Research Area: Algebra & Number Theory

Research Area: Algebra & Number Theory

Research Area: probability-statistics

Research Area: probability-statistics

Research Area: Algebra & Number Theory

Reserach Profile:  Algebra and Number Theory
My background is in number theory and algebra. Currently I solve functional equations; examples of these are:
f(x + y) = f(x) + f(y) (Cauchy) g(xy) + g(1/x) + g(1/y) = 0 (Kepler-Napier)
D(xy x) = xy D(x) + x D(y) x + D(x) yx (Herstein-Kaplansky)
F(x, y) + F(x + y, z) = F(x, y + z) + F(y, z) (Cocycle)
The methods I use are, in the main, algebraic: I do not consider continuity, analyticity and so on of solutions.

Research Area: Algebra & Number Theory

Reserach Profile:  Algebra and Number Theory
My background is in number theory and algebra. Currently I solve functional equations; examples of these are:
f(x + y) = f(x) + f(y) (Cauchy) g(xy) + g(1/x) + g(1/y) = 0 (Kepler-Napier)
D(xy x) = xy D(x) + x D(y) x + D(x) yx (Herstein-Kaplansky)
F(x, y) + F(x + y, z) = F(x, y + z) + F(y, z) (Cocycle)
The methods I use are, in the main, algebraic: I do not consider continuity, analyticity and so on of solutions.

Research Area: probability-statistics

Research Profile:Stochastic models, genetics and medicine, probability bounds 
Stochastic processes are models for describing phenomena which exhibit random fluctuations in their behaviour. Although individual predictions cannot be made, there is remarkable regularity over time which is then exploited in an effort to explain and predict natural phenomena.

My work has been mainly in the following areas:

• population genetics: to develop tools for making inferences about the evolutionary forces in the past which have led to the observed genetic variations among species. In May 1999, I gave a graduate course on this topic at the Fields Institute for Research in Mathematical Sciences. These will be published by the American Mathematical Society.
• combinatorics: to understand the models leading to various related combinatorical structures arising in genetics and ecology.
• large deviations
• probability bounds: to determine optimal upper and lower bounds on the probability of an event using only limited information.
• bootstrap: statistical analyis of channel and bundle power data from Ontario Power Generation CANDU reactors.

In my teaching I have recently begun to experiment with Excel for my undergraduate courses. This represents a departure from the traditional use of specialized statistical software. I have written the Excel Manual which accompanies Introduction to the Practice of Statistics by David Moore and George McCabe. This text is the most widely_used statistics text in North America.

Research Area: probability-statistics

Research Profile:Stochastic models, genetics and medicine, probability bounds 
Stochastic processes are models for describing phenomena which exhibit random fluctuations in their behaviour. Although individual predictions cannot be made, there is remarkable regularity over time which is then exploited in an effort to explain and predict natural phenomena.

My work has been mainly in the following areas:

• population genetics: to develop tools for making inferences about the evolutionary forces in the past which have led to the observed genetic variations among species. In May 1999, I gave a graduate course on this topic at the Fields Institute for Research in Mathematical Sciences. These will be published by the American Mathematical Society.
• combinatorics: to understand the models leading to various related combinatorical structures arising in genetics and ecology.
• large deviations
• probability bounds: to determine optimal upper and lower bounds on the probability of an event using only limited information.
• bootstrap: statistical analyis of channel and bundle power data from Ontario Power Generation CANDU reactors.

In my teaching I have recently begun to experiment with Excel for my undergraduate courses. This represents a departure from the traditional use of specialized statistical software. I have written the Excel Manual which accompanies Introduction to the Practice of Statistics by David Moore and George McCabe. This text is the most widely_used statistics text in North America.

Research Area: Algebra & Number Theory

Research Profile: Algebraic Number Theory, Algebraic K-Theory
Some highlights in classical algebraic number theory are the development of class field theory, the study of L-functions and Zeta-functions, and their relationship to class groups and unit groups of number fields. New powerful tools have been invented since the 1960s and 1970s, in particular Iwasawa-theory, ätale cohomology, and last not least, algebraic K-theory, whose applications go far beyond the classical framework and lead into arithmetic algebraic geometry. My current research focuses on applications of this machinery to classical problems in number theory (e.g. Leopoldt’s Conjecture, Vandiver’s Conjecture), on generalizations of classical results about class groups and units to higher algebraic K-groups, and on their calculation by means of special values of L-functions.

