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Faculty Research

Seung-Yeop Lee
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.

Alexandre Tartakovsky

Research interests

Computational methods:

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

Ghanim Ullah
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/

David Rogers
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/

Miller

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.


Motta

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 http://shell.cas.usf.edu/millerlab/index.html



Motta

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.


Duncan

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 http://chemistry.usf.edu/faculty/acevedo/


Arendash

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.


Beneteau

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, Discrete Wavelets, Real Analysis I. For more on Dr. Beneteau, visit her faculty page at http://math.usf.edu/faculty/cbeneteau/


Motta

Mohamed Elhamdadi, Low dimensional Topology, Knot Theory, K-Theory

Dr. Elhamdadi has been widely published for his research. Recent publications include: "Cohomology 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 groupoid cocycles.

"Cohomology 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.

"Virtual 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.

"Tangle 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.

"Cohomology 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 thus obtained.

"Cocycle 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/