Partial Differential Equations and Mathematical Physics
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 differential-geometry 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 d-bar 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 Korteweg-deVries 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.
Many objects of interest cannot be studied directly -- for example, if they are not transparent. Such problems arise in medical diagnostical imaging, non-destructive 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 condensed-matter physics and form a focus of the spectral theory done here. Quantum field theory has always required and stimulated cutting-edge mathematics. Currently, vacuum (Casimir) energy is the field-theory topic of primary interest in our department. Quantum graphs are one-dimensional 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.
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 spectral-theory questions we are addressing are important in many areas of physics, chemistry, and other applications, including Anderson localization, carbon (and other) nano-structures, topological insulators, and metamaterials (e.g., photonic crystals and invisibility cloaks).
This list includes persons whose primary identification is with another group. There are many overlaps with the Applied Mathematics and Interdisciplinary Research group.
J. M. Landsberg
Postdocs and Other Visitors
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