# 2018-2019 Colloquium Schedule

## Winter 2018

## Arithmetic of Apollonian Circle Packings

**Speaker**: Holley Friedlander (Dickenson College)- February 13th at 4:00 pm in Chavis Hall 105
- Refreshments at 3:40 pm in Chavis Hall foyer
**Abstract**: The concept of an Apollonian circle packing traces back to the Greek astronomer Apollonius of Perga who discovered the following theorem: to any three mutually tangent circles, there are precisely two circles mutually tangent to all three. Starting from four mutually tangent circles and continually adding newer circles tangent to three of the previous circles yields an infinite configuration known as an Apollonian circle packing. If each of the four initial circles has integer curvature, then amazingly all circles in the packing will have integer curvature. Given such a packing, there are many questions a number theorist might ask. For example, are there infinitely many circles with prime curvature? (The answer is yes!) In this talk, we introduce and discuss the arithmetic of Apollonian circle packings.

## My life and times with dots and lines

**Speaker**: Wayne Dymacek (Washington & Lee University)- January 31st at 4:00 pm in Chavis Hall 105
- Refreshments following the talk in Chavis Hall foyer
**Abstract**: In this lecture we briefly discuss the Society of the Cincinnati from which this professorship is named and why the professorship is in the Mathematics Department. We then talk about the philosophy of mathematics known as constructivism and it more extreme versions, finitism and ultra-finitism. The talk will conclude with examples from my mathematical career.

## Fall 2018

### Variations on Kuratowski’s 14-set theorem

**Speaker**: David Sherman (University of Virginia)- November 14th at 4:00 pm in Chavis Hall 105
- Refreshments at 3:40 pm in Chavis Hall foyer
**Abstract**: Kuratowski’s 14-set theorem from 1920 says that in a topological space, the maximal number of distinct sets that can be generated from a fixed set by taking closures and complements (in any order) is 14. I’ll present this — the proof is surprisingly simple — and some variations that I published in the American Mathematical Monthly a few years ago. Although the topic is “point-set topology,” the methods are algebraic, with cameos by logicians and an obsessed trucker. I will start at the beginning, explaining basic topological notions for the real line.

### Knots vs. Graphs: An Epic Tale of Survival

**Speaker**: Hugh Howards (Wake Forest University)- November 1st at 4:00 pm in Robinson Hall 105
- Refreshments at 3:40 pm in Robinson Hall foyer
**Abstract**: A famous old result says that the complete graph on 5 vertices (5 vertices with edges running between every pair of vertices) cannot be drawn in the plane without edges crossing. Conway and Gordon asked the related question of when you can expect to find a knot or link in a graph. Their paper has inspired a huge quantity of student research over the last 20 years. We will look at a new twist that Howards and his student Natalie Rich came up with springing from Conway and Gordon's beautiful result.

### What we still don't know about addition and multiplication.

**Speaker**: Carl Pomerance (Dartmouth College)- October 9th at 4:00 pm in Robinson Hall 105
- Refreshments at 3:40 pm in Robinson Hall foyer
**Abstract**: How could there be something we don’t know about arithmetic? It would seem that subject was sewn up in third grade. But here’s a problem: What is the most efficient method for multiplication? No one knows. And here is another: How many different numbers appear in a large multiplication table? There are many more such problems, largely unsolved, and for which we could use some help!

### What We Did Last Summer (Parents and Family Weekend Event)

**Speakers**: W&L Math students: Max Masaitis, Isabell Russell, Corinne Joireman, Allison Young and Phuong Mai- September 28th at 4:00 pm in Robinson Hall 105
- Refreshments at 3:40 pm in Robinson Hall foyer
**Abstract**: The students will talk about their summer research on one-dimensional packing and covering, claw-free Steinhaus graphs, and the mathematics of tie knots.