## Spring 2019 SCHEDULE

**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 7-12) 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 clean-slate 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!

**Cobwebs, Maps and a Route to Chaos**

*Abstract:* I will introduce one-dimensional maps (you can simulate some by pressing the same
calculator button many times). They can easily be defined and analyzed and are used
as models in many applications. Looking deceptively simple, the maps give rise to
complex behaviors, including chaos. As we follow the route to chaos in these systems,
we will find universal constants (as basic to this route as π is to circles): δ =
4.669..., and α = -2.5029...

** 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 fractal-type 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 one-hundredth. 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 Rayleigh-Taylor instability drives pattern formation
from this dual viscous layer.

**Instant Insanity**

*Abstract:*Fifty years ago a new puzzle was unleashed upon the ever-patient public. It consists of four cubes whose faces are colored one of four colors: red, green, blue, and white. The objective of the puzzle is to arrange the cubes in a row so that along the front, back, top, and bottom each color appears once among the four cubes. The colors of the left and right faces of the cubes are of no consequence (indeed, except for the left of the first cube and the right of the final one, those faces are hidden).

** 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 likely-to-be-landed-upon
properties. We'll also discuss how to answer the question I ask myself whenever I
play my three-year old in Hi Ho Cherry-O: ``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.

## Math 3000 (Receive Credit for Attending)

The Undergraduate Colloquium is open to anyone to attend; however, if students would
like to receive credit, you may register for **Math 3000**.

This is a 1 credit hour CR/NC course. To receive credit:

- You may not miss more than 2 of the colloquia
- You will need to write a short paper on one of the topics presented during the semester. (Report Specifications)

Course Coordinators

Peter Alfeld | Lisa Penfold |

Course Instructor | Administrative Coordinator |

pa@math.utah.edu | ugrad_services@math.utah.edu |