Faculty Research Areas


Rakhim Aitbayev

Many phenomena in sciences and engineering are described by partial differential equations (PDEs). Exact solutions of most of real world problems are often very difficult or impossible to obtain. Solutions to such problems can be approximated using numerical methods. Numerical Methods for Solving PDEs is a vast area which deals with numerical errors, computational stability, parallel algorithms, efficient computation, and solution of challenging multi-physics problems. My research interests are in the following areas of Numerical PDEs: orthogonal spline collocation method, iterative methods for solving large sparse systems of equations, numerical solution of nonlinear elliptic PDEs, numerical solution of non-self-adjoint or indefinite problems, and interface problems.
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Ivan Avramidi

I work on problems at the intersection of global analysis on manifolds, Riemannian geometry and mathematical physics. The main part of my research is devoted to the study of spectral asymptotics of elliptic partial differential operators on manifolds via heat kernel methods and to the application of these methods to various problems of spectral geometry, mathematical physics, quantum field theory, quantum gravity and financial mathematics.
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Brian Borchers

My research interests are in optimization and applications of optimization techniques in inverse problems.  In optimization my research has involved interior point methods for linear and semi-definite programming with applications in integer programming.  The work on parameter estimation and inverse problems has involved optimization, statistics, and Tikhonov regularization.
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Anwar Hossain

My research focuses mostly on diagnostic methods and their applications to reliability and engineering models. These two areas are crucial within the statistical research community, and are also favorable at technical institutions. In the past few years, I have focused on other fields of statistics; the most attention has been given to cover theory and applications of multivariate analysis, survival estimation, Bayesian estimation, and reliability.
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Bert Kerr

My research focuses on determining analytic solutions of Boundary Value Problems. For example, the types of mixed BVP's which arise in analyzing the stress field in the vicinity of a crack, or in the study of flow past an obstruction. I am currently working with the Electrical and Microsystems modeling group at Sandia National Lab. I work on solving BVP's, referred to as compact models. The development of compact models that can simulate the excess carrier dynamics in undepleted semiconductor regions are important for many applications, including photonic and power devices.
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Oleg Makhnin

Modeling complex multidimensional data; in particular, spatial data. Precipitation modeling. Geostatistics. Bioinformatics. Statistical consulting.
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Michael Maroun

Currently, there exists a successful physical theory known as the Standard Model, which gives accurate predictions of physical phenomena. Yet it remains a fact that despite decades of research, there is no mathematically precise definition for a quantum field. Consequently, my research is focused on the mathematical analysis of Schrodinger operators with physically important but mathematically difficult potentials, the calculus of distributions and generalized functions as they pertain to quantum field theory, and mathematically precise extensions of the Feynman integral from quantum mechanical systems to the realm of quantum field theory.
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John Starret

My main area of interest is the topology of strange attractors, particularly the topological types of the interlocked periodic orbits. The template construction of Birman and Williams provides a theoretical framework with which the periodic orbit structure can be transferred to the study of knots and links on branched two-manifolds supporting a semi-flow. Some of my recent publications explored various suspensions of maps to flows, so that the simpler dynamics of periodic and chaotic orbits in a map can be mapped onto a flow generated by a smooth vector field. Recently I have been working with large coupled chaotic systems, investigating the synchronization of identical units.
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W.D. Stone

My research interests are Mathematical Biology and Mathematical Modeling. Recently I have been increasingly working with modeling of Fluids. In the past few years I have had students working on modeling of glaciers, size-structured populations, age-distribution of groundwater, resonances in the plasmosphere, and other areas of modeling.
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Bixiang Wang

My research is centered at partial differential equations and dynamical systems along with their applications. I am particularly interested in the infinite-dimensional dynamical systems generated by deterministic and stochastic PDEs defined in bounded or unbounded domains. My current work has involved random attractors, invariant manifolds, random periodic solutions and random almost periodic solutions for equations driven simultaneously by deterministic and stochastic forcing.
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