how slide rules work, why they work, and what you can do with them. A typical slide rule has anywhere from ten to thirty scales, rather than just two, and there are thousands of mathematical expressions that you can evaluate just as easily as you can multiply or divide two numbers. On the other hand, you can't use a slide rule to add or subtract two numbers, and you need to understand your problem well enough to be able to figure out on your own the location of the decimal point in your answer.
You'll be able to examine several slide rules, and I'll tell you what's involved in being a slide rule collector.
Desargues's theorem
Abstract: Desargues's theorem, published in 1648, is a result in geometry about triangles in perspective. Desargues was one of the fathers of projective geometry, and his theorem is best understood in that setting. We'll talk about Desargue's theorem, projective space, and how to prove the theorem.
Representation Theory and the Hydrogen Atom
Abstract: This talk is an advertisement for a new course that will be running in Spring 2018 called Representation Theory Techniques in Quantum Physics. The course will tell the story of representation theory's predictive role in quantum mechanics by closely examining the most basic example: the hydrogen atom. This talk will start by examining the history of representation theory and exploring how representation theory found its way to the forefront of 20th century mathematics and physics. Then it will give a preview of the material that we will cover in Math 5750. The talk will assume a background in linear algebra, but no background in group theory or quantum mechanics is necessary.
No Talk  Happy Thanksgiving
There will be an optional meeting for any students enrolled in Math 3000 who want to talk about the required report due at the end of the semester.
Waves in strings
Abstract: In this talk, we will study the problem of a vibrating string. How does a wave propagate in a homogeneous string when the string is plucked? This is a standard problem in mechanics and it has a wellknown solution. So we will change some of the basic assumptions to turn it into a very challenging problem. First, we will see how the wave propagates when the string is made of two different materials. The key point is to understand what happens at the spaceinterface between the two materials. Then we will try to solve the following problem: what if the properties of the string suddenly change in time? In other words, what happens if a string made of a certain material, suddenly turns into a string made of another material? What happens at the timeinterface? How is the propagation of the wave affected by this timeinhomogeneity?
Why can you tune a guitar, but not a piano?
Abstract: Musicians rely on being able to tune their instruments to create music that sounds pleasing and harmonious. In this talk, we'll discuss the mathematics behind the music created by vibrating strings and how guitarists use these frequencies, called "harmonics," to get in tune. Exploring this topic further leads to a surprising and unsettling result: it is impossible to tune a piano! Once we prove this, I'll explain what strategy piano tuners use to get around it. Partial differential equations and boundary conditions will make an appearance, but no knowledge of differential equations or music is required.
Past Colloquia
(links to Fall 1999  Spring 2017)