# Undergraduate Colloquium

**Fall 2015**

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

**LCB 225**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- August 26 No Talk

- September 2 Andrejs Treibergs
**Analysis Meets Topology: Gauss Bonnet Theorem**

*Abstract:*This talk is about one of the most important results about the geometry of surfaces. The Gauss Bonnet Theorem equates the integral of the curvature, which is an analytic invariant with the Euler Characteristic, which is a topological invariant. We begin by describing some calculus of surfaces from an intrinsic point of view. We assume we have a Riemannian metric so that we can measure lengths and angles of vectors. We explain how to differentiate vector fields, compute the geodesic curvature which tells how much a curve turns, and the Gauss curvature which gives the shape of the surface. By applying Green's Theorem, we relate Gauss curvature to the angles and geodesic curvature of a curvilinear triangle. This generalizes the fact that the interior angles of a Euclidean triangle add up to pi. By combining the formulas for triangles that subdivide a surface yields the result. Some applications will be mentioned.

Slides for the talk are available: www.math.utah.edu/~treiberg/GaussBonnetSlides.pdf- September 9 Jon Chaika
**Metric number theory, continued fractions and dynamics.**

*Abstract:*As you probably know, the rational numbers are dense. However, if you only consider rationals with denominator at most n you do not get a dense set. Given an irrational number, a natural object to consider is the sequence of rational numbers closest to it and a natural question is how close a point in this sequence is to the irrational number when compared to the size of its denominator. This talk will discuss this and other questions.- September 16 Karl Schwede - 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 23 Liz Fedak
**Interdisciplinary Model Building: Constructing, Validating and Analyzing ODE Models For Tumor-Immune System Interaction**

*Abstract:*Mathematical oncologists seek to describe how tumor cells and the immune system interact over time, then use such knowledge to gain insight that would benefit clinicians. As useful as they are, understanding these models requires one to be well-versed in both mathematical and biological methodology, and are often intimidatingly complex to the layperson. Liz Fedak's talk attempts to rectify this by demystifying large systems of ODEs piece by piece. By drawing on previous works by local researchers, recent advances in the field, and her own previous and current research, she hopes to provide her audience with the specialized knowledge needed to parse models of this type. The first part of the talk will focus on the origin of terms in a simple ODE model of tumor-immune system interaction, while the second part includes a thorough description on how these models are built, validated, and analyzed. Afterwards, attendees will be given the chance to demonstrate what they have learned.

This talk is intended for an audience familiar with systems of ODEs, and further alludes to immunology, numerical analysis, and dynamical systems analysis.- September 30 Peter Alfeld
**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.- October 7 Adam Boocher
**The Weird and Wonderful Chemistry of the Look-and-Say Sequence**

*Abstract:*1,11,21,1211,111221,312211,13112221,... Do you see the pattern? The defining rule can be explained to children and is a great off-the-wall example for Calc II students. Despite a simple definition, this sequence has surprising mathematical properties. We'll talk about eigenvalues of large matrices, roots of degree 71 polynomial equations, and something John Conway calls the "Cosmological Theorem." In the spirit of Calc II we might even do the ratio test!

A PDF of Conway's paper can be found at: http://www.math.utah.edu/~boocher/writings/ConwayLook.pdf- October 14 No Talk - Fall Break

- October 21 Sean McAfee
**Mathematical Paradoxes**

*Abstract:*The word "paradox" is used in a mathematical sense to describe a statement that is extremely counterintuitive but true. For example, the Banach-Tarski paradox states that a 3-dimensional ball can be decomposed into five disjoint subsets which can be reassembled to form two balls, each having the same volume as the original ball! We will discuss this and other historical paradoxes such as Zeno's Paradox, Gabriel's Horn, and the Potato Paradox.- October 28 Christopher Miles
**How the Zebra Got its Stripes: The Mathematics of Pattern Formation**

*Abstract:*Although found prevalently throughout nature, patterns are a uniquely complex and diverse phenomenon. For instance, seashells have a distinctly different pattern than cheetahs, which also have different patterns than zebras or cacti. Can a single mathematical theory explain how all of these patterns arise? In 1952, Alan Turing answered this question, particularly in the context of developmental biology. In this talk, we'll discuss the primary mechanism proposed by Turing: a reaction-diffusion system. By looking at an example, the Gierer-Meinhardt model, we'll attempt to develop intuition for how these partial differential equations behave and why this mechanism does such a good job explaining pattern formation. Along the way, we'll hopefully see some math and pretty pictures.

Slides for the talk are available: slides.com/cmiles/undergrad_colloq_2015#/ - November 4 Anna Macquarie Romanova
**A Glimpse of the Fourth Dimension: Exploring Higher Dimensional Polytopes**

*Abstract:*Platonic solids are the most symmetric objects that we can construct in three dimensions. Built from regular polyhedra, they look the same at each vertex. One of the major revelations of classical Greek mathematics was that the fact that there are only five solids possessing this maximal symmetry - the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. But what happens if we expand our view to four dimensions? In this talk, I will introduce the four dimensional analogues to the platonic solids and discuss how we can understand objects that we cannot see. Along the way, we'll encounter the princess of polytopes, a hypercube of monkeys, and the view from inside a hyperdodecahedron. Bring your iOS or Android devices for full immersion.

Slides for the talk are available: http://www.math.utah.edu/~romanova/talks/polytopes.svg

(Please be patient while slides load)- November 11 Shiang Tang
**Continued fractions and music theory**

*Abstract:*Continued fraction is an alternative way of expressing a number as opposed to decimal expansion. It provides better rational approximation to real numbers. In this talk, I will show you how to express some important irrational numbers by continued fraction and an application to piano theory. No preliminary knowledge is required other than elementary algebra.- November 18 Ian Shipman
**Sometimes you just get lucky**

*Abstract:*The probabilistic method is a technique that can be used to prove existence results in a wide variety of discrete contexts by analyzing a random example. I will explain the technique through examples from graph theory, elementary number theory, coding, and discrete geometry.

Notes from Ian's talk are available: Lecture Notes- November 25 Damon Toth
**Stochastic branching process theory and Ebola**

*Abstract:*Stochastic branching processes describe the probabilistic propagation of offspring across generations and are well suited for modeling outbreaks of diseases that spread from person to person. In most countries, incoming travelers infected with Ebola caused only small outbreaks. I will show how data from these extinguished Ebola outbreaks can be represented by a "subcritical" branching process model, in which outbreaks will eventually extinguish with probability 1, but perhaps not before a large number of transmissions occur. I will derive the final outbreak size distribution equations and show how they can be used by public health officials to assess the risk of "worst-case scenario" outbreaks after an Ebola introduction under different conditions.- December 2 Michael Elifritz, Actuary & Math Department Alumnus
**Advice on entering actuarial and business careers for math majors**

*Abstract:*I will talk about my transition from a mathematics student to an actuary. There will also be a discussion about my insights on how people with quantitative skill sets can make a similiar transition from student to becoming a working professional.- December 9 Vira Babenko
**Main Problems and Applications of Approximation and Interpolation.**

*Abstract:*In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to estimate the value of that function for an intermediate value of the independent variable. A different but closely related problem is the approximation of a complicated function by a simple function. We will discuss how to deal with these problems and where do we apply approximation everyday.