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Faculty Individual Research Interests
Department of Mathematics & Statistics
Faculty Individual Research Interests
S. Ejaz Ahmed  Stephen Anco  H. BenElMechaiekh  Chantal Buteau
Babak Farzad  Henryk Fukś  Mei Ling Huang  Omar Kihel  Yuanlin Li
Alexander Odesskii  Bill Ralph  Jan Vrbik  Thomas Wolf
Xiaojian Xu  Wai Kong (John) Yuen
My area of expertise includes statistical inference, high dimensional data analysis Shrinkage estimation, statistical quality control, and asymptotic theory and its application. The high dimensional data analysis is a hot topic for the statistical research due to continued rapid advancement of modern technology that is allowing scientists to collect data of increasingly unprecedented size and complexity. Examples include epigenomic data, genomic data, proteomic data, highresolution image data, high frequency nancial data, functional and longitudinal data, and network data, among others. Simultaneous variable selection and estimation is one of the key statistical problems in analyzing such complex data. This joint variable selection and estimation problem is one of the most actively researched topics in the current statistical literature. More recently, regularization, or penalized, methods are becoming increasingly popular and many new developments have been established. The shrinkage estimation strategy is playing in important role in this arena. Currently, I am working on the following problems including:
 Shrinkage Estimation for High Dimensional Data Analysis
 Difference Based Shrinkage Analysis in High Dimensional Partially Linear Regression
 Improved Estimation Strategies in Generalized Linear Models
 Shrinkage Estimation and Variable Selection in Multiple Regression Models with Random Coecient Autoregressive Errors.
My research lies in several areas of nonlinear differential equations, integrability and solitons, mathematical physics and analysis.
Some problems I am currently working on include:
 new exact solutions of radial nonlinear Schrodinger equations and wave equations in n dimensions,
 "hidden" conservation laws of fluid flow equations and related potential systems,
 integrable groupinvariant soliton equations and their derivation from curve flows in geometric manifolds,
 symmetry and conservation law structure of wave maps and Schrodinger maps,
 symmetries and conservation laws in curved spacetime for Maxwell's electromagnetic field equations, gravity wave equations, and other fundamental physical field equations,
 exact monopole, plane wave, Wittenansatz solutions in a nonlinear generalization of YangMills/wave map equations,
 novel nonlinear generalizations (deformations) of YangMills equations for gauge fields, and Einstein's equations for gravitational fields
In addition I am coauthoring two books with G. Bluman in the area of symmetry methods and differential equations, in the Applied Mathematical Sciences series of SpringerVerlag. The first book provides an introduction to symmetry methods for both ordinary and partial differential equations, as well as a comprehensive treatment of first integral methods for ordinary differential equations. The second book will cover conservation laws (local and nonlocal) and potential systems for partial differential equations, and Bluman's nonclassical method of finding exact solutions.
I also have an active interest in symbolic computation using Maple and some of my research in symmetry and conservation law analysis makes use of this software and involves development of algorithmic computational methods.
My research interests are in topological methods in nonlinear analysis with focus on setvalued analysis and its applications to fixed point theory, mathematical economics, game theory and optimization. I am particularly interested in the solvability of nonlinear inclusions where classical hypotheses of convexity fail. Methods include a blend of topology, functional analysis, and nonsmooth analysis.
My research interests are in Mathematics Education and Mathematical Music Theory. In terms of Mathematics Education, I'm particularly interested in the use of technology in mathematics learning and teaching, university mathematics education, and mathematics teacher education. My recent and current work in this area include:
 University mathematics student learning through designing, programming, and using interactive computer environments for the investigation of mathematics concepts, theorems or conjectures, or realworld situations (i.e., microworlds)
 Computer Algebra System (CAS) use in University Instruction, and departmental integration
 Epistemic mathematics computer game
In terms of Mathematical Music Theory, I'm particularly interested in modeling motivic (melodic) structure and analysis of musical compositions through a topological approach. My ongoing interdisciplinary research mainly involves:
 concrete applications to a music corpus (using a model implementation in JAVA)
 categorical extension of the model
 issues in Computational Music Analysis (CMA).
Broadly speaking, my research interests lie in the diverse area of Discrete Mathematics and Theoretical Computer Science. In particular, I am interested in graph colouring and algorithmic graph theory. Structure of colourcritical graphs, a variety of choosability problems in planar graphs, Vizing's conjecture and improper colourings are some of my current projects in this area.
Recently, I have become interested in algorithmic game theory. Some projects that I am currently working on include: a variety of network flow problems, sponsored search, online auctions with perishable goods and game theoretical aspects of small world models.
My research interests can be summarized as theory and applications of spatiallyextended discrete dynamical systems. I am interested in mathematical modeling of physical, biological, and social systems using cellular automata (CA), lattice gas automata (LGA), and agentbased systems.
I am also studying dynamics of discrete complex systems with nontrivial topologies, such as random graphs, smallworld networks, and scalefree networks.
All these areas can be categorized as follows:
 Theory:
 Generalization of CA
 Additive invariants in CA
 Phase transitions in CA
 Computation in CA
 Modeling:
 Growth of complex networks
 Models of granular and traffic flow
 Models of language acquisition
 Discrete models of diffusion and spread
 Software:
 Agentbased simulations of complex systems
 Algorithms for CA/LGA simulations
Some problems I am currently working on include:

