Skip to content

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 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.

Inverse Problems

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.

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 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).


Faculty

This list includes persons whose primary identification is with another group. There are many overlaps with the Applied Mathematics and Interdisciplinary Research group.

Guy Battle
Constructive quantum field theory, wavelets

Gregory Berkolaiko
Quantum graphs, spectral theory, stochastic DE, dynamical systems, quantum chaos

Goong Chen
Control theory, molecular quantum mechanics, applied and computational PDE, chaotic dynamics

Andrew Comech
Stability of solitary waves, harmonic analysis

Prabir Daripa
Fluid Mechanics, computation of free surfaces, Hele-Shaw and porous media flows, applied and computational PDE

Yalchin Efendiev
Heterogenous porous media and phase-flow

Ciprian Foias
Navier-Stokes equations

Stephen Fulling
Quantum field theory, asymptotics, eigenfunction expansions

Aditi Ghosh
Numerical solvers for elliptic PDE

Peter Howard
Stability of nonlinear waves

David Kerr
Dynamical systems

Thomas Kiffe
Integral equations, mathematical biology

Peter Kuchment
Inverse problems, spectral theory, PDEs, mathematical physics

J. M. Landsberg
Differential geometry, string theory

Bruce Lowe
Inverse problems

Francis Narcowich
Quantum mechanics, operator theory, approximation theory

Dmitry Panchenko
Spin glasses

Lee Panetta
Geophysical fluid dynamics

Guergana Petrova
Hyperbolic PDEs and conservation laws

Michael Pilant
Inverse problems, scientific computation

Alexei Poltoratski
Harmonic analysis, spectral theory

Bojan Popov
Conservation laws and linear transport

Eric Rowell
Topological quantum field theory

William Rundell
Inverse problems

Michael Stecher
Partial differential equations

Dustin Steinhauer
Inverse problems

Steven Taliaferro
Nonlinear partial differential equations

Thomas Vogel
Capillary surfaces

Jay Walton
Solid and continuum mechanics

Mariya Vorobets
Cauchy problems, Maxwell equations

Yaroslav Vorobets
Dynamical systems

Philip Yasskin
General relativity, twistor theory

Guoliang Yu
Noncommutative geometry

Igor Zelenko
Control theory, differential geometry

Jianxin Zhou
Control theory, nonlinear elliptic equations


Postdocs and Other Visitors

Jeffrey Bouas
Branimir Cacic
Alexey Sergeev
Alim Sukhtayev

Graduate Students

Hamilton Carter
Ngoc T. Do
Minh Kha
Ruomeng Lan
Alexis Olson
Fatma Terzioglu
Tracy Weyand


Seminars

The seminars that most closely align with the PDE and Mathematical Physics group are

Other related seminars at TAMU and vicinity include