## Fall 2018 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.

**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 signal-receptor 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 phosphorylation-dephosphorylation
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**

*Abstract:*The Labouchere betting system was popularized in the late 19th century by British politician and journalist Henry Labouchere. The system is easy to understand but surprisingly difficult to analyze. In their 2001 book, One Thousand Exercises in Probability, Grimmett and Stirzaker posed the problem of proving that the maximum bet size in the Labouchere system applied to a fair game has infinite expectation, and they provided a solution. Five years later it was discovered that their proof is incorrect, so the claim became a conjecture. We will provide evidence suggesting that the conjecture is true, but the fact remains that it has defied resolution for 12 years. That is, until July 2018 when it was finally settled by two graduate students at Stanford University.

**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), 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 Rice-Eccles stadium, there is a strategy that guarantees their freedom 99.996948242% percent of the time.)

No talk, Thanksgiving Break.

## 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 |