Enumeration of preimages in elementary cellular automata rules

Search for nonadditive invariants in cellular automata rules

The problem of structural stability in discrete systems

Density classification problems in dimensions higher than 1

Shiftdynamical systems in topologies other than Cantor topology

Models of language vocabulary growth (involving scalefree networks)
I am also working on development of HCELL library for manipulation and simulation of onedimensional cellular automata
My research interests are in exploration of new efficient, optimal methods of statistical inference for distribution function, quantile and regression, and development of computing and simulation methods with applications to survival analysis, network and stochastic models. I am working on the following topics.

Nonparametric distribution, quantile and regression estimation and testing are important research directions with many applications. I have been studying several methods in this field.Study weighted empirical distribution function to develop more efficient estimation and testing methods. For example, explore more efficient nonkernel quantile estimation methods. Study properties of these estimators and tests: consistency, rate of convergence, efficiencies. Computational methods and simulation methods also are developing.Develop new prediction methods for stochastic processes. For example, use sample path of martingales and Markov processes. Apply these methods to economics, quality control, queueing networks, insurance and biostatistics.

Studies of truncated and censored data have important applications in biostatistics, industrial engineering and other fields. The topics are:Search efficient estimation methods for truncated data of types of heavy tail distribution for example, simulating and estimating waiting time of using Internet or other stochastic models by using Pareto distribution.Develop efficient estimation methods in survival analysis and its applications. For example, predicting recovery times of cancer patients, estimating the value at risk of stocks.
My research lies in finite field functions and their applications to coding theory and cryptography; existence of primitive polynomials over finite fields; exponential sums over finite fields.
My research interest lies in the areas of groups, rings, group rings and combinatorial number theory. The group ring of a group G over a commutative ring K is the ring KG of all formal finite sums: u= \sum a_g g, and is an attractive object of study. Here group theory, ring theory, commutative algebra and number theory come together in a fruitful way, and moreover the study of group rings has important applications in coding theory. My recent research work has thrown light on structures of group rings and their unit groups. I am also interested in studying homological properties of modules and rings. In addition, I investigate the interplay between rings and their graphs (such as zerodivisor and annihilating ideal graphs). A few years ago, I started a new exciting research initiative and extended my research interest into the combinatorial number theory by investigating a few combinatorial problems (e.g. zerosum problems) in that filed. Some of my ongoing research projects are listed below:
1. Zassenhaus conjectures and related problems.
2. The normalizer problem and Coleman automorphisms.
3. Generators of large subgroups of (central) unit groups of group rings.
4. Index of a sequence of a _nite cyclic group.
5. The ErdosGinzburgZiv Theorem and its improvement.
6. Morphic groups and related problems.
7. Zerodivisor (annihilating ideal) graphs of (group) rings.
8. Morphic and reversible group rings.
9. Combinatorial problems in group theory and ring theory.
10. Injectivity of modules and related topics.
My main research interests are in Mathematical Physics in the sense of Mathematics inspired by ideas that come from Theoretical Physics. More precisely, I am interested in algebraic and geometric structures which come from quantum field theory, statistical mechanics and the theory of integrable systems.
My mathematical research began in algebraic topology with the study of exotic homology and cohomology theories and their connections with Banach Algebras. After that, I developed a transfer for finite group actions and studied a number that appears in the transfer that I call the "coherence number" of the group. Lately, I've been using the Hausdorff dimension of the orbits of dynamical systems to generate mathematical art.
