Last Saturday, we attended WIMIN15, a conference sponsored by the Center for Women in Mathematics at Smith College. We heard two invited speakers, Linda Chen of Swarthmore College and Mariel Vazquez of University of California at Davis. Chen gave the Alice Dickinson Lecture in Mathematics, “Enumerative Geometry, Combinatorics, and Algebra,” explaining Schubert’s enumerative calculus through connections to algebra and representation theory. Vazquez gave a talk titled “DNA Unlinking in Bacterial Cells” in which she detailed the study of DNA replication and recombination using techniques from knot theory and low-dimensional topology. Both talks were fascinating and introduced unfamiliar areas and applications of math.

In addition to the invited talks, we attended several student talks on topics ranging from climate modeling of winter storm Juno to the study of orbits in representation theory, and in the afternoon, we gave a talk on the research we did this summer with Professor Susan Loepp as a part of SMALL. There was also a short panel on the experience of applying to and being in graduate school with six speakers, all women currently in graduate school: Saray Bray (Tufts), Heidi Goodson (UMN), Siddhi Krishna (Boston College), Elizabeth Sheridan-Rossi (UCT), Emily Gunawan (UMN), and Eva Goedhart (Smith College). The speakers shared general advice as well as anecdotes about the process and the mentors who helped them along the way. Overall, the conference was a great experience, and we would definitely recommend it to any students interested in math research or graduate school!

]]>When I got my new MacBook Air, I had to switch from my old email client, Eudora, to MacMail, which seemed like the best alternative. I often work offline, especially when traveling, because the internet can be expensive and often unavailable, so I need some such client. MacMail has the following inferiorities:

]]>- We have a weekly problem solving dinner at 5:30pm in the 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.
- There are several math competitions each year. On October 31st we’ll travel to Middlebury and defend the Green Chicken. There’s also the Virginia Tech math competition (we’ll do this remotely Saturday October 24th), the Putnam exam (on Saturday December 5th) and the University of Rochester Math Olympiad (we’ll do this remotely mid February).
- We also frequently field teams for the Mathematical Contest in Modeling.

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. Come join us and experience a wonderful “aha” moment when you solve a problem.

For some more puzzles take a look at Professor Miller’s math riddles page (see http://mathriddles.williams.edu/). 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 him a line at sjm1@williams.edu.

]]>Congratulations to Matthew and Stephanie!

]]>Devadoss will be giving a Family Weekend talk (all welcome) on “The Shape of Nature: Bee, Tree, Origami,” 2-2:45 pm, Saturday October 24.

He was most lately featured in a Maclean’s article on chalk.

]]>The centennial summer meeting (“MathFest“) of the Mathematical Association of America, August 5-8, 2015, in Washington DC, featured Hedrick Lectures on “Algebra Over Finite Fields” by Karen Smith, Centennial Lectures by Erik Demaine, Jennifer Chayes, Ingrid Daubechies, Carlos Castillo-Chavez, Karen Parshall, and Manjul Bhargava, and other invited lectures by Jeffrey Lagarias, David Bressoud, Erica Walker, Joseph Gallian, Noam Elkies, and Terrence Blackman. SMALL undergraduate research students Mia Smith (Williams) and Nat Mayer on won an excellent student talk award, and Matt Dannenberg placed fifth in the student mathematics competition. Alum Andrew Beverige won an Allendoerfer Award.

Presentations by Williams/SMALL folks included the following:

Colin Adams | Panel 11: Congratulations on Getting Tenure! Now What? | Wednesday, August 5 | 4:10-5:30 | Washington 6 |

Colin Adams | Cirque de Mathematiques | Wednesday, August 5 | 7-9 pm | Salon 2/3 |

Frank Morgan | What’s It Like to Be Editor–in–Chief of the Notices of the American Mathematical Society? | Thursday, August 6 | 8:15-8:25 am | Maryland B |

Frank Morgan | Albert’s Bridge: A Tragicomedy by Tom Stoppard | Friday, August 7 | 9-10 pm | Salon 2/3 |

David Schwein and Nina Pande | A Metric for Local Rings | Thursday, August 6 | 9:50-10:05 am | Virginia A |

Lena Ji, Sarah Fleming, and Peter McDonald | The Relationship Between a Local Ring and its Completion | Thursday, August 6 | 10:10-10:25 am | Virginia A |

Nathaniel Mayer and Mia Smith | When is a Knot Hyperbolic? | Thursday, August 6 | 3:20–3:35 pm | Virginia B |

Xinyi Jiang, Alex Kastner, and Greg Kehne | Totally Geodesic Surfaces in Hyperbolic Knot Complements | Thursday, August 6 | 3:40-3:55 pm. | Virginia B |

Aaron Calderon | Surfaces in Hyperbolic Knot Complements | Friday, August 7 | 10:10-10:25 am | Delaware B |

John Berry | Optimal Pentagonal Tilings | Thursday, August 6 | 4:00pm | Virginia B |

Jason Liang | The Double Gaussian Isoperimetric Problem | Thursday, August 6 | 4:20pm | Virginia B |

Yingyi Zeng | Relaxed Disk Packings | Thursday, August 6 | 5:00pm | Virginia B |

Matthew Dannenberg | The Convex Body Isoperimetric Conjecture | Thursday, August 6 | 5:20pm | Virginia B |

N/A | MAA Ice Cream Social and Undergraduate Awards Ceremony | Saturday, August 8 | 12:30-2 pm | Salon 3 |

For more information, please see http://www.imagescinema.org/events

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Studying this and related problems has led me to examining many properties of the Fibonaccis. The following result is well-known, but I hope you’ll enjoy the method of proof. Annoyingly, for this result we must set f_1 = 1, f_2 = 1 and f_{n+1} = f_n + f_{n-1} (if we had two 1’s we of course could not have a unique decomposition for all numbers).

**Theorem: The sum of the squares of the first n Fibonacci numbers is the product of the n-th and (n+1)-st Fibonacci numbers. In other words, f_1^2 + … + f_n^2 = f_n * f_{n+1}.**

A beautiful way to prove this is through the Fibonacci spiral, which provides our **third** equivalent definition for the Fibonaccis. We start with a 1×1 square. We place another 1×1 square along a side, forming a 2×1 rectangle. We now place a 2×2 square next to that, forming a 3×2 rectangle. We now add a 3×3 square, giving us a 5×3 rectangle. The pattern continues. Notice that at each stage our next choice is forced upon us, and they’re the Fibonacci numbers.

We can now prove the theorem by using one of the most powerful proof techniques in combinatorics: calculate something two different ways. Let’s say we’ve spiraled out and included the first n Fibonacci numbers. We obtain a rectangle of dimensions F_{n+1} x F_n; thus the area of this rectangle is F_{n+1} * F_n. We can, however, calculate the area another way. We built it by adding squares whose side lengths were F_1, F_2, …, F_n, and thus the area is also equal to F_1^2 + F_2^2 + … + F_n^2, which completes the proof!

I built this out of fuse beads with my daughter Kayla and my son Cameron. It was a lot of fun, and led to a nice visual proof that I can pack up and take with me to classes and schools.

For more reading / viewing, here are some clickable links:

- Professor Arthur Benjamin talks about the Fibonacci numbers (and gives this proof of the above theorem).
- Kologlu, Kopp, Miller and Wang’s paper on Fibonacci numbers and Zeckendorf decompositions.
- Survey article on Zeckendorf decompositions and the power of the combinatorial perspective (Miller and Wang).
- Talk by Miller on the Fibonacci numbers and Zeckendorf’s theorem (and generalizations).
- Perler fuse beads (note it took us four BIG baseboards to do this!).