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.


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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.