# Brett C. Smith

I am a teaching postdoc in the math department at Yale University. I also teach at Housatonic Community College and collaborate with the Yale Center for Teaching and Learning.

I earned a PhD in mathematics at Wesleyan University. My advisor was Karen L. Collins and my thesis was On minimality of planar graphs with respect to treewidth.

I double majored in mathemaics and visual art at Bowdoin College.

I taught mathematics at St. Mark's School for two years. I also coached JV basketball and lacrosse.

I grew up in Mystic, Connecticut.

# Research

I am interested in graph theory (i.e. networks). Specifically, I have expertise with a measure called the treewidth of a graph. Treewidth is at the heart of the proof of Robertson and Seymour's deep Graph Structure Theorem and has far reaching implications, contributing to proofs of the Graph Minor Theorem and the Strong Perfect Graph Theorem, and a simplified proof of the Four Color Theorem.

The Graph Structure Theorem says that for any fixed graph, say G, every graph either reduces to G through a sequence of edge deletions, edge contractions and vertex deletions (i.e. contains G as a minor) or can be built by piecing together graphs of bounded genus in a tree structure. The result rests on the fact that planar (genus zero) graphs can have arbitrarily large treewidth. In particular, the proof makes use of square grids because for n > 1 the (n x n)-square grid has treewidth n.

We generalize the structures in square grids that guarantee large treewidth. Actually, square grids are inefficient for this outcome. Our more general structures, which we call nets, are defined by identifying three sides of a graph embedded in the plane. These nets turn out to be particularly fruitful for understanding treewidth by using properties of the embedding.

## If you have never heard of treewidth (or graph theory), here is a teaser ...

Make a list of all the people who have ever sent you a text message. On a big piece of paper, draw a point representing each person on that list. You can place the points wherever you like. For any two people on your list who have texted each other, draw a line connecting the points representing them. Your drawing now describes your text message network.

This network of people is going to protect a password. The password can have as many characters as you want, and you are going to divide up the password so that no one person knows the whole thing.

Here is your strategy. Group your friends into several (possibly overlapping) sub-networks. Each sub-network will come up with one character in the password. The sub-networks need to be connected so that every member can learn the character by a chain of text messages that stays within the sub-network.

Individuals may need to recover the password quickly, so each sub-network needs to be able to communicate directly with every other sub-network. That is, for any two sub-netwoks, either you have a friend who is in both of them or there is a member of one sub-network who can text a member of the other sub-network.

A spy agency is trying to learn your password by hacking into individual phones. Hacking reveals each character on the phone, so it is preferable if no individual is in too many different sub-networks.

## Question:

Given your text message network, is there a way to design sub-networks so that as long as there are no more than four hacked phones, the spy agency will not uncover your full password? Five hacked phones? One hundred hacked phones?

## ... Bramble Number

For any graph, a collection of subsets of its vertices is called a bramble if (1) every subset is connected in the graph, and (2) the union of any pair of subsets is also connected. The order of a bramble is the smallest number of vertices needed to get a representative of each vertex subset in the collection. And the bramble number of a graph is the maximum order among all brambles in a graph.

The teaser is asking for the bramble number of your text message network. Moreover, bramble number is a dual problem to the treewidth of a graph.

# Teaching

Add up the first five odd numbers: 1+3+5+7+9 = 25. That's 5 x 5. If you add up the first ten odd numbers, you get exactly 100 = 10 x 10. The first twenty odd numbers add to 400 = 20 x 20. That can't be a coincidence... Learning starts with curiosity. We care about questions when the answer seems simple, surprising or useful.

Do you feel like telling someone this odd-number trick (even if you already know it)? Maybe we stumbled on one of the universe's greatest secrets, or maybe we just added wrong. Talking to other people validates our understanding, uncovers our misconceptions and gives us new ideas. Our learning community moves us forward.

I drew the picture on the right to help me understand why the odd numbers sum to square numbers. Left to your own devices, you would undoubtably come up with a (maybe slightly) different way of describing this phenomenon. Deep understanding requires creativity and a personal connection. So what is your explaination? How do you plan to use this knowledge?

## Teaching programs and projects

I was a Project NeXT fellow (red dot).

I co-organize a STEM Education journal club and seminar series through the Yale Center for Teaching and Learning.

I participated in the teaching track in the Metric Geometry and Gerrymandering workshop at Tufts.

I have been both a participant and a facilitator at the northeast Summer Institute for Scientific Teaching.

I coordinated several sections of a flipped version of Calculus II at Yale.

I taught a summer course in Wesleyan University's Center for Prison Education.

I am an alumnus of the PROMYS for Teachers program at Boston University.

## Video game project: Cartes

James S. Rolf and I created a video game called Cartes to help students better understand function properties. We designed the game with PreviewLabs. We also had advice from the Yale Center for Health and Learning Games, and our funding came from the Yale Center for Teaching and Learning. Check out a demo of the gameply captured during a talk to the Math Club at the University of Bridgeport:

Brett C. Smith
Helmsley Postdoctoral Teaching Scholar

Department of Mathematics
Yale University
10 Hillhouse Ave
New Haven, CT 06511

EMAIL: brett.c.smith (at) yale.edu

OFFICE: 423 Dunham Labs