Partial Differential Equations and Mathematical Physics
Research fields
This broad area has many different facets. The fields described here are not exhaustive (some faculty do not fit into any of the bins) and not exclusive (there are personnel and even research topics that belong in more than one). The orderings are alphabetical.
Geometric Analysis and Nonlinear PDEs
Control theory at Texas A&M has a strongly geometrical flavor. Multiple solutions with different performance indices exist in many nonlinear partial differential equations and dynamical systems. Morse theory and other nonlinear functional analysis tools are used to find multiple critical points in a stable numerical way. Also, many problems of control theory boil down to challenging differentialgeometry issues, going back to the work of Elie Cartan but presenting many difficulties still today, which are under study here. Existence and number of solutions of nonlinear PDEs continue to be important questions (and are related to the multiple critical points mentioned above). Various aspects of geometric analysis on manifolds are considered, such as integral geometry, Liouville theorems, positive solutions, representations of solutions, and the Neumann dbar problem. Solitary waves are found in many areas of physics and mathematics, often under the names of solitons and nonlinear waves. Originally found in the KortewegdeVries equation, they were later discovered in many other nonlinear systems. Do these waves exist in a particular system of equations? Are they stable? Are they asymptotically stable? Does the set of nonlinear waves form an attractor of all finite energy solutions? Studying these questions involves tools from harmonic analysis and complex analysis, spectral theory, and numerical simulation.
Inverse Problems
Many objects of interest cannot be studied directly  for example, if they are not transparent. Such problems arise in medical diagnostical imaging, nondestructive industrial testing (e.g., the determination of cracks within solid objects), finding material parameters such as the conductivity of inaccessible objects, cargo inspection at harbors and border crossings, geophysical imaging, oil prospecting, and many other practical areas. In mathematical terms, one usually obtains differential equations containing unknown coefficients, which one attempts to determine using exterior (boundary) measurements.
Quantum Theory and Relativity
Periodic potentials are important in condensedmatter physics and form a focus of the spectral theory done here. Quantum field theory has always required and stimulated cuttingedge mathematics. Currently, vacuum (Casimir) energy is the fieldtheory topic of primary interest in our department. Quantum graphs are onedimensional networks that combine some properties of multidimensional systems with the analytical simplicity of ordinary differential equations; their theory has been actively developed at TAMU. Topological quantum field theories model exotic states of matter such as those appearing in fractional quantum Hall systems and topological insulators; these materials are being studied for their potential use in quantum computing devices.
Spectral Theory
Many problems of mathematical physics reduce to spectral analysis for differential (or other) operators. Among the issues arising one can mention the structure of the spectrum (e.g., absolute continuity), existence and location of spectral gaps, behavior of Green functions, spectral asymptotics, expansions into (generalized) eigenfunctions. The spectraltheory questions we are addressing are important in many areas of physics, chemistry, and other applications, including Anderson localization, carbon (and other) nanostructures, topological insulators, and metamaterials (e.g., photonic crystals and invisibility cloaks). Problems of uniqueness and existence in spectral theory often can be shown to be equivalent to particular cases of questions of completeness, Riesz bases, and frames in harmonic analysis. Methods from the area of gap and type problems in Fourier analysis can be applied in spectral theory via analogy between Fourier and Weyl transforms.
Faculty
This list includes persons whose primary identification is with another group. There are many overlaps with the Applied Mathematics and Interdisciplinary Research group.
Dean Baskin
Guy Battle
Gregory Berkolaiko
Andrea Bonito
Michael
Brannan
Goong Chen
Andrew Comech
Prabir Daripa
Ronald DeVore
Yalchin Efendiev
Ciprian Foias
Quantum field theory, asymptotics, eigenfunction expansions
JeanLuc
Guermond
Boris Hanin

Peter Howard
Junehyuk Jung
David Kerr
Thomas
Kiffe
Peter Kuchment
J. M.
Landsberg
Francis
Narcowich
Lee Panetta
Guergana Petrova
Michael Pilant
Alexei
Poltoratski
Bojan Popov

Eviatar
Procaccia
Topological quantum field theory, quantum computation
William Rundell
Michael
Stecher
Steven Taliaferro
Edriss Titi
Thomas Vogel
Jay Walton
Mariya
Vorobets
Yaroslav
Vorobets
Philip
Yasskin
Guoliang
Yu
Igor Zelenko
Jianxin Zhou

Postdocs and Other Visitors 
Graduate Students
Weston Baines 
Seminars
The seminars that most closely align with the PDE and Mathematical Physics group are
 Mathematical Physics, Harmonic Analysis, and Differential Equations Seminar
 Analysis/PDE Reading Seminar
Other related seminars at TAMU and vicinity include
 Other TAMU Mathematics Department seminars
 TAMU Physics Department seminars and colloquium (Look at all menu items under the "Events"tab.)
 Mathematical Physics Seminar at UTAustin (see also here)
 Analysis Seminar at UTAustin
 GeometryAnalysis Seminar at Rice University