## Applied Mathematics

Real world applications have steered research in mathematics for centuries, and at McMaster applications involving differential equations are a central theme of active research. There are research groups in *Applied Partial Differential Equation* (nonlinear optics, phase boundary motion in materials, superconductivity, viscous flow, and water waves), *Dynamical Systems* (KAM theorems, bifurcation theory), *Mathematical Biology* (population ecology, epidemiology, neural networks), *Mathematical Finance*, and *Numerical Analysis* (wavelets and spectral methods).

## Information Box Group

### Stanley Alama

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.

### Ben Bolker

Professor, Director of School of Computational Science and Engineering

**Research Area:** mathematical-biology

**Research Profile:** *mathematical-biology.*

I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

### Ben Bolker

**Professor, Director of School of Computational Science and Engineering**

### Blaise Bourdin

Professor, Canada Research Chair

### Lia Bronsard

Professor

**Research Area:** Analysis, applied-mathematics

**Research Profile:***Nonlinear partial differential equations, interface dynamics.*

Interface dynamics is a study of the qualitative behaviour of solutions to certain nonlinear partial differential equations known as singular reaction-diffusion systems. These systems are usually models developed by material scientists and mathematicians in order to understand the properties of interfaces or phase boundaries in alloys, polymers or glasses. Under stress, the alloy divides into several regions where the orientation of the molecules differs. The boundary between these regions evolve in time in order to decrease surface tension. The mathematical models developed to study this evolution lead to geometrical problems for the interfaces. An important example is mean curvature evolution, where the normal velocity of the interface is given by its mean curvature. A formal asymptotic tool called the method of matched asymptotic expansion is often used to predict the behaviour for the evolution of the interfaces. Once this formal study is done, the problem is then to develop analytical and geometrical tools to verify rigorously the expected behaviour. Since the models involve nonlinear partial differential equations, new tools are often needed to successfully complete the work.

### David Earn

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Epidemiology, Ecology, Evolutionary Game Theory*

Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

### Nicholas Kevlahan

Professor

**Research Area:** applied-mathematics, Fluids & Turbulence

**Research Profile:** *
applied-mathematics, Turbulent fluid flow, wavelet methods *My research focuses on the theory and computation of fluid turbulence, with a special interest in dynamically adaptive numerical methods based on the wavelet transform. There are numerous problems that remain unresolved in the theory of turbulence, despite more than 100 years of research on the subject. A complete and precise theory of turbulence would be useful in areas as diverse as aerodynamics, combustion, urban pollution modelling, weather prediction and climate modelling. Although we are still far from being able to formulate such a theory, much progress has been made in the last few decades. The aim of my research is to combine several recent discoveries in order to develop a new approach to turbulence modelling. These discoveries include wavelet transforms (which are used to compress data and solve partial differential equations), penalisation methods (which can be used with any numerical method to simulate complex geometries, such as an airplane), and coherent vortices (flow structures that control turbulence dynamics). Current research projects include adaptive multiscale climate modelling, fluid-structure interaction, compressive sampling and the role of turbulence in star formation.

### Dmitry Pelinovsky

Professor

**Research Area:** applied-mathematics

**Research Profile:** *Partial differential equations, solitons*

My research concentrates in the areas of applied nonlinear partial differential equations, spectral analysis, nonlinear waves (solitons), mathematical modeling in nonlinear optics, photonic crystals and Bose-Einstein condensates.

### Bartosz Protas

Professor and Department Chair

**Research Area:** applied-mathematics, Fluids & Turbulence

**Research Profile:** *applied-mathematics, Fluids & Turbluence*

applied-mathematics, Fluids & Turbluence I am an applied mathematician interested primarily in problems combining the theoretical and computational fluid dynamics with the theory of optimization and control. In addition, I also have a broad interest in applied-mathematics and scientific computing. My research is a blend of fundamental investigations in core problems of hydrodynamics and more applied studies relevant for industrial fluid mechanics. I focus on the use of advanced mathematical methods to develop efficient computational algorithms and models useful for studying problems in fluid mechanics. My administrative responsibilities include directing McMaster’s School of Computational Science and Engineering.

### Gail S.K. Wolkowicz

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology*

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.

### Stanley Alama

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.

### Stanley Alama

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.

