Faculty and students in Applied Mathematics are an integral part of the Department of Mathematics and Statistics at UNM.

Research in applied mathematics is active and broad, benefiting greatly from close collaborations with scientists at the Sandia and Los Alamos National Laboratories, the Air Force Research Laboratory, and several Science and Engineering Departments at the University.

There is also close interaction with the programs in pure mathematics and statistics, and our students are encouraged to take courses in these programs and in other departments.

Research topics explored by our faculty and students include analysis and numerical simulation of wave propagation; dynamics of fluids; mathematical biology, in particular dynamics of infectious diseases and modeling of protein folding; applications of mathematics to particle accelerators, nonlinear optics, continuum mechanics, and electromagnetic lens design; numerical and symbolic computing for partial differential equations.

Alejandro Aceves

My main area of research is in Nonlinear Wave Phenomena as it applies to problems in Nonlinear Optics. Concretely, I have worked on the modeling of optical pulse and beam dynamics as they propagate in a medium that has a nonlinear response to the electric field. My most recent work include: (with former PhD student Dohnal) light bullet trapping in a nonlinear photonic waveguide; (with PhD student Sukhinin and Physics colleague Dr. JC Diels) Ultraviolet laser light filamentation in air and (with PhD student Srinivasan) nonlinear Anderson localization in a disordered optical fiber array system.

On a limited basis I have done research on modeling in epidemiology and in neuroscience. I hope, time permitting, to engage more in these research areas.

Asymptotics and singular perturbations, high fidelity numerical methods, especially spectral methods. The study of bifurcation and transition phenomena in fluids and plasmas by the blending of asymptotic, variational and numerical techniques. Scientific computing and modeling. Past areas of application include vortex dynamics, the interaction of vortical structures with rigid walls and the formation and evolution of coherent vortices in forced shear layers and in decaying turbulent flows in wall bounded domains.

More recent interests include the development of techniques for high-accuracy modeling of the structure of macromolecules, such as proteins and nucleic acids. My work has focused on developing computational geometry algorithms for the problem of loop closure in proteins. I am currently applying these to the refinement of homology models and the design of efficient Monte Carlo methods for local structure refinement, using both physical and knowledge based force fields.

Jim Ellison

I'm an Applied Mathematician actively involved in Dynamical Systems, Probabilistic Methods, Stochastic Processes, Perturbation Theory, and Scientific Computing. I have worked on both fundamental and formal aspects of averaging methods in both deterministic and stochastic dynamical systems, but always with an eye for proofs of asymptotic validity.  Our main current focus is on problems defined by mean field Vlasov, Vlasov Fokker-Planck and Vlasov-Maxwell systems. The applications I have worked on are related to Beam Dynamics in Modern Particle Accelerators and Particle Channeling in Crystals .

For example, we are investigating coherent radiation and space charge in the context of charged particle bunches evolving under the mean field Vlasov-Maxwell dynamics.  This involves both analytic and numerical (e.g., meshless) approaches and the development of fast numerical algorithms in a high performance computing environment. In addition, we are beginning a study of the probabilistic Large Deviation Theory as a tool to study large N particle dynamics and the above mean field approximations.

Pedro Embid

My research interests lie in partial differential equations, functional and asymptotic analysis, and its applications to the physical sciences. I have done work in fluid dynamics, including compressible flows, multi-phase flows, reactive flows at low and high Mach numbers, and geophysical fluids.  Currently I am involved in projects concerning the modeling of rapidly rotating and strongly stratified fluids, advection diffusion equations in hydrology, and image reconstruction in optics.

Alexander Korotkevich

My research interests are primarily concentrated on nonlinear waves in different media. In many cases, for such interesting and complex phenomena, the only available tool for solution of the equations is a numerical simulation. I have extensive experience in the simulation of nonlinear interaction of waves on the surface of fluids and wave turbulence. I was also involved in the creation and implementation of the parallel algorithms for simulation of light propagation in high bandwidth optical fibers. I prefer problems of applied mathematics which have motivation in theoretical physics. Recently I participated in research of high energy laser beam propagation in a nuclear fusion plasma, which was motivated by the creation of the National Ignition Facility.

Recent development of metamaterials --- artificial optical materials having properties which are absent in natural light propagation media, proposed a challenging problem of simulation of light propagation in such a nanoscale optical structures. Usually numerical code for simulation of such a nanocomposite involves solution of Maxwell equations for very specific geometry and boundary conditions. Solution of 3D Maxwell equations were always a challenging problem and this case is not an exception. Currently I am involved in the creation of algorithms and implementation of such specialized Maxwell solver.

Jens Lorenz

My areas of reseach are Applied Analysis and Scientific Computing. Specific problems that I have worked on lately include: the stability of shock and spiral waves; the modeling of ocean circulations for climate studies; the transition from smooth to rough attractors and Lyapunov-type numbers.

I am broadly interested in partial differential equations and their numerical approximation. Most of the applications I have worked on are related to fluids, but a recent interest is a nonlinear Black-Scholes equation in financial mathematics. The Black-Scholes equation is used extensively for option pricing; it becomes nonlinear if one assumes dependence of the volatility on the asset price.

My research interests include a wide range of topics in applied mathematics, nonlinear waves and theoretical physics. Among them are laser fusion and laser-plasma interaction; dynamics of fluids with free surface, Kelvin-Helmholtz instability and nonlinear interactions of surface waves; theory of the wave collapse, singularity formation and its application to plasma physics, hydrodynamics, biology and nonlinear optics; bacterial aggregation, chemotaxis, cell-cell interactions; collapse of bacterial colonies, stochastic Potts model of biological cell; pattern formation in photorefractive crystals and other nonlinear optical media; high-bit-rate optical communication; dispersion-managed optical fiber systems; soliton propagation in optical systems; high performance parallel simulations of optical fiber systems; Bose-Einstein condensation of ultracold dipolar gases.

Stanley Steinberg

My research focuses on the use of numerical and symbolic computing techniques for solving partial differential equations (PDEs) and most recently on the modeling of signal transduction in cells. I am co-director of the Center for the Spatiotemporal Modeling of Cell Signaling (STMC), a NIH/NIGMS Center of Excellence in Complex Biomedical Systems Research located in the Health Sciences Center (HSC) at the University of New Mexico , and with co-leaders and members in the College of Engineering and College of Arts and Sciences and at Sandia National Laboratory.

We work on integrating mathematical, statistical and computational modeling into ongoing research on complex cell signaling networks. My current numerical interest is in developing mimetic numerical methods for PDEs. These methods are based on discretizing a mathematical theory such as vector calculus or differential forms, rather than discretizing individual problems.  

Deborah Sulsky

Helen Wearing

Mathematical biology: I'm broadly interested in using mathematical models to understand the biological processes that shape population and community dynamics, with a particular interest in the ecology and evolution of infectious diseases. My research uses a combination of analytical, statistical and computational tools that are drawn from the fields of dynamical systems and stochastic processes.