Student Research

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Waverly Harden ’19

Dynamics of Hollow Elliptical Cylinder Arrays

Stipend and travel support by the NSF through grant DMS-1615037; additional travel support provided by the Grua/O’Conell Fund

Advisor: Christopher Chong

Strain-softening materials have gained significant interest in the physics, mathematics and material science communities recently. Most materials that we are familiar with are strain-hardening, meaning as they deform, they become harder to deform. Strain-softening materials have the opposite property; as they deform, they become easier to deform. The dynamics of strain-softening materials in an experimental setting have been rarely studied. For seven weeks, Waverly worked on the modeling and simulation of a system that exhibits strain-softening behavior. It consists of a chain of hollow elliptical cylinders (HEC). She visited the laboratory of her advisors collaborator, Professor Jinkyu Yang, at the University of Washington for the final three weeks of her project to work with the team who developed the concept of the HEC and the experimental set-up. Using the experimental data, she was able to refine her mathematical model. She also worked on an experimental apparatus that controls the impact frequency of the chain, and worked on processing of the experimental data that was acquired through digital image correlation.

Patrick Blackstone ’17 (Kibbe Institutional Research Fellowship)

Lp-Harmonic Analysis on SO(3,1)

Advisor: William Barker

The long term goal of this project was to determine the image of the Lp-Schwartz space (0 < p  2) under the operator-valued Fourier transform on the semi-simple Lie group SO(3,1), the proper Lorentz group. This would extend research conducted in the 1980s for the Lie group SL(2,R). Patrick’s work on SO(3,1) mirrored that of Andrew Pryhuber and Justin Dury-Agri for SU(2,1), though SO(3,1) presents a slightly more tractable group since it is only six dimensional as opposed to SU(2,1) which is eight dimensional. Of more significance, SO(3,1) has no “discrete series” of representations, a major simplification over SU(2,1). Patrick concentrated on the more algebraic foundations of the project, working out the necessary algebraic structure theory of SO(3,1). As a sophomore Patrick did not have the background to explore the more analytic aspects of the representation theory of SO(3,1), a requirement for tackling the ultimate Lp-Schwartz space problem. 

sharma.pngParikshit Sharma '17

Neural networks and transfer learning models in image subject classification

Advisor: Thomas Pietraho

In transfer learning, machine learning models trained to perform a specific task are retrained and repurposed to perform complementary tasks.  Often, lower order features learned by a neural network in one context can be reassembled into higher-order features that are useful in another.  In this project, we set out to classify the subject area of a book based solely on the image of its cover. Using transfer learning from  a number of pretrained image classification neural nets such a Google's Inception, the classification accuracy of our new models exceeded 70 percent.  This is near the accuracy of a human asked to perform the same task.

Sophie Bèrubè '16 and Tara Palnitkar '16

Diestel-Leader Groups are Graph Automatic

Supported by the NSF through grant DMS-1105407

Advisor: Jennifer Taback

Automatic groups were introduced in the 1990's in order to classify the groups associated with three dimensional "shapes," or manifolds.  A group is a set with an operation which satisfies certain basic properties, such as the integers under addition, and in general captures the symmetry of some, perhaps complicated, object.  In an automatic group, simple computational machines called finite state automata are used to describe the group elements as well as the group multiplication.  These automata streamlined computation within the group, leading to expedited solutions of geometric and computational questions in group theory.  Unfortunately, not all three manifold groups fell into this category, and so the definition was extended to that of graph automatic groups.  The latter retained many of the computational advantages of automatic groups while enlarging the class considerably.  

Sophie and Tara worked to show that a new infinite family of groups, called Diestel-Leader groups, or higher rank lampligher groups, are graph automatic.  This required gaining a deep geometric and algebraic understanding of these groups, as well as learning about different types of finite state automata.  Along the way, they proved that some related infinitely generated groups were also graph automatic.  This work will appear as an article in the Journal of Algebra.

jake.jpegJacob Hart ’17 and Carina Spiro ’18carina.jpeg  

Nonlinear Energy Harvesting

Stipends supported by the Maine Space Grant Consortium, travel supported by the NSF through grant DMS-1615037

Advisor: Christopher Chong

The last several decades as seen a burst of research activity in vibration energy harvesting, which deals with the conversion of mechanical energy (such as vibration) into electrical energy. Most commercial energy harvesters are based on linear responses, which have a relatively restricted range of vibration frequencies where energy harvesting is efficient. Jake and Carina explored the integration of novel aspects of nonlinear wave equations to try to harvest energy in new ways. In particular,they studied an array of cantilevered beams that are coupled via magnetic links. For the first six weeks of the project, they worked on the modeling of the system, its numerical simulation, and the computation of periodic orbits. They visited the laboratory of their advisors collaborator, Professor Chiara Daraio, at the ETH Zurich for their final two weeks to work with the team developing the experimental set-up of the array of cantilevers. Using the experimental data, they were able to refine their mathematical model.