Research Area: Algebra & Number Theory

Research Profile: Algebraic Number Theory, Algebraic K-Theory
Some highlights in classical algebraic number theory are the development of class field theory, the study of L-functions and Zeta-functions, and their relationship to class groups and unit groups of number fields. New powerful tools have been invented since the 1960s and 1970s, in particular Iwasawa-theory, ätale cohomology, and last not least, algebraic K-theory, whose applications go far beyond the classical framework and lead into arithmetic algebraic geometry. My current research focuses on applications of this machinery to classical problems in number theory (e.g. Leopoldt’s Conjecture, Vandiver’s Conjecture), on generalizations of classical results about class groups and units to higher algebraic K-groups, and on their calculation by means of special values of L-functions.

Research Area: probability-statistics

Research Profile: Statistics, biological and medical applications, mixture distributions, nonlinear estimation, computing
New applications demand new statistical methods. Statisticians must ensure that new methodology is available to those who need it in the form of reliable, well-documented computer software, especially when the methods involve intensive computation.

While my interests in statistical applications are very general, much of my research has been in collaboration with fisheries biologists: enumerating migrating populations of juvenile Pacific salmon and inferring the age composition of populations from length-frequency data. For the latter problem I use computer graphics and non-linear estimation methods to resolve age groups from overlapping peaks in the length-frequency distribution. This is called mixture distribution analysis and my software developed at McMaster is now used world-wide for diverse applications. With the assistance of graduate students, I have implemented this methodology as the mixdist package in the open source R environment. Future work will develop the interactive graphical interface and bootstrap calculation of standard errors.

I served on Scientific Advisory Panels for the United States Environmental Protection Agency from 2000 to 2014, evaluating models to estimate human uptake of pesticide residues.

I am now an associate member of the GERAS Centre for Aging Research and collaborate with the GERAS team in study design and analysis and in medical education.

Research Area: probability-statistics

Research Profile: Statistics, biological and medical applications, mixture distributions, nonlinear estimation, computing
New applications demand new statistical methods. Statisticians must ensure that new methodology is available to those who need it in the form of reliable, well-documented computer software, especially when the methods involve intensive computation.

While my interests in statistical applications are very general, much of my research has been in collaboration with fisheries biologists: enumerating migrating populations of juvenile Pacific salmon and inferring the age composition of populations from length-frequency data. For the latter problem I use computer graphics and non-linear estimation methods to resolve age groups from overlapping peaks in the length-frequency distribution. This is called mixture distribution analysis and my software developed at McMaster is now used world-wide for diverse applications. With the assistance of graduate students, I have implemented this methodology as the mixdist package in the open source R environment. Future work will develop the interactive graphical interface and bootstrap calculation of standard errors.

I served on Scientific Advisory Panels for the United States Environmental Protection Agency from 2000 to 2014, evaluating models to estimate human uptake of pesticide residues.

I am now an associate member of the GERAS Centre for Aging Research and collaborate with the GERAS team in study design and analysis and in medical education.

Research Area: geometry-topology

Research Profile: Differential Geometry, Geometric Analysis, Finance
Currently, I am working on the following problems:
1. Properties of Large Scale Networks and Graphs
2. Efficient Manifold Learning Algorithms
3. Asymptotic properties of Heat Kernels
4. Implementing geometric ideas in Probability and Statistics

Research Area: geometry-topology

Research Profile: Differential Geometry, Geometric Analysis, Finance
Currently, I am working on the following problems:
1. Properties of Large Scale Networks and Graphs
2. Efficient Manifold Learning Algorithms
3. Asymptotic properties of Heat Kernels
4. Implementing geometric ideas in Probability and Statistics

Research Area: probability-statistics

Research Area: probability-statistics

Research Area: geometry-topology

Research Profile: Topology and geometry
Topology and geometry Geometry is the study of shapes and spaces. Most people are aware of the standard objects of Euclidean geometry: lines, circles, polygons, and of familiar notions such as angles, parallel lines, and congruent figures. In its modern form geometry has a much wider scope, reaching into higher dimensions and encompassing a broad range of current ideas. The subject of Topology is concerned with those features of geometry which remain unchanged after twisting, stretching or other deformations of a geometrical space. It includes such problems as colouring maps, distinguishing knots, classifying surfaces and their higher dimensional analogs. The influence of topology is also important in other mathematical disciplines such as dynamical systems, algebraic geometry (the study of polynomial equations in many variables) and certain aspects of analysis and combinatorics.

