Graduate Fellows

• Rolando Estrada
  Bayesian Tree Analysis: Topology and Geometry Estimation via Stochastic Models

  Tree-like structures occur in retinal vessels, rivers, plant roots, and many other natural phenomena. When these structures appear in images, it is often desirable to tell the exact path of every branch and the topology of the entire tree. However, the reconstruction of tree topology and geometry from images is difficult, due to imaging limitations and the inherent complexity of trees, which includes branch occlusions, ambiguous cross-sections and varying local textures. These difficulties can be overcome by casting the image analysis problem in the framework of Bayesian estimation, thereby allowing the use of prior knowledge about typical trees in the domain of interest.

  The optimization problem resulting from a Maximum a Posteriori (MAP) formulation is computationally expensive,but may yield to Markov Chain Monte Carlo (MCMC) methods. In this thesis, we plan to adapt the mathematics of these stochastic methods to the peculiarities of tree-like structures, develop efficient algorithms for analyzing images of these structures, and apply the resulting methods to a sample of scenarios from medicine and biology.

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• Shishi Luo
  Mathematically modeling the antigenic evolution of influenza

  A major challenge in controlling rapidly evolving RNA viruses like influenza is that their evolutionary dynamics occur on the same time scale as their epidemiological dynamics. Understanding the overall dynamics of such viruses requires interfacing disease dynamics, traditionally modeled by nonlinear systems of differential equations, with molecular sequence evolution, traditionally studied using the tools of population genetics. While these processes have each been extensively studied, the study of their interaction in rapidly evolving diseases is relatively new, with researchers in the newly named field of ‘phylodynamics’ (Grenfell, et al., 2004) specifically focused on considering the feedbacks between population ecology and viral evolution. As part of my thesis and proposed work under the CTMS fellowship, I hope to integrate the knowledge and observations in biology with the tools of mathematics to address open biological questions about phylodynamics in the specific case of influenza.

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• Samuel Stanton
  Nonlinear Dynamical Systems for Renewable Power Generation in Complex Spectral Environments

  Advances in small-scale electronics have reduced power consumption requirements such that mechanisms for harnessing ambient kinetic energy for self-sustenance are a viable technology. Devices utilizing linear resonance to generate energy comprise the state-of-the-art but are fundamentally limited by their narrow-band response. Environmental energy sources typically exhibit spectral content distributed over a much broader range than is available to a resonant harvester. Next generation energy harvesting technology must capably perform in complex multi-frequency and stochastic mechanical environments. Accordingly, this research proposes new designs and analyses for devices exploiting purposefully nonlinear phenomena to meet modern challenges in vibration-based power generation. Electromechanical devices with nonlinear characteristics require more intricate analytical methods for performance prediction and design. Therefore, the focus of this research is to apply local and global predictive techniques to determine new performance metrics for purposefully nonlinear multi-degree-of-freedom electroelastic power generators operating in broadband, multi-frequency, and stochastic environments. Furthermore, we will study the influence of typically neglected intrinsic material nonlinearities that have been shown to manifest in a wide array of experimental devices.

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