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Computational
Science
(Amundson, Dean, Glowinski, He, Kuznetsov,
Pan, Sanders, Hoppe) The computational science
program is currently the most active in the training
of PhD students in our department. Graduates from
this program are in demand in both industry and
academia. This group runs a weekly interdisciplinary
seminar.
The specialties in this group include numerical
linear algebra (K), finite element methods for
partial differential equations including domain
decomposition methods and fictitious domain methods,
optimal control and large scale optimization (D,G,H,Ho),
computational fluid dynamics (D,G,H,Ho,K,P,S) and
computer graphics (S).
There are many areas of computational science
of interest to prospective graduate students:
 Direct numerical
simulation of particulate flow (D,G,H,K,P); large
scale parallel computation
for particulate flow in visco-elastic, Newtonian
and non-Newtonian fluids (e.g., fluidization and
sedimentation of solid particles (G,K,P) and electro-
and magnetorheological fluids (Ho)); investigation
of advanced interior-point optimization methods
for the treatment of collisions (D,G) or in structural
optimization (shape and topology optimization (Ho));
efficient parallel solvers for diffusion problems
in heterogeneous anisotropic media (K); accurate
numerical algorithms for geophysical electrodynamics
(K); supersonic and hypersonic flows, particularly
for spacecraft reentry simulation
with equilibrium and non-equilibrium chemistry
(S); computational electromagnetics and numerical
methods for the simulation of flow in porous media
(G,Ho,K); Molecular Dynamics with applications
in materials science (Ho); biomimetics, in particular
biotemplating (Ho); radiosity methods for global
illumination in computer graphics (S).
Partial
Differential Equations
The group is active in training PhD students
and is a regular host of the Texas PDE Conference.
Current research areas that involve students and
postdocs include
- Conservation Laws (Canic, Christoforou,
Keyfitz, Wagner) Self-similar solutions of
multidimensional problems: degenerate elliptic
equations; classical solutions of free-boundary
problems and the development of efficient numerical
algorithms; existence theorems for systems
of mixed type (coupled hyperbolic and elliptic
equations); shock reflection problems~(Can,
K; postdoc Kim). Vanishing viscosity solution
to systems of conservation laws such as balance
laws and non-local problems with fading memory;
construction of admissible weak solutions as
limits of viscous and self-similar viscous
approximations. Continuous dependence of BV
weak solutions on physical parameters; applications
to Euler equations (Chr). Mathematical properties
of nonhyperbolic equations: singular shocks;
influence of viscosity~(K, Can; recent PhD
student Reiff; current PhD student Mora). Reacting
compressible flows; the dynamics of hyperelastic
materials~(W).
- Reaction-Diffusion Equations (Auchmuty,Fitzgibbon,Sanders).
Well-posedness and a priori estimates, convergence
of singular perturbations, and long-term dynamics
of abstract nonlinear evolution equations, integral
and partial integro-differential equations, and
parabolic and elliptic systems~(F); Stefan-Maxwell
diffusion, nonlinear boundary value problems,
and Hamilton-Jacobi equations~(S); analysis of
vortices and div-curl boundary value problems,
and eigenvalue problems for operators of classical
field theories and linear stability analyses~(A).
- Models from Engineering, Ecology and Environment (Auchmuty,Fitzgibbon).
A hybrid computational and analytical approach
to developing models in population dynamics;
spatially distributed ecological systems;
spatio-temporal
spread of infectious diseases; photo-chemical
production and atmospheric dispersion of
pollutants; rotating waves; and cardiac modeling~(A,
F; recent
PhD students Berry, Gross; current student
Martynenko).
Modern
Analysis
The Analysis Research Group has an excellent record of producing PhD students
and placing them in research positions. The
weekly functional analysis seminar is regularly
attended by faculty from area universities.
This seminar presents current research at a level
accessible to graduate students and is often used
as to present short courses. This group is a regular
host of {the Functional Analysis Day} at which
graduate students and postdocs from area universities
present their research.
Specific areas of current research include:
- C*-algebras associated to discrete and dynamical structures (Tomforde): This
project involves associating C*-algebras to discrete and dynamical systems (e.g., graphs, matrices, bimodules, shift spaces) in order to study the structure of the C*-algebra and its relation to the defining system. It deals with the questions: When can a given C*-algebra be modeled as the C*-algebra of more than one type of discrete or dynamical system? Can one translate well-known properties of the system into properties of the associated C*-algebra? At what level of generality can one use these methods to model classifiable C*-algebras?
- Classical field theories, particularly those
of electromagnetics and mechanics (Auchmuty):
Topics include representation theorems, trace
and embedding theorems, analysis of linear
operators, and variational principles. Related
issues include nonconvex duality in the calculus
of variations and convex analysis.
- Operator spaces and completely bounded maps (Blecher,Paulsen):
Operator spaces is a new area of functional analysis
providing powerful tools with applications to
unitary representations of groups, C*-algebras,
von Neumann algebras, and operator theory.
- Interpolation theory (Paulsen): existence
and construction of functions, belonging to a
special family, whose graphs pass through given
data points. A typical application is the construction
of an electrical circuit that at given frequencies
has given responses.