My research in recent years has focused on the study of the topology of manifolds and cell complexes by means of algebraic K-theory, pseudoisotopy theory, gauge theory, and representation theory.

Research Area: geometry-topology

Research Profile: Topology and geometry
Topology and geometry Geometry is the study of shapes and spaces. Most people are aware of the standard objects of Euclidean geometry: lines, circles, polygons, and of familiar notions such as angles, parallel lines, and congruent figures. In its modern form geometry has a much wider scope, reaching into higher dimensions and encompassing a broad range of current ideas. The subject of Topology is concerned with those features of geometry which remain unchanged after twisting, stretching or other deformations of a geometrical space. It includes such problems as colouring maps, distinguishing knots, classifying surfaces and their higher dimensional analogs. The influence of topology is also important in other mathematical disciplines such as dynamical systems, algebraic geometry (the study of polynomial equations in many variables) and certain aspects of analysis and combinatorics.

My research in recent years has focused on the study of the topology of manifolds and cell complexes by means of algebraic K-theory, pseudoisotopy theory, gauge theory, and representation theory.

Research Area: Combinatorics

Research Area: Combinatorics

Research Area: probability-statistics

Research Profile: Statistical Inference, reliability, Industrial Statistics, Quality Control

An area of industrial research that has motivated many practical and theoretical statistical procedures is the study of the reliability of products and processes. One of my interests is the development of statistical methods to model and analyze reliability data. Another aspect focuses on the modeling and analysis of discrete data. Specifically, I have studied models and methods for the estimation of population size with application to animal abundance, the use of adaptive rules in clinical trials, and the study of correlated Bernoulli trials.

Currently, I am working on quality control methods, specifically of the multivariate type. Here I study methods to monitor processes where the number of quality indicators varies from small to quite large. In the latter case, dimension-reduction techniques such as principal components are first used, followed by control charts on leading components. The methods have been applied to the monitoring of product profiles.

Research Area: probability-statistics

Research Profile: Statistical Inference, reliability, Industrial Statistics, Quality Control

An area of industrial research that has motivated many practical and theoretical statistical procedures is the study of the reliability of products and processes. One of my interests is the development of statistical methods to model and analyze reliability data. Another aspect focuses on the modeling and analysis of discrete data. Specifically, I have studied models and methods for the estimation of population size with application to animal abundance, the use of adaptive rules in clinical trials, and the study of correlated Bernoulli trials.

Currently, I am working on quality control methods, specifically of the multivariate type. Here I study methods to monitor processes where the number of quality indicators varies from small to quite large. In the latter case, dimension-reduction techniques such as principal components are first used, followed by control charts on leading components. The methods have been applied to the monitoring of product profiles.

Research Area: applied-mathematics

Research Profile:  Transforms, Digital Signal Processing
My major research interests are in the area of transforms and transform algorithms in digital signal processing. Also of interest to me are the applications of non-linear dynamical systems to the problems of multi-user communications.

Digital signal processing is a wide field of research, in which analysis and synthesis of digital signals are studied, with the final purpose of improving communications and data transmissions. The use of transforms represents a very important tool in this field. Signals in the time or spatial domain are transformed, analyzed, modified, and reconstructed. By this process, noise may be reduced, signals from different users may be better separated and targets may be more accurately identified. The proper algorithms provide speedy numerical implementations of these transforms, and make real time applications possible.

Non-linear dynamical systems are finding numerous applications in the fields of signal processing and communications. In particular, deterministic chaotic systems promise to bring new and exciting ideas to these areas. Many of these are currently being explored with my colleagues in the Communications Research Laboratory.

Research Area: applied-mathematics

Research Profile:  Transforms, Digital Signal Processing
My major research interests are in the area of transforms and transform algorithms in digital signal processing. Also of interest to me are the applications of non-linear dynamical systems to the problems of multi-user communications.

Digital signal processing is a wide field of research, in which analysis and synthesis of digital signals are studied, with the final purpose of improving communications and data transmissions. The use of transforms represents a very important tool in this field. Signals in the time or spatial domain are transformed, analyzed, modified, and reconstructed. By this process, noise may be reduced, signals from different users may be better separated and targets may be more accurately identified. The proper algorithms provide speedy numerical implementations of these transforms, and make real time applications possible.

Non-linear dynamical systems are finding numerous applications in the fields of signal processing and communications. In particular, deterministic chaotic systems promise to bring new and exciting ideas to these areas. Many of these are currently being explored with my colleagues in the Communications Research Laboratory.