1. In ideal Hele-Shaw flows, the evolution can be tracked down by the pole singularities of a conformal mapping. A question is whether the cusp singularities are generic when there are large number of poles. One needs to solve a (big) system of ODEs.
2. Statistical properties of two-dimensional electrons has been studied using random matrix theory for a certain special temperature. For other generic temperature, it is much less studied due to theoretical difficulty. From physical argument, it is conjectured that the conformal field theory appears in the thermodynamic limit. Numerical simulation will help finding this important fact. Numerics will involve integrating over large number of variables therefore and will require Monte-Carlo simulation.
3. Complex dynamics of Schwarz function: It is well known that iterating a quadratic function leads to a beautiful fractal shapes called Julia sets and the Mandelbrot set. There is a generalization that replaces the quadratic function by a Schwarz reflection. Extremely beautiful fractal shapes arise and there are interesting properties to find out in the emerging shapes.
In addition to basic mathematics about complex variables, it is essential to use some computational software (such as Mathematica) and some programming language.
Multiscale mathematics for complex systems, computational fluid dynamics, stochastic differential equations, uncertainty quantification, Lagrangian particle methods.
Models of natural systems
Multiphase flow and contaminant transport in subsurface, ice sheet dynamics, sediment transport, geological CO2 sequestration, geochemistry, saltwater intrusion.
Computational needs of students
Scientific computing; FORTRAN, C++, MATLAB, MPI
Numerical methods for PDEs (finite difference, finite elements)
Numerical Particle Methods (MD, SPH, DPD, DEM)
Stochastic methods (statistics, Monte Carlo methods)
Parameter estimation, optimization
I use computational methods to gain deep insight into the function and dysfunction of biological systems. Particular areas of interest include neuronal disorders such as epilepsy, spreading depression, and Alzheimer’s disease, neuronal networks, ion channels, Markov chains, stochastic dynamics, differential equations, bifurcation theory, diffusion, nonlinear dynamics, calcium signaling, cell signaling pathways, and application of control theory to biology. I program in Fortran, Matlab, and Mathematica. Specialized tools that are required for my research include XPPAUT, QUB, and Neuron.
Personal Website: http://faculty.cas.usf.edu/gullah/
My group develops predictive models for new physics and chemistry that appear
when moving up from the atomic to the nano and micro-scale. To support this
goal, we are developing the thermodynamics of far-from equilibrium systems,
building functional data structures for supercomputing and applying Bayesian
inference to mine simulation data. Work in these topics builds on recent
advances in fundamental computer science, applied statistics, and
nonequilibrium physics and chemistry. Together, new developments in these
fields will allow unprecedented access to electron through device-level
simulations and analysis for materials design grounded in fundamental physics.
For more please visit: http://chemistry.usf.edu/faculty/drogers/
Casey Miller, Spintronics
The grand challenges of nanomagnetism are the creation, exploration, and understanding
of collective, emergent magnetic phenomena exhibited by nanoscale materials. My
interests focus on facets of these challenges related to spintronics, specifically
spin transport phenomena in multilayer magnetic heterostructures. My laboratory
employs two tools of nanoscience to investigate spin-dependent transport: geometric
confinement, and physical proximity. New properties arise when well known materials
are confined in one or more dimensions, for example, by growing them as thin films
and/or via nanolithography techniques. Similarly, novel phenomena originate from
the proximity of dissimilar materials in multi-component systems, such as thin film
heterostructures. Materials and devices are fabricated by a combination of thin
film growth and lithography, then characterized via cryogenic transport measurements
in high magnetic fields.
Spin-Dependent Tunneling for Spin-Electronics.
This figure shows an over simplified band structure representation of magnetic tunnel
junctions (MTJs) fabricated in Prof. Miller's Spintronics Lab. The parallel state
(a) has low tunneling resistance, while adensity of states bottleneck causes a relatively
high resistance in the antiparallel state (b). These different states serve as the
binary logic basis for data storage, sensing, and other spintronics applications.
For more on Casey Miller and his Spintronics lab, visit
Phil Motta was just notified that his NSF collaborative grant with Woods Hole, and
Mote Marine Lab will be funded starting in early 2009. The title of his grant is:
“Collaborative Research: Multi-sensory guidance of marine animal navigation and
prey capture”. The total award is for about $600K for all institutions for 2 years.
