Research Opportunities
Faculty Research Projects
There are several faculty research projects that students could get involved with.
Phyllotaxis
with Pau Atela and Chris Golé
Phyllotaxis is the study of plant organ arrangements. These arrangements are initiated at a microscopic level at the apex (tip) of the plant, in the form of cell bulges called primordia. Our work gives a mathematical framework explaining the striking phenomena of spiral configurations involving Fibonacci numbers (see our phyllotaxis Web site).
One aim of the project is to understand mathematically the universe of all possible phyllotactic configurations. On the other hand, we want to determine empirically and formally why some configurations are more common than others in plants. Research projects in this area could be either theoretic (exploring the geometry and topology of the set of configurations), numerical (creating computer experiments in search of new structures or to create global pictures of the set of structures, or creating JAVA applets for the website), and could be more connected to the biology (data analysis of pictures produced in collaboration with Jacques Dumais' laboratory in Harvard)
Posets of Involutions in Weyl Groups
with Ruth Haas
Weyl groups generalize permutation groups and are useful in many branches of mathematics. They are used to describe motions of objects in space... Understanding which elements are in this set is important for many applications of Weyl groups to symmetric spaces. The elements of .... are characterized by sequences in ... which induce a poset relationship between them called the Bruhat Lattice. While there is an algorthim to find this poset for any particular case, it would be nice to have an understanding of the structure in general, without having to make direct calculations. Relatively little is known about the structure of this poset. For example, what is the number of longest paths (from the top to the bottom of the poset)? Checking some simple examples has suggested that this sequence of numbers does not correspond to any known sequence (as currently listed in The On–Line Encyclopedia of Integer Sequences). Another question is to determine if there are Gray Codes (Hamilton Paths) on this poset. While these are interesting combinatorial questions in their own right, a better understanding of the combinatorial structure will lead to better algorithms for symmetric spaces.
These problems can be tackled by exploring examples, both with computation by hand, and writing computer code to look at bigger examples. Some examples have been calculated by Smith students, an example is in the figure below.
Bouncing Inside a Triangle
with Jim Henle
Given an integer N, draw an equilateral triangle with side N.
Now start on the bottom edge, a distance of 1 from the left vertex. Go straight up until you hit a side of the triangle. Bounce off, but bounce at a 90-degree angle from the side.
I am investigating the question: How many times do you bounce before you stop? It depends on N, of course, but how? The problem involves, at the very least number theory, combinatorics, algebra. Ultimately, it might involve much more.
Infinite and Infinitesimal Numbers
with Jim Henle
There is a simple way to add to the real number system. The new numbers are useful for understanding and proving the theorems of the calculus. They are also very strange. Unlike the real numbers, which lie on a line, these numbers can't be ordered nicely. There are numbers that infinitely large, larger than all reals. There are number that are infinitely small, closer to 0 than all real numbers except 0. But there are also numbers that are vague. There's a number, for example, that's greater than 2 and less than 17, but that's all we can say about it.
I'm exploring these numbers. It's a task that involves calculus, logic, and set theory.
Statistics Applied to Mental Health
with Nick Horton
My research program involves the development of new statistical methods in the fields of mental health and substance abuse. This research includes ways to account for dropout (or missingness), which can lead to wrong answers from studies. In addition, certain factors are not easy to measure precisely (such as psychiatric problems or substance use). Researchers may collect multiple reports of such factors (i.e. from parents, kids and teachers), but then are faced with the problem of how to reconcile potentially discordant reports. I plan to develop and apply new statistical methods to address missingness and multiple reports in mental health and substance abuse research. This will include applying these new methods and disseminating new techniques through expository and tutorial papers in ways that are accessible to researchers in these fields.
The Geometry of Protein Chain Reconfigurations
with Joe O'Rourke
I am exploring the configuration space of polygonal chains that model the backbone of a protein molecule. The ultimate goal is to speed up protein folding simulations by reducing the search pace. With colleagues I have proven that the "producible" protein hains live in a space of fewer dimensions than do all protein chains. This shrinking of the configuration space depends on whether or not the chains can "lock," that is, can be in an unflattenable configuration.
Because protein amino acids are about the same length, it has become important to explore whether or not equilateral (or nearly equilateral) protein chains can lock. This is an unsolved problem. Nadia Benbernou, has been exploring as part of her senior honors thesis the "maximum span" of an equilateral chain, which is related to lockedness. She just proved that the maximum span of an equilateral chain with all equal angles is achieved via a planar configuration of the chain. This is not true for non—equilateral chains, nor for chains of unequal angles. The two figures attached illustrate different aspects of the proof.
Student Presentations, Projects and Papers
Flat knotted ribbons in the plane
Shivani Aryal, Shorena Kalandarishvili and Sarah Meyer studied flat knotted ribbons (knots and links constructed from a rectangle of fixed width which is then folded flat in the plane). The ribbonlength of a knot is the ratio of its length to its width. The goal of the research is to minimize ribbonlength for a given knot or link type. In previous work, other authors calculated the ribbonlength for some classes of knots. In addition, they conjectured that the minimum ribbonlength trefoil has the shape of a pentagon. The students attacked this conjecture. Key steps included providing rigorous definitions and proving results about the local structure of knotted ribbons. The students also gave other examples of conjectured minimum ribbonlength knots and links. The unknot proved to have many interesting cases (faculty advisor Elizabeth Denne).
