Emeritus Professor
Research Area: Analysis, applied-mathematics
Research Profile: Nonlinear partial differential equations, mathematical physics. I work in the areas of elliptic partial differential equations, the calculus of variations, and mathematical physics. Elliptic PDEs often arise as stationary equilibria in physical problems or in describing curved surfaces in differential geometry. The calculus of variations is concerned with extrema (critical points) of functions defined on infinite dimensional spaces. For example, solutions to the Dirichlet problem minimize an associated integral among all functions with the given boundary data (“Dirichlet’s Principle”). This observation, known to Gauss and Riemann, introduced variational methods as a tool in studying elliptic PDEs. Today we use a combination of classical variational techniques, real and functional analysis, and topology to study existence, multiplicity, smoothness, stability, and other qualitative properties of solutions to PDEs. Of particular interest are those problems (arising in physics and geometry) where minimizing sequences may not converge, due to the natural symmetries of the problem.
Research Area: Algebra & Number Theory
Reserach Profile: Algebra and Number TheoryMy 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: geometry-topology
Research Profile: Algebraic and geometric topology One of the unifying principles in geometry is that complex systems, such as configurations of planets and stars can often be understood by means of their symmetries. Familiar symmetries include the rotations or reflections of solids in space and the Lorentz transformations of space- time. Discrete invariants and groups of symmetry of continuous motions are studied in algebraic topology, while geometric topology is concerned with the properties of differential manifolds, or higher-dimensional surfaces. Topology developed rapidly in the period between 1950 and 1980 and is now one of the most active fields of mathematics. Symmetries of differentiable manifolds are related to algebra and number theory through group theory, and to partial differential equations and analysis through differential forms. My recent work deals with finite group actions on 4-dimensional manifolds and their connections to gauge theory, and infinite discrete group actions on high-dimensional manifolds. Both areas provide many opportunities for interesting research problems.
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:
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 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.
Associate Professor
Research Area: Financial Mathematics
Research Profile: Financial Mathematics My research interests concern the mathematical modelling and analysis of problems in the area of quantitative finance. Of particular interest to me are problems that arise in credit risk, including issues of correlation, counterparty exposure, and capital assessment and allocation.
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 Profile: Differential Geometry, Geometric Analysis, FinanceCurrently, 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 Profile: Topology and geometryTopology 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: Mathematical Logic
Research Profile: Mathematical logic,universal algebra and computational complexity
Mathematical logic,universal algebra and computational complexity I am involved in the study and classification of general algebraic systems. This area of mathematics is often called Universal Algebra and got its start in the 1930s. In order to compare and classify algebras they are often grouped together according to the equations that they satisfy.
Borrowing and expanding on techniques and ideas from mathematical logic, classical abstract algebra, and also from newer branches of mathematics such as lattice theory and category theory, powerful tools have been developed to help organize and understand the structure of varieties (classes of algebras defined by equations) and the algebras they contain. Recent advances in the field have opened up a new area of study dealing with the local structure of finite algebras. This new local theory of finite algebras has not only been useful in solving several longstanding problems but it has also suggested a number of new and challenging research problems.
My current research program involves studying the computational complexity of subclasses of the Constraint Satisfaction Problem (CSP). Many well known complexity problems, such as graph coloring or Boolean satisfiability, can be naturally presented within the vast CSP framework. Recent work of Bulatov, Jeavons, Krokhin and others has established a strong connection between the CSP and universal algebra and some of the important open problems in the field can be expressed in purely algebraic terms.
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.