This new NSF project will help reinforce the link between USF and Mote and a fisheries
initiative with USF- Marine Science, and of course fits in nicely with our School
of Natural Sciences and Mathematics clusters.
Mildred Acevedo-Duncan, Signal Transduction Pathways
With current funding from Alaska Run for Women,
the Greenberg Breast Cancer Foundation, and the William & Ella Owens Medical
Research, Dr. Acevedo-Duncan's laboratory investigates signal transduction pathways
involved in cell cycle control in high-grade gliomas, one of the most radioresistant
forms of cancer for which there is no current successful therapy. These studies
explore whether certain protein kinase C (PKC) isozymes regulate specific cell cycle
phases in human glioma cells, since earlier studies have implicated elevated levels
of PKC in growth control of gliomas. In particular, recent findings suggest that
PKC may regulate the cell cycle through its interaction with cyclins and cyclin-dependent
kinases (CDKs). Activation of the CDKs is through phosphorylation on a threonine
residue by CDK-activating kinase (CAK).
To evaluate if PKC is involved in the mechanism of CAK activation, the phosphorylation
of CAK and one CDK family member, cdk2, was monitored throughout the cell cycle.
Results indicate that PKC inhibitors prevent hyperphosphorylation of CAK and cdk2
at specific points in the cell cycle, suggesting that PKC is involved in regulating
CAK. Current studies are exploring the mechanisms of CAK regulation by PKC, for
the ultimate purpose of identifying new molecular targets for development of novel
therapies to treat gliomas.
For more on Dr. Acevedo-Duncan’s lab, check out her faculty page at
Gary Arendash, Neuroscience Research
The Arendash laboratory is interested in understanding the pathogenesis of Alzheimer's
Disease, as well developing effective protective strategies and treatments against
the disease. My colleagues and I currently have 10 transgenic mouse lines that develop
Alzheimer's pathology (amyloid plaques or neurofibrillary tangles), with most of
these mouse lines also exhibiting age-related cognitive impairment. My laboratory
has emphasized cognitive assessment of Alzheimer's transgenic mice and correlating
cognitive performance to neuropathologic and neurochemical measures taken from the
same animals. To that end, we have developed an extensive behavioral battery of
cognitive, anxiety, and sensorimotor tasks that collectively characterize the behavioral
abilities of a given mouse. Recently, my laboratory has utilized high level multi-metric
analyses (function analysis, discriminate function analysis, and neural networks)
that look at behavioral performance across many behavioral measures to determine
cognitive impairment and/or treatment effects. Our use of multiple behavioral measures
and advanced statistical analysis in Alzheimer's transgenic mice have resulted in
a sensitive behavioral methodology for evaluating protective measures and therapeutics
against the disease.
Utilizing our Alzheimer's mice, we have evaluated a variety of possible protections
against the disease. In 2000, my colleagues and I published the initial report indicating
that vaccinations (with the human protein beta-amyloid) can protect Alzheimer's
mice against development of cognitive impairment. Most recently, we have developed
a novel vaccine approach against Alzheimer's Disease that is long-lasting and without
any deleterious side effects. An additional protection-based approach we have studied
is environmental enrichment. Given the epidemiologic evidence that a life-long pattern
of high cognitive activity may protect against Alzheimer's disease, our studies
have determined that Alzheimer's mice raised in an enriched environment are indeed
protected against cognitive impairment later in life.
In addition to "protection-based" studies, we are also investigating a
variety of treatments against Alzheimer's disease in our transgenic mice that already
have cognitive impairment. Both vaccinations and cognitive stimulation result in
improved cognitive performance that often attains normal levels. Several dietary
strategies are currently being investigated to protect against or treat Alzheimer's
disease in our transgenic mice.
Since last year, I have been closely associated with the Byrd Alzheimer's Center
and Research Institute, which is on the USF campus. The Byrd Institute is a state-wide
research center, which is bringing Alzheimer's researchers from throughout the state
of Florida together in attacking the disease. USF and the Byrd Institute were just
awarded an Alzheimer's Disease Research Center (ADRC) grant from NIH, which is the
first and only ADRC in the state. Through USF and the ADRC, I will continue my investigations
of environmental factors that may protect against or treat Alzheimer's disease.
