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Faculty Individual Research Interests
Faculty Individual Research Interests
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 group-invariant 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, Witten-ansatz solutions in a nonlinear generalization of Yang-Mills/wave map equations,
- novel nonlinear generalizations (deformations) of Yang-Mills 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 Springer-Verlag. 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 set-valued 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 non-smooth analysis.
My research field is the Mathematical Music Theory. I am particularly interested in modeling motivic (melodic) structure and analysis of musical compositions through a topological approach.
The motivic analysis of a music composition consists of identifying the short melody, called a motif, that unites the composition through its strict repetitions, the so-called imitations, and its variations and transformations which are heard throughout the whole composition. Mainly using group theory, linear algebra and general topology concepts, we construct a (T_0-) topological structure corresponding to the motivic hierarchy of a composition. Our program (JAVA) Melos can analyse music compositions such as Schumann’s Dreamery from Kinderszenen .
My ongoing interdisciplinary research mainly concerns:
- Concrete applications to a music corpus;
- A categorical extension of our model including e.g. continuous functions between 2 motivic spaces, products of different spaces, natural transformations (gestalt spaces);
- Visualisation of Melos’ multiple outputs in order to show and hear, and to explore mathematics and music results.
Regarding mathematics education I’m interested in developing tools using music for the exploration of mathematics concepts.
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 colour-critical 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, on-line auctions with perishable goods and game theoretical aspects of small world models.
My research interests can be summarized as theory and applications of spatially-extended discrete dynamical systems. I am interested in mathematical modeling of physical, biological, and social systems using cellular automata (CA), lattice gas automata (LGA), and agent-based systems.
I am also studying dynamics of discrete complex systems with non-trivial topologies, such as random graphs, small-world networks, and scale-free networks.
All these areas can be categorized as follows:
- Generalization of CA
- Additive invariants in CA
- Phase transitions in CA
- Computation in CA
- Growth of complex networks
- Models of granular and traffic flow
- Models of language acquisition
- Discrete models of diffusion and spread
- Agent-based 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 non-additive invariants in cellular automata rules
The problem of structural stability in discrete systems
Density classification problems in dimensions higher than 1
Shift-dynamical systems in topologies other than Cantor topology
Models of language vocabulary growth (involving scale-free networks)
I am also working on development of HCELL library for manipulation and simulation of one-dimensional 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 non-kernel 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 focus over the past decade has been the Sobolev and Besov spaces of differentiable functions, which are vital in the study of partial differential equations and statistical estimation.
Now I am looking at the recently developed wavelet expansions (of Hermite-type) in these smoothness spaces. Together with an undergraduate student at Brock, I am looking to using them to rigorously account for some intriguing experimental (MATHEMATICA) computations Professor Vrbik has made involving the Central Limit Theorem. Professor Mei Ling Huang and I propose to explore the application of wavelets to probabilty density estimation.
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.
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 munber theory come together in a fruitful way, and moreover the study of group rings has important applications in coding theory. Units play a very important role in the investigation of the relation between the group-theoretic structure of G and its group ring. My recent research work has thrown light on the structure of the unit groups in the group rings. I am also interested in studying the homological properties of modules and rings.
Currently, I am working on the following problems:
Zassenhaus conjectures and related problems.
The normalizer problem and class-preserving automorphisms.
The central units and hypercentral units in arbitrary group rings.
Units in alternative loop rings.
CS-property and related problems.
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 decade has been developing and perfecting an analytical (even though iterative) solution to the Kustaanheimo-Stiefel formulation of the Kepler problem. The resulting algorithm has been then applied to a series of traditional (and sometimes outstanding, e.g. the formation of Kirkwood gaps) problems of Celestial Mechanics, to demonstrate its utility. The technique proves to be particularly useful when dealing with resonances, a topic I intend yet to explore in more detail.
Quite recently, I have become interested in Statistics-related issues concerning asymptotic behaviour of sampling distributions of various least-square estimators. In particular, this involves developing computational techniques to find the corresponding moments (expanded in negative powers of n, the sample size) and, based on these, reconstructing the sampling distribution as accurately as possible.
I am also involved in collaborative research with B. M. Singh in the area of elasticity, cracks and fractures (with engineering applications), and with S. M. Rothstein in Monte Carlo simulation of various molecular properties (Quantum Chemistry).
My research interests include differential equations and integrability, computer algebra, General Relativity and special aspects of optimization and artificial intelligence.
Work in computer algebra concerns algorithms to simplify and solve overdetermined systems of equations (linear/non-linear), (algebraic/ordinary differential (ODEs)/partial differential (PDEs)). These basic algorithms are applied in higher level programs for the determination of symmetries, conservation laws or other properties of differential equations. Applications include the classification of integrable systems of evolutionary scalar PDEs, vector PDEs, single and systems of supersymmetric evolutionary PDEs and recently integrable quadratic Hamiltonians with higher degree first integrals.
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.
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.
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.
Ontario Association for Mathematics Education Golden Section Spring Conference
May 23, 2013 - 3:30pm - 7:00pm
Canadian Mathematics Education Study Group 2013 Annual Meeting
May 24, 2013 - 9:00am - May 28, 2013 - 5:00pm
Mathematics Education Research and Mathematics Teaching: Illusions, Reality, and Opportunities
May 24, 2013 - 9:00am - 5:00pm