- Wavelet analysis (Papadakis): multiresolution
theory, frame theory and sampling
using techniques from functional and harmonic
analysis. See Featured Project.
Dynamical Systems and Ordinary
Differential Equations
 The dynamical systems group specializes in systems
with symmetry, ergodic theory, and applications.
This group runs the weekly interdisciplinary Nonlinear
Dynamics seminar jointly with colleagues in Physics
and Engineering and has hosted many visitors and
postdocs. Ian Stewart, University of Warwick, is
an adjunct professor at UH and a frequent visitor.
Current
theoretical interests include ergodic theory
and central limit theorems,
in particular for symmetric dynamical systems
(F,T); the structure of symmetric attractors (F,G,S);
coupled cell systems (F,G,J,S); generic properties
of diffeomorphisms and flows including transitivity,
ergodicity, and rapid mixing (F,T); classification
of cocycles over actions of certain groups with
application to rigidity properties (T); the extension
of spectral theory for selfadjoint differential
operators to the non-selfadjoint case (W).
Applications of equivariant dynamics
include pattern formation (G,S,T), bursting (G,J),
hypermeander of spirals based on ergodic-theoretic
techniques in symmetric systems (F,T), and neuroscience
(G,J,T).
Differential Geometry
Bao works on Riemann-Finsler geometry
and its applications to the physical sciences. Current
investigations concern constant curvature metrics
(the space forms problem), and conformal deformations
of Finsler metrics (encompassing issues from the
existence of Ein\-stein-Finsler metrics to the
flow of air traffic).
Topology
Friedberg studies topological algebra
and topological semigroups, specifically the representation
of uniquely divisible semigroups
as cone-like structures and the representation
of such semigroups by matrices.
Complex
Geometry and Complex Analysis
Ji
works on complex analysis and complex (CR)
geometry. Current research is on the algebraicity
problem of strongly pseudoconvex real analytic
hypersurfaces and on proper holomorphic mappings
between balls in different dimensional spaces (classification
of such mappings, and estimate of degrees of rational
maps).
Ru studies complex analysis and
(complex hyperbolic) geometry. Topics include the
geometric, function theoretic, and number theoretic
properties associated with an algebraic variety
and their connections. Faltings's work on the Fermat
equation is a notable illustration of such questions.
Automated Discovery and
Graph Theory
Fajtlowicz has developed GRAFFITI, an
interactive computer program capable of making
mathematical conjectures. This program generates
conjectures in graph theory, number theory, and
geometry. GRAFFITI has also made conjectures about
fullerenes, the new form of carbon. Conjectures
by GRAFFITI have inspired papers by mathematicians
(Alon, Chung, and Kleitman), computer scientists,
and chemists.
Both undergraduate and graduate
students can do research based on GRAFFITI, either
by continuing its development or by investigating
conjectures generated by the program. The accessibility
by undergraduates to original research is a special
feature of this work.
Algebra
Fields of interest of the algebra group include
number theory, group and module theory, rings and
nearrings, lattice and abstract ideal theory, logic
and foundations.
Hausen's research includes the relationship between
algebraic structures and associated sets of mappings
(e.g.~between groups and their automorphism groups).
Current research topics include determining whether
all nonzero abelian 2-groups with isomorphic automorphism
groups are isomorphic; characterizing the rings
$R$ such that every $R$-homogeneous map between
any two nonsingular $R$-modules is a homomorphism;
and generalizing the concept of $E$-rings and $E$-modules.
Kaiser studies lattices for logic and various
versions of normality in the context of universal
first order sentences, and epistemic logic (reasoning
about knowledge). The department cross-lists his
courses on automata and advanced graduate logic
with Computer Science.
Financial
Math
Financial mathematics is concerned
with mathematical/statistical modeling and
analysis of issues relating to risk management
in the financial and energy markets. Kao's
research interests include the applications
of computational probability and statistics in
pricing derivative products for risk management.
His recent work have evolved around problems such
as pricing NYMEX Gas Futures contracts, alternative
approaches for valuation of spread options, Markov
regime switching models for electricity spot prices,
and volatility estimation and its implications
in energy risk management.
Stochastic processes are a common model for systems
whose past history influences but does not determine
its future state. M. Nicol is interested
in the theory and application of stochastic processes.
For example, finding observations of Benford's law
in many types of multiplicative stochastic processes,
investigating the fine structure (such as Hausdorff
dimension) of invariant measures for stochastic processes
satisfying a contraction-on-average condition, and
determining statistical behavior of models arising
from economics, physics and biology, in particular, limit laws such as the
central limit theorem and Brownian-motion like behavior
in a wide variety of stochastic models.
Statistics
Decell's specialty is applied statistics: parameter
and density estimation, statistical pattern recognition,
the theory of generalized inverses, and statistical
signal processing of remote sensing data. His expertise
also includes applications to powered flight guidance
and orbital mechanics.
Peters has interests in Bayesian statistics and
deconvolution density estimation with applications
in geology.
Mathematical Biology
There are several faculty members who have done
work in mathematical biology, or whose work is
directly applicable to mathematical biology. Please
click here to
find more information.
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