Students' Research Experiences

In this page, we include information about Smith Math majors research opportunities at Smith.There are research projects that faculty are opening to students either during the Spring Semester of their junior year (including visitors in the new Junior Year & Post BA Visiting Programs), or during the summer. Availability and project description may change. Contact the professors to find out conditions under which you can work with them.Click here for opportunities outside of Smith.

Also available are some research abstracts from previous research experience by students.

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Research Projects

Here are a few faculty research projects that students could get involved with. They are listed in alphabetic order of the faculty names.

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Distinguishing Geometric Graphs Mike Albertson Begin with a set of vertices, points in general position in the plane. Connect some pairs of points with straight line segments. The resulting object is called a geometric graph. Recently, I've been interested in the symmetries of geometric graphs. Formally, an automorphism of a geometric graph is a permutation of the vertex set that preserves its essential structure: edges, non edges, crossings, and non crossings. Given a geometric graph G, the distinguishing number, denoted by Dist, is the minimum number of colors needed to color the vertices so that the identity is the only geometric automorphism that preserves the colors on the vertices of G. Dist is a measure of the relative symmetry of G. A graph with k vertices and every possible edge is called a k-clique. Debra Boutin and I have shown that every geometric k-clique has Dist < 5, and we conjecture that every geometric k-clique with k > 6 has Dist < 3. Open problems include determining Dist for other families of geometric graphs. The figures to the right exhibit two geometric 6-cliques with Dist = 3 and one geometric 6-clique with Dist = 2.

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Phyllotaxis 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 website). 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)

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Posets of Involutions in Weyl Groups Ruth Haas

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Bouncing Inside a Triangle 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.

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Infinite and Infinitesimal Numbers
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.

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Statistics Applied to Mental Health
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.

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The Geometry of Protein Chain Reconfigurations 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.

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Power Bases for Cyclotomic Integer Rings
Leanne Robertson

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Research Abstracts

The list below is not complete but should be a good sample of students' research at Smith in the last few years.

Summer 2006

RHOMBIC TILINGS AND PHYLLOTAXIS by Jordan Crouser, Erich Kummerfeld and Duc Nguyen (rtf)

A CAUTIONARY NOTE REGARDING COUNT MODELS OF ALCOHOL CONSUMPTION IN RANDOMIZED CONTROLLED TRIALS by Eugenia Kim (MS word doc)

Summer 2005

CURRICULAR DEVELOPMENT PROJECTS IN STATISTICS by Lula Abasha (MS word doc)

COMPUTING THE INVOLUTION POSETS OF WEYL GROUPS by Kate Brenneman (MS word doc)

WRITING OF SWEET REASON by Penka Kovacheva (MS Word doc)

LOGIC PEDAGOGY by Juan Li (MS Word doc)

THE INCREASING SOPHISTICATION OF STATISTICAL METHODS IN THE NEW ENGLAND JOURNAL OF MEDICINE by Suzanne S. Switzer (MS Word doc)

PROPERTIES OF INVOLUTIONS AND TWISTED INVOLUTIONS IN WEYL GROUPS by Nicole Rizki (pdf)

Other Summer projects from 2005 or before, by Stephanie Jakus, Elizabeth McAninch, Amanda Schack, Laurel Miller-Sims, Caitlin Brady and Irene Song.

Honors Theses in Statistics, by Mariel Finucane, Suzanne Switzer, Emily Shapiro and Linjuan Qian