During my sabbatical at Stanford last year, I spent a part of my time having coffee with Owen Schuh, an artist in the SF bay area. The goal of this collaboration was to engage mathematics and the visual arts in a direct manner, with concrete outputs, that does not insult either field. In other words, new mathematical questions need to be formulated and new artworks need to be produced for the success of this venture. Our work should be a true collaboration, with the mathematician involved in the drawings and the artist involved in the mathematics.

In the end, we created a triptych of works, titled **Cartography of Tree Space**, involving acrylic, watercolor, and graphite on 108cm x 108cm wood panels. Our work was picked up for an inaugural gallery showing in Germany by Satellite Berlin. One of the pieces, titled “Underground”, is shown below.

The particular object of our study is a configuration space of phylogenetic trees, originally made famous by the work of Billera, Holmes, and Vogtmann. Each point in our space corresponds to a specific geometric, rooted tree with five leaves, where the internal edges of the tree are specified to be nonnegative numbers. From a global perspective, this “tree space” is made of 105 triangles glued together along their edges, where three triangles glue along each edge. This results in 105 distinct edges, and 25 distinct corners. This space of trees appears in numerous areas of mathematics, including algebraic topology, enumerative combinatorics, geometric group theory, and biological statistics.

Although the space is only two-dimensional (made of numerous triangles), the natural world for this tree space to inhabit is in four-dimensions, where the full symmetry of its structure will be made transparent. Our goal was not to describe the space in mathematical terms. Instead, we wanted to describe what it feels like to live in this tree space, to inhabit it as a world like any other world. To this end, we use the world of cartography and map making to invite the viewer to understand tree space.

The collaboration took place during an 18-month timeframe, from September 2013 until February 2015. Roughly, the first six months were spent in understanding the goals of the project and choosing a point of collaboration. The second six months were spent at coffee shops and studios, where Owen and I would meet and go through both mathematics and sketches. The final six months were focused on crafting extreme details and formulating a unifying vision to the project.

In all of this, the process behind the mathematics and the art was quite similar. Ideas were conjectured, tested, and evaluated, both visually and analytically. And there was a sense of incredible freedom to explore these worlds, with a strong instinct guiding the collaborators, as to the right road to pursue.

]]>*The past fifty years have seen computing move from a fringe activity in universities to a central part of academic life. Today’s university students never knew a world without personal computers, networks, email, and the web. Nowadays, the computers and software that we use in universities are commercial products, but this was not always the case. When the computer industry failed to meet our needs, we took the initiative. For thirty years academic computing diverged from the mainstream. We built our own state-of-the-art systems and ran them successfully. Universities led the development of timesharing and local area (campus) networks. The distributed computing, email, national networks, and the web that everybody uses today are direct descendants of systems that universities and scientific researchers built for themselves. As a student, faculty member, and administrator, I lived through many of these developments and for seventeen years I was in charge of computing at two of the leading universities, Dartmouth College and Carnegie Mellon. This talk tells the story.*

Events continue with a lunch at the Faculty House at noon and talks by the national PBK president and secretary, our visiting scholar William Arms of Cornell, the president of the NY PBK Association, and others from 4-8pm in Griffin 4 (refreshments and dinner provided). All are welcome; if possible please email Steven Miller (sjm1@williams.edu) so we can get an accurate headcount. A complete schedule of the talks is online here: http://web.williams.edu/Mathematics/sjmiller/public_html/pbk/ (videos of the talks will be posted later on YouTube and linked to this page).

]]>- iBerkshires.com: pre-build article
- iBerkshires.com: story on successful build
- Berkshire Eagle: story on successful build
- Berkshire Eagle: photographs of successful build (zipped photos superstarsuccessfulbuild2015)
- Berkshire Eagle: time lapse video of build (camera above table 1)
- WABC story: http://wamc.org/post/piece-piece-williams-students-embark-light-speed-voyage
- Lego instructions: click here

**We hope to have other events along these lines in the future — if you’re interested email Steve Miller at sjm1@williams.edu.**

If you have a solution you would like to post here, email it to me and I’ll add it.

- B2: Anonymous submission: Putnam20142B2
- B3: Jesse Freeman: FreemanPutnam2014B3

I use homemade sweet treats as prizes for going above and beyond the course work. This semester in my Stat 101 class, the first person in each section to explain the solution to the “Birthday Problem” in front of the class without notes got a Triple Chocolate Espresso Brownie. I’ll give you the premise of the question and and then the recipe for these great, easy brownies.

The “Birthday Problem” is a fascinating probability problem that can be answered with a few basic probability rules that you learn in introductory statistics. The problem statement is *how many people do you need to have in a room such that there is at least a 50% chance of at least two people in the room having the same birthday?* To answer this, we need to simplify our world a bit. Let’s ignore Leap Day and assume there is an equal chance of being born on each of the 365 days of the year (which is probably not true these days with modern maternity medical practices).