My educational research concerns the use of multimedia and computer software to teach mathematical concepts. I recently published a complete multimedia treatment of first year calculus called "Journey Through Calculus."
My main research interest over the past two decades has been developing and perfecting an analytical solution to the perturbed Kepler problem. The resulting algorithm has been then applied to a series of both traditional and outstanding (e.g. formation of Kirkwood gaps) problems of Celestial Mechanics. The technique proves to be particularly useful when investigating resonances and the ensuing weak chaos.
More recently, I have become interested in Statisticsrelated issues concerning Edgeworthseries improvement over the basic Normal approximation. This involves developing computational techniques to find the first four cumulants of the sample statistic (usually a parameter estimator) whose sampling distribution is being investigated.. One can also correspondingly correct the chisquare distribution of the goodnessoffit test, etc. A further extension is possible to situations involving sampling (and subsequent parameter estimation) of TimeSeries (where the individual observations are no longer independent).
My research interests include differential equations and their integrability, computer algebra, General Relativity and mathematical aspects of the game of Go.
Work in computer algebra concerns algorithms to simplify and solve overdetermined systems of equations (linear/nonlinear), (algebraic/ordinary differential (ODEs)/partial differential (PDEs)). These basic algorithms are applied in higher level programs for the determination of symmetries, conservation laws, Lax Pairs, Recursion operators or other properties of differential equations. Applications include the classification of a wide range of integrable systems, for example, systems of evolutionary scalar PDEs, vector PDEs, single and systems of supersymmetric evolutionary PDEs and integrable quadratic Hamiltonians with higher degree first integrals.
A new method and its implementation for solving extremely large sparse linear algebraic systems is applied in the study of noncommutative Laurent ODEsystems.
Attempts to increase the efficiency of related programs lead to a study of the parallelization of my algorithms and programs. Currently I am the Brock site leader of the SHARCNET consortium that operates large computer clusters.
An old hobby of mine concerns the computerization of the Asian game of Go. My algorithms and programs specialize on life and death problems in Go. Recent work includes the modelling of Gopositions as discrete dynamical systems and the mathematical analysis of Semeai and Seki positions.
My research lies in several areas of experimental designs, robust inferences, and survey sampling.
Some problems that I am currently working on include:

Constructing exact designs that provide optimal solutions for a variety of inferences.

Analysis for robustness of experimental designs against different model violations.

Optimal planning for accelerated life testing experiments.

Optimal designs for mixed models.

Robust designs for nonlinear models.

Optimal methods for statistical inferences in indirect sampling.
My research interests are in bounding the convergence rate and optimizing the efficiency of various Markov chain Monte Carlo (MCMC) algorithms in high dimensions with the aid of extensive and computationally intensive simulation studies. Over the last 15 years, MCMC algorithms are widely used, largely due to their general applicability to Bayesian inference problems. Therefore, monitoring the convergence of these algorithms has become an important topic. One important class of algorithms called local MCMC algorithms refers to one with the property that the transition of the underlying Markov chain is local, e.g. random walk Metropolis algorithms. My research focuses on local MCMC algorithms, typically on unbounded Euclidean state spaces, motivated by distributions such as those encountered in Bayesian analysis. Currently, I am working on the following problems:
 Bound the convergence rate of these algorithms quantitatively, using techniques developed on discrete state space.
 Optimize the convergence rates by proper scaling of the underlying Markov chain, with significant implications to algorithms in high dimension.
 Apply MCMC algorithms to a Bayesian model for baseball predictions.
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