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.
Scientific computing and numerical analysis; Parallel multigrid and multilevel methods for large-scale coupled systems; Efficient numerical methods for reservoir simulation, fluid-structure interaction, and other applications.
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.
To each point on a curve, one can often associate in a natural way a line or plane (or higher dimensional linear variety) that moves with the point in the curve. This set of linear spaces is called a vector bundle. Vector bundles appear in a variety of questions in Physics (like the computation of Gromov-Witten invariants) . Moreover, they provide new insights into old mathematical problems and have been used to give beautiful proofs to long standing conjectures as well as striking counterexamples to some others.
data science, statistical signal processing, inverse problems, compressed sensing, information theory, convex optimization, machine learning, algorithms for geophysical signal processing, compressed sensing architectures and evaluation, video and image data acquisition and processing
The psychology of mathematical thinking, teachers' and students' understanding and use of inscriptions, multiplicative reasoning, applications of psychometric modeling for assessment and research in mathematics education.
Signal and image processing, tomographic image formation and object characterization, inverse problems, regularization, statistical signal and imaging processing, and computational physical modeling. Applications explored include medical imaging and image analysis, environmental monitoring and remediation, landmine and unexploded ordnance remediation, and automatic target detection and classification.