Analysis and PDEs

The Analysis cluster at Tufts includes a range of specialties that cross over between pure and applied math, with common interests in harmonic analysis, PDEs, and integral geometry.

Harmonic analysis is an abstract generalization of Fourier theory that includes the study of Laplacians and their spectra on manifolds and in many other geometric settings. Partial differential equations (PDEs) are omnipresent in applications of mathematics to physics, from the time of Fourier and Maxwell onwards. Integral geometry is the study of integral transforms that integrate functions over curves or surfaces in Euclidean space or manifolds.

Fulton Gonzalez works in harmonic analysis on homogeneous spaces and in integral geometry, using tools from representation theory, functional analysis, differential geometry, and algebraic geometry. Todd Quinto's pure research involves the use of microlocal analysis in integral geometry, with applications to harmonic analysis and PDEs. Bruce Boghosian studies periodic solutions to PDEs that exhibit spatiotemporal chaos, such as the Kuramoto-Sivashinsky equations that govern the dynamics of flame fronts. James Adler's research uses tools from numerical analysis (including finite elements) for studying the solution of PDEs. Xiaozhe Hu's research studies the qualitative properties of PDEs such as stability and well-posedness for developing efficient and robust numerical methods. James Murphy uses methods from harmonic analysis, spectral graph theory, and high-dimensional statistics for theoretical machine learning.

Faculty

James Adler

Scientific computing and numerical analysis: Efficient computational methods for complex fluids, plasma physics, electromagnetism and other physical applications.

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.

Fulton Gonzalez

Noncommutative harmonic analysis, representations of Lie groups, integral geometry, and Radon transforms

Xiaozhe Hu

Associate Professor

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.

James Murphy

Assistant Professor

Machine learning, harmonic analysis, statistical learning, graph theory, data science, computational mathematics, image processing, signal processing

Kasso Okoudjou

Time-frequency analysis, pure, applied, and numerical harmonic analysis; analysis and differential equations on fractals and graphs.

Eric Quinto

Robinson Professor of Mathematics

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.