First off, we notice that it is easier to calculate the chance that you don’t have at least two people with the same birthday in the room. The first person in the room could have any of the 365 days for their birthday (365/365) and then the second person in the room has to have a birthday on one of the 364 days left (364/365) and then the third person in the room has a birthday on one of the 363 days that aren’t called for yet (363/365). If there are n people in a room, the probability of getting at least two people with the same birthday is

This probability surpasses 0.5 when n = 23. One of my students eloquently explained this in class and got a homemade Triple Chocolate Espresso Brownie. The brownie recipe that I used, adapted from Cooks Illustrated, is listed below.

**Triple Chocolate Espresso Brownies **

5 oz. semisweet chocolate, chopped

2 oz. unsweetened chocolate, chopped

8 T. unsalted butter, cut into quarters

3 T cocoa powder

1.5 T instant espresso powder

3 large eggs

1 1/4 C. sugar

2 t. vanilla extract

1/2 t. salt

1 C. all-purpose flour

1. Heat oven to 350 with oven rack in lower-middle position. Spray 8-inch square pan with cooking spray. Take one 12-inch piece of foil, fold it and lay it horizontally and press it into the corners. Take a second 12-inch piece of foil, fold it, and lay it vertically on the pan (making an cross with the other piece) and press it into the corners. Spray foil with cooking spray.

2 . In medium saucepan, melt chocolates and butter on low, stirring frequently until mixture is smooth. Take off heat. Whisk in cocoa and espresso until smooth. Set aside to cool.

3. Whisk together eggs, sugar, vanilla, and salt in medium bowl until combined, about 15 seconds. Whisk warm chocolate mixture into egg mixture; then fold in flour until just combined. Pour mixture into prepared pan, spread into corners. Bake until slightly puffed and toothpick inserted in center comes out with a small amount of sticky crumbs clinging to it, 35-40 minutes.

4. Cool pan on wire rack to room temperature, about 2 hours, and then remove brownies from pan using foil handles. When ready to serve, cut into 1-inch squares. Enjoy!

]]>I didn’t quite know what to expect from a small college tucked away in the Berkshires, but it didn’t take long for me to fall in love with this place and be proud to call it home. There are many reasons for this, although it’s impossible to pinpoint a single defining one. Perhaps it’s the students, who are intelligent, motivated, and teach me just as much as I teach them. Or perhaps it’s the faculty in the math department, who are inspiring teachers and brilliant individuals, but are also fun to be around and always have some interesting stories to tell over lunch (come join us one day, you’ll see!). Maybe it’s the fall colors (check out the view from my office!) and watching the mountains change from season to season.

Either way, happy Mountain Day!

]]>- We have a weekly problem solving dinner at 5:30pm in Dennett Private Dining Room at Mission on Wednesdays. There’s no prep work; feel free to drop in any time (and if you’re not on the meal plan we’ll provide a swipe). The way it works is we print out a math competition from somewhere in the world, and then brainstorm and attack the problems together.
- We’ll also meet for lunch on Wednesdays (in Mission at noon, room TBD) to do Project Euler problems.
- There are several math competitions each year. Some time in October or November we’ll defend the Green Chicken when Middlebury travels here. There’s also the Virginia Tech math competition (we’ll do this remotely Saturday October 25th), and the Putnam exam (which will probably be Sat Dec 6th).
- We also frequently field teams for the Mathematical Contest in Modeling.
- I’m also teaching a class on math puzzles and problem solving, Math 331: The little Questions. Feel free to check out the homepage for resources, as well as lectures (each class is recorded and uploaded to YouTube).

Many people love math puzzles or riddles. They’re often fun, frequently illustrate a beautiful concept or perspective, and unlike real world research problems they typically have an elegant answer. Below is one of my favorites. It can be solved by brute force but only at great cost (and a high probability of forgetting a case); however, if you look at it the right way it’s just one line (and this proper perspective illustrates a powerful technique which is of use in research mathematics).

**Problem: Imagine you have 10 distinct cookies but 5 distinguishable, hungry graduate students (so they’re not engineering students working away in a lab!). How many ways can you divide all the cookies among the people? In other words, we only care about how many cookies each person gets, not which ones.**

I run a math riddles page (see http://mathriddles.williams.edu/ ), and this was the first problem I added; the problems and resources posted there for students and teachers are used in schools throughout the world, and if you’re interested in helping with the site drop me a line at sjm1@williams.edu. It turns out that this problem is equivalent to Waring’s Problem with exponent 1; I’ve used this idea with SMALL students in research problems related to Fibonacci numbers (this is discussed in another post, To Bead or Not To Bead). Enjoy!

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