## Faculty Research Areas

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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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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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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My main research interests are in the area of mathematical
modeling. In particular, I have been involved in the
mathematical modeling of processes that emerge from life
sciences such as epidemics, diseases, social behaviors, viruses
and cellular systems. It is important to remark that, life
sciences are expanding and the related topics are increasing
exponentially. The life sciences problems nowadays have
become a huge source of mathematical problems for different
areas of mathematics. Thus, it has been necessary to create
interdisciplinary groups to solve the challenge problems that
arise. The main mathematical tools that I have been using for
research are nonlinear dynamical systems, differential
equations, numerical methods, discrete mathematics, probability,
stochastic processes, scientific computing, and numerical
analysis. In particular I have applied these tools to study the
dynamics of influenza, Chagas, respiratory syncytial virus
(RSV), Chikungunya, Toxoplasmosis, obesity, crime, rumors and
others.

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Profile
My research is to develop and apply effective and efficient
mathematical approaches to solve the problems arising in systems
under uncertainty with an emphasis on interdisciplinary
research. Specifically, I am interested in uncertainty
quantification (UQ) in simulations, including model correction
based on physical constraints, parametric uncertainty analysis,
nonprobabilistic uncertainty modeling, and the application of UQ
techniques to various disciplines.

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

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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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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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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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My research interests are dynamical systems, nonlinear partial
differential equations and their applications. Currently, I
focus on the application of Geometric Singular Perturbation
Theory to Poisson-Nernst-Planck (PNP) models for ionic flows
through membrane channels. For the PNP problem, together with
bifurcation analysis and numerical simulations, I mainly focus
on (i) the current-voltage (I-V) relations, of particular
interest are the effect on I-V relations from finite sizes, ion
valences, small permanent charges and channel geometry including
random perturbation; and (ii) multiple solutions of the PNP
system and the corresponding stability problem.

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