Computational and Applied Mathematics
Our faculty in this area work on a number of themes, including Numerical Analysis, Inverse Problems, Computational Science and Engineering, and the application of mathematical and computational tools to problems in Biology and Medicine. Research in these directions actively involves faculty as well as graduate and undergraduate students, with regular informal meetings and seminars and the Schlumberger-Tufts Computational and Applied Mathematics Seminar. Our work is highly interdisciplinary, with collaborations within the broader Tufts community, the many universities in the Boston area, and worldwide.
James Adler's expertise is in the study of complex fluids, including computational plasma physics and magnetohydrodynamics. Bruce Boghosian's work currently focuses on wealth inequality. Christoph Börgers' research interests include anomalous diffusion processes and differential equations describing nerve cells and neuronal networks. Misha Kilmer's work is in the area of numerical linear algebra and inverse problems, as well as in multilinear algebra. Xiaozhe Hu focuses on adaptive, parallel, and multilevel methods for partial differential equations and graph problems such as multiphase flow in porous media and magnetohydrodynamics. Todd Quinto uses microlocal analysis to understand inverse problems and develop and improve his tomography algorithms. James Murphy leverages methods of applied harmonic analysis, machine learning, and nonparametric statistics to develop new algorithms and analyze large, high-dimensional datasets. Our graduate students are closely involved in all aspects of this research.
Faculty
Bruce
Boghosian
Applied dynamical systems, applied probability theory, kinetic theory, agent-based modeling, mathematical models of the economy, theoretical and computational fluid dynamics, complex systems science, quantum computation
Current research emphasis is on mathematical models of economics in general, and agent-based models of wealth distributions in particular. The group's work has shed new light on the tendency of wealth to concentrate, and has discovered new results for upward mobility, wealth autocorrelation, and the flux of agents and wealth. The group's mathematical description of the phenomenon of oligarchy has also shed new light on functional analysis in general and distribution theory in particular.
Secondary projects include new directions in lattice Boltzmann and lattice-gas models of fluid dynamics, kinetic theory, and quantum computation.
Eric
Quinto
Tomography is an inverse problem, and the goal of tomography is to map the interior structure of objects using indirect data such as from X-rays. Integral geometry is the mathematics of averaging over curves and surfaces, and it is the pure math behind many problems in tomography. Integral geometry combines geometric intuition, harmonic analysis, and microlocal analysis (the analysis of singularities and what Fourier integral operators do to them). I have proven support theorems and properties of transforms integrating over hyperplanes, circles and spheres in Euclidean space and manifolds.
Because of the mentorship of Tufts physics professor and tomography pioneer, Allan Cormack (Tufts' only Nobel Laureate) I developed X-ray tomography algorithms for the nondestructive evaluation of large objects such as rocket bodies, and this motivated my research in limited data tomography
In limited data tomography problems, some tomographic data are missing. I developed a paradigm to describe which features of the object will be visible from limited tomographic data and which will be invisible (or difficult to reconstruct). I proved the paradigm using microlocal analysis. Often artifacts are added to tomographic reconstructions from limited data, and colleagues and I recently used microlocal analysis to prove the cause of these added artifacts and to predict where they will occur.
Collaborators and I have developed local algorithms for electron microscopy, emission tomography, Radar, Sonar, and ultrasound. In each case we use microlocal analysis to determine the strengths and weaknesses of the problem and to refine and improve the algorithms.
