# Undergraduate Colloquium

**Fall 2014**

**Wednesdays 12:55 - 1:45**

**LCB 225**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- August 27 No Talk

- September 3 Ivan Sudakov
**Fractals in Nature**

*Abstract:*We discuss the mathematical concept of fractals and how fractals relate to issues within the Earth’s climate system. A fractal is any equation or pattern, that when visualized, produces a similar image when viewed at any spatial scale. We describe how fractal-like patterns occur widely in nature and introduce the fractal dimension, a number which describes the fractal-like nature of an object. Specifically, we will highlight the ubiquitous nature of fractals within many facets of the climate system. We provide a real example of fractals in nature and involve our students to solving a few entertaining problems.- September 10 Andrejs Treibergs
**Mapping the Earth**

*Abstract:*Can regions on the surface of the earth be mapped to the plane in such a way as to preserve areas, angles and lengths, i.e., by local isometries? We develop notions from the differential geometry of the sphere such as curvilinear coordinates, metric and length. We show that local isometries from the sphere to the Euclidean plane are impossible because if they existed, a natural compatibility condition on the metric would have to be satisfied: the curvature would have to vanish. We give examples of maps that preserve areas but not angles such as the one discovered by Lambert, and examples of maps that preserve angles but not areas such as the one first drawn by Mercator. We consider a notion of map distortion and discuss the theorem of Milnor, that for maps of spherical caps, the least distortion occurs for the azimuthal equidistant projection.

Lecture slides are available: http://www.math.utah.edu/~treiberg/MappingtheEarthSlides.pdf- September 17 Kenneth Bromberg - Director of Graduate Studies
**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?

- September 24 Peter Trapa
**The Hat Problem**

*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), placed on their heads. They cannot see their own hat, but can see the hats of the others. They win when at least one prisoner guesses the color of his hat without any incorrect guesses being made. The prisoners may work together to devise a strategy before the game begins, but cannot communicate once the game starts. What is an optimal strategy for the prisoners to secure their freedom?- October 1 Kelly MacArthur
**Equivalent Inequalities**

*Abstract:*Start with a < b . Does this necessarily mean f(a) < f(b)? What about g(a) < g(b)? Or is h(a) < h(b) true? It's likely that you learned to “flip” the sign in an inequality when you multiply or divide both sides of an inequality by a negative number. Perhaps you were even left believing that's the only operation applied to both sides of an inequality that requires switching the sign. But, is that really the only time we need to switch the sign? What is the underlying mathematical reason that we switch the sign anyway? And, is there a mathematical operator that we could apply to both sides of the inequality where we can't determine whether to switch the sign or not? Are there operators that we are allowed to “do to both sides” in an equation that are off limits in inequalities? We'll explore these questions and uncover the answers together. Be prepared to think and work!- October 8 Aaron Bertram
**Lines in Space**

*Abstract:*If you choose four lines in space “at random,” there are exactly two other lines that meet all four of them. Why is that? There is an easy answer, and an interesting answer. We’ll spend a few minutes on the easy answer and then look for coordinates for the manifold that parametrizes all the lines in space. This will allow us to “see” the space of lines as (most of) the zero locus of a quadratic polynomial in five variables. We can also “build” the space of lines and “complete” it, with some thought and some simple Young diagrams.- October 15 No Talk - Fall Break

- October 22 Vira Babenko
**When Zombies attack, or Mathematical model of doomsday scenario**

*Abstract:*Halloween is just around the corner, and what can be a better way to welcome it than to build a math model of a zombie invasion? They are usually portrayed in movies as being brought about through an outbreak or epidemic. We will discuss how we can model a zombie attack, using biological assumptions based on popular zombie movies. Then we will discuss model for zombie infection, determine equilibria and their stability. After we will tune the model to introduce a latent period of zombification. Also, we will then modify the model to include the effects of possible quarantine or a cure. Finally, we will show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.- October 29 Ben Trahan (Math Department Alumnus, NSA)
**Life Outside Academia**

*Abstract:*Many mathematicians who leave academia find themselves labelled as "Data Scientists" and are told to do things like "Machine Learning". In this talk I will try to define at least one of those terms, and then explain the role that mathematicians and computer scientists have had in attacking some of the most exciting new problems in humanities research. The talk will focus on a particular algorithm, Latent Semantic Analysis, which is useful for studying broad trends in large text corpora -- for instance, finding topics that span all of English Victorian literature.