Snake cube puzzles: Hamilton paths in grid graphs
Alison McDonough studied snake cube puzzles, which can be thought of as Hamilton paths in grid graphs. The puzzle traditionally consists of a string of 27 small cubes that, when folded correctly, form a larger 3 by 3 by 3 cube. A large number of Hamilton Knots were found during my work. The existence of these cycles through three-dimensional grid graphs suggests further questions. Of any knot configuration, we can ask, "What is the smallest cube that can contain such a knot? Can the knot span the entire cube? (faculty advisor Pau Atela).
Fibonacci sequences and the golden angle in plants
Hillary Sackett, Samantha Oestreicher, Cheryl Milton and Megan Sawyer continued study of the golden angle and Fibonacci spirals in plants. Specifically, they refined previous results that found plant structures where the ties between Fibonacci numbers and the golden angle were severed. Furthermore, this team has explored all possible orderings for the (2,3) case (faculty advisor Christophe Golé).
Conjugacy classes in Coxeter groups
Anna Boatwright, Emily Gunawan and Jennifer Koonz studied Coxeter groups, a type of group that is generated by involutions. Two finite Coxeter groups were studied, those of type Bn and those of type Dn. The group was primarily interested in the conjugacy classes of the involutions of each group and the relationships between these conjugacy classes. They observed distinct patterns in the interrelationship between the involution conjugacy classes (faculty advisor Ruth Haas).
Constructing a tragedy: geometric analysis of The Daughter of Jephthah
Katrina Greene investigated the underlying structure of The Daughter of Jephthah, Edgar Degas’ most ambitious history painting and one of the prized unfinished canvases at the Smith College Museum of Art. The goal of the project was to determine if Degas used a grid system to arrive at the current composition and whether a series of pinholes along the edges of the canvas is physical evidence of geometry at work in the image (faculty advisor Pau Atela).
Using experimental design to facilitate technology transfer in laser applications
Zehui Chen, Xiaoshuang Chen, and Portia Parker have been working with Preston Macy, CEO of Applied Light, LLC of Holyoke, MA to design and analyze studies of the quality of laser-inscribed identification codes on metal and glass. Applied Light, LLC employees conduct the experiments using the designs we provide. Goals for this research include identifying laser settings and process characteristics of the bonding material that create permanent, legible marks on industrial parts (faculty advisor Katherine Halvorsen).
The use and abuse of multiple outcomes in randomized controlled trials of depression
Kristin Tyler studied the use of multiple outcomes in randomized clinical trials. These are commonly collected and analyzed, though, a fully effective and efficient method for analysis is not yet established. The most common analysis methods for multiple outcome data involves separate testing of each individual outcome, global testing of all outcomes or single testing of a composite outcome. We describe and report the results from a comprehensive review (n=55) of multiple outcomes from depression trials published between January 2007 and October 2008 from 6 widely read journals. Parallel data collection was undertaken for each paper to determine the number of primary and secondary outcomes, as well as the methods used for analysis. We found that the use of multiple outcomes was quite common, yet the methodology utilized to synthesize these reports was often quite simplistic. Results were presented at the Joint Statistical Meetings in Washington DC in August 2009 (faculty advisor Nicholas Horton).
Combinatorics and characterizations for involutions and twisted involutions
Constance Baltera, Ashley Hatfield, and Rebecca Tramel examined posets of involutions and twisted involutions in several families of Weyl groups. The location of an element in a poset can be described by its rank. Formulae for determining the rank of a given involution or twisted involution provide useful structural information about the poset. We present rank formulae for the posets of twisted involutions in An, D2n+1, and D2n (faculty advisor Ruth Haas).
Knot theory
Reagin Taylor McNeill has been researching knots. A mathematical knot is much like a knotted piece of string where the ends are joined together and the piece of string has no thickness. I have done a study of knot theory, learning the foundations of the subject including knot and link polynomials, knot coloring, bridge and stick number, braids, Seifert surfaces and genus. As the study of knot theory is closely related to abstract algebra and topology some topics in these fi elds were investigated as well. In addition, my research emphasized the study of the crossing and supercrossing number1 of knots through examination of the crossing map (faculty advisors Elizabeth Denne and David Cohen).
Generating DeBruijn sequences and linear recurrences
Priscah Cheruiyot, Alison McDonough and Rebecca Tramel have been studying De Bruijn sequences. A De Bruijn sequence of order k over A is a word of length n to the k that, when viewed cyclically, contains each possible word of length k as a subword exactly once. For example 00010111 is a De Bruijn sequence of order 3 over the alphabet A = {0, 1} since moving from left to right we see that it contains every possible word of length 3 as a subword: 000, 001, 010, 101, 011, 111, 110, 100. De Bruijn sequences are known to exist for all values of n and k and have been studied by mathematicians, computer scientists and electrical engineers. They have many interesting algebraic and combinatorial properties. Another aspect of our research involved studying certain connections between recurrence equations and De Bruijn sequences. A recurrence equation is an equation that defines the nth term of a sequence in terms of the k proceeding terms (faculty advisor Michael Bush).
Modeling income inequality with a single parameter
Stephanie Jakus tackled a problem in econometrics. The distribution of income is a multifaceted phenomenon. One-dimensional mathematical models based on the general Lamé curves or super-ellipses provide a remarkable fit to data from different countries in different years. Stephanie's paper was published in Modeling Income Distributions and Lorenz Curves, edited by Duangkamon Chotikapanich (faculty advisors James Henle and Nicholas Horton).
