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 Kurimoto-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.
Recent postdoc: Christensen