Afterward there will be a discussion of job opportunities at the National Security Agency.

- *Location Change to LCB 215 (this talk only!)

- November 5 Sean McAfee
**Does Your Vote Count?: Voting, Game Theory, and the Shapley-Shubik Power Index**

*Abstract:*Voting comes in many forms, from presidential elections (where each citizen gets one vote), to dictatorships (where only the vote of the dictator counts), to a company's board of directors (where votes are weighted based on share ownership). In 1953, Lloyd Shapley developed a process for quantifying the worth of a person's vote. We will discuss this process (and the closely related Shapley-Shubik Power Index) and describe its applications to the Electoral College, workers' unions, and bribery in the New York legislature.- November 12 Peter Alfeld
**What is a slide rule?**

*Abstract:*There was a time when calculators did not exist. That did not stop us from building the Boeing 747, or going to the moon. In those days engineers, scientists, and students used slide rules on a routine and daily basis in place of calculators. I will show several slide rules, explain how they work, and describe what kind of mathematical expressions can be evaluated with a slide rule. (There are tens of thousands.) We'll also have a drawing. The lucky winner will get a slide rule to keep.- November 19 Stewart Ethier
**College Admissions and the Stability of Marriage**

*Abstract:*Suppose we have*n*men and*n*women who are looking for mates. Each man ranks the set of women, and each woman ranks the set of men. Is there a matching of men to women that is*stable*, that is, for which there do not exist a man and a woman who prefer each other to the mates to whom they are matched? This question was asked and answered positively by Gale and Shapley in a 1962 article in the*American Mathematical Monthly*. Fifty years later, Shapley shared the Nobel Prize in Economics for this work, which has application to more serious topics such as matching college applicants to colleges and matching medical school graduates to hospitals.- November 26 Shiang Tang
**Designing a super varied necklace and polynomials over Galois field**

*Abstract:*Consider a necklace with pearls of four different colours: red, green, blue and yellow. We call it super varied if one can find all possible combination of colours on adjacent pearls. There are 4^2=16 possible combination, which means the number of pearls on a super varied necklace should be at least 16. But is it possible to design such a 16-pearl necklace? And if so, how to arrange those red, green, blue and yellow pearls to get a super varied necklace not just by trials and errors? The study of polynomials with certain "numbers" as coefficients will answer our question. Those "numbers" are actually elements of Galois fields, which is a beautiful object discovered in 19-th century and has found its applications in cryptography, coding theory, computer science, etc.- December 3 Braxton Osting
**Ranking rankings and active ranking methods**

*Abstract:*In our meritocratic society, the concept of rank is paramount. Consumers seek the best product, search engines recommend the most relevant document, and sports fans demand to know the standing of their favorite sports team! The need for rankings in various contexts has led to the development of many different ranking algorithms. In the first part of this talk, we’ll review some of these algorithms and discuss a statistical method to compare their predictive power using sports data. In the second part of the talk, we'll consider the dependency of the ranking problem on the available data and discuss methods for actively collecting pairwise comparison data for ranking. In the context of sports rankings, this is equivalent to scheduling games that will be most informative to the ranking.- December 10 Anna Macquarie Romanova
**Platonic Solids and the ADE Correspondence**

*Abstract:*A Platonic solid is a convex polyhedron with congruent regular polygons for faces and the same number number of faces meeting at each vertex. The Greeks discovered that only five such polyhedra exist – the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. The symmetry and beauty of these five objects has fascinated mathematicians for thousands of years, and they have shown up across various fields of mathematics in surprising ways. In this talk, we will introduce these objects, convince ourselves that there are only five, and discuss their appearance throughout history. Then we will study their symmetries to associate them with a collection of special graphs called Dynkin diagrams. This association is one example of a mysterious phenomenon called the ADE correspondence, where a variety of seemingly unrelated objects in mathematics and physics are classified by the same Dynkin diagrams. This talk should be accessible to any student with some background in college algebra.