Catherine Beneteau, Complex Function Theory
A native of Canada, Dr. Beneteau brings to USF her passion for research and education.
With current research funding from the National Science Foundation for her project
“Collaborative Research: A Phase II Expansion of the Development of a Multidisciplinary
Course on Wavelets and Applications.” Dr. Beneteau’s research interests include
Complex analysis, analytic function spaces, Hardy and Bergman spaces, interpolation,
non linear extremal problems. She is also very interested in mathematics education
and in curriculum development issues. She teaches
Intermediate Analysis II,
Real Analysis I. For more on Dr. Beneteau, visit her faculty page
Mohamed Elhamdadi, Low dimensional Topology, Knot Theory, K-Theory
Dr. Elhamdadi has been widely published for his research. Recent publications include:
of the adjoint of Hopf algebras " (with J. Scott Carter, Alissa
Crans & Masahico Saito), Journal of Generalized Lie Theory and Applications,
vol 2 (2008), no 1, 19-34. Abstract: A cohomology theory of the
adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic
techniques. Explicit calculations are provided in the cases of group algebras, function
algebras on groups, and the bosonization of the super line. As applications, solutions
to the Yang-Baxter equation are given and quandle cocycles are constructed from
of Categorical self-distributivity" (with J. Scott Carter,
Alissa Crans & Masahico Saito), Journal of Homotopy and Related Structures,
vol 3 (2008), no 1, 13-63. Abstract: We define self-distributive
structures in the categories of coalgebras and cocommutative coalgebras. We obtain
examples from vector spaces whose bases are the elements of finite quandles, the
direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive
operations of these structures provide solutions to the Yang-Baxter equation, and,
conversely, solutions to the Yang-Baxter equation can be used to construct self-distributive
operations in certain categories. Moreover, we present a cohomology theory that
encompasses both Lie algebra and quandle cohomologies, in analogous to Hochschild
cohomology, and can be used to study deformations of these self-distributive structures.
All of the work here is informed via diagrammatic computations.
knot invariants from group biquandles and their cocycles "
(with J. Scott Carter, Masahico Saito, Dan Silver & Susan Williams), arXiv:math.GT/0703594,
To appear in Journal of Knot Theory and its Ramifications. Abstract:
A group-theoretical method, via Wada's representations, is presented to distinguish
Kishino's virtual knot from the unknot. Biquandles are constructed for any group
using Wada's braid group representations. Cocycle invariants for these biquandles
are studied. These invariants are applied to show the non-existence of Alexander
numberings and checkerboard colorings.
embeddings and quandle cocycle invariants ," (with Kheira
Ameur, Tom Rose, Masahico Saito & Chad Smudde), to appear in Experimental Mathematics.
Abstract: To study embeddings of tangles in knots, we use quandle
cocycle invariants. Computations are carried out for the table of knots and tangles,
to investigate which tangles may or may not embed in knots in the tables.
of Fronenius algebras and the Yang_baxter equation " (with
J. Scott Carter, Alissa Crans & Masahico Saito), arXiv:math.GT/0801.2567, To
appear in Communications in Contemporary Mathematics, vol. 10 (1), (2008), pp 1-24.
Abstract: A cohomology theory A cohomology theory for multiplications
and comultiplications of Frobenius algebras is developed for in low dimensions,
in analogy with Hochschild cohomology of bialgebras, based on the deformation theory.
Concrete computations are provided for key examples. Skein theoretic constructions
give rise to solutions to the Yang-Baxter equation, using multiplications and comultiplications
of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices
deformations of Algebraic Identities and R-matrices " (with
J. Scott Carter, Alissa Crans & Masahico Saito), arXiv:math.GT/08. Abstract:
For an arbitrary identity L=R between compositions of maps
L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition
is given. These 2-cocycles correspond to those obtained in deformation theories
of algebras. The construction is applied to a canceling pairings and copairings,
with explicit examples with calculations. Relations to the Kauffman bracket and
knot invariants are discussed.
For more on the research and scholarly products of Dr. Elhamdadi, please visit his
faculty page at http://math.usf.edu/faculty/melhamdadi/