### Ben Bolker

Professor, Director of School of Computational Science and Engineering

**Research Area:** mathematical-biology

**Research Profile:** *mathematical-biology.*

I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

### Ben Bolker

Professor, Director of School of Computational Science and Engineering

**Research Area:** mathematical-biology

**Research Profile:** *mathematical-biology.*

I work on a wide variety of problems in theoretical ecology and evolution. Mathematical interests: the use of “moment equations” (equations for the mean, variance, and second-order spatial statistics such as spatial correlation) to understand the qualitative dynamics of biological communities (e.g. hosts and parasites, competing plant species)interacting in continuous spatial arenas. Biological interests: host/parasite interactions, broadly defined to include any “intimate association” among natural enemies and their victims: viruses, bacteria, and classical macroparasites. I work on the general dynamics of infectious disease epidemics, and particularly on the evolution of virulence. Statistical interests: fitting of parameters for ecological and evolutionary models, especially in a maximum likelihood settings. My recent work has focused on improvement and dissemination of tools for fitting generalized linear mixed models.

### Blaise Bourdin

Professor, Canada Research Chair

### Blaise Bourdin

Professor, Canada Research Chair

### Lia Bronsard

Professor

**Research Area:** Analysis, applied-mathematics

**Research Profile:***Nonlinear partial differential equations, interface dynamics.*

Interface dynamics is a study of the qualitative behaviour of solutions to certain nonlinear partial differential equations known as singular reaction-diffusion systems. These systems are usually models developed by material scientists and mathematicians in order to understand the properties of interfaces or phase boundaries in alloys, polymers or glasses. Under stress, the alloy divides into several regions where the orientation of the molecules differs. The boundary between these regions evolve in time in order to decrease surface tension. The mathematical models developed to study this evolution lead to geometrical problems for the interfaces. An important example is mean curvature evolution, where the normal velocity of the interface is given by its mean curvature. A formal asymptotic tool called the method of matched asymptotic expansion is often used to predict the behaviour for the evolution of the interfaces. Once this formal study is done, the problem is then to develop analytical and geometrical tools to verify rigorously the expected behaviour. Since the models involve nonlinear partial differential equations, new tools are often needed to successfully complete the work.

### Lia Bronsard

Professor

**Research Area:** Analysis, applied-mathematics

**Research Profile:***Nonlinear partial differential equations, interface dynamics.*

Interface dynamics is a study of the qualitative behaviour of solutions to certain nonlinear partial differential equations known as singular reaction-diffusion systems. These systems are usually models developed by material scientists and mathematicians in order to understand the properties of interfaces or phase boundaries in alloys, polymers or glasses. Under stress, the alloy divides into several regions where the orientation of the molecules differs. The boundary between these regions evolve in time in order to decrease surface tension. The mathematical models developed to study this evolution lead to geometrical problems for the interfaces. An important example is mean curvature evolution, where the normal velocity of the interface is given by its mean curvature. A formal asymptotic tool called the method of matched asymptotic expansion is often used to predict the behaviour for the evolution of the interfaces. Once this formal study is done, the problem is then to develop analytical and geometrical tools to verify rigorously the expected behaviour. Since the models involve nonlinear partial differential equations, new tools are often needed to successfully complete the work.

### David Earn

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Epidemiology, Ecology, Evolutionary Game Theory*

Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

### David Earn

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Epidemiology, Ecology, Evolutionary Game Theory*

Biological systems involve intricate interactions on many spatial and temporal scales. Mathematical models often make it possible to identify mechanisms behind complex biological processes and to predict the outcomes of environmental changes. My current research can be classified according to the time scale over which biological changes occur. Short time scales: Population dynamics of ecological and epidemiological systems. This work, which has implications for conservation of endangered species and eradication of infectious diseases, makes use of modern bifurcation theory. Analytical investigations of discrete maps and differential equations are usually supplemented by extensive numerical studies. Long time scales: Evolutionary dynamics of behavioural traits. This work, which is mainly based on game theoretical analysis, clarifies the adaptive significance of animal behaviour, ranging from cooperation and parental care to foraging and cannabilism.

### Nicholas Kevlahan

Professor

**Research Area:** applied-mathematics, Fluids & Turbulence

**Research Profile:** *
applied-mathematics, Turbulent fluid flow, wavelet methods *My research focuses on the theory and computation of fluid turbulence, with a special interest in dynamically adaptive numerical methods based on the wavelet transform. There are numerous problems that remain unresolved in the theory of turbulence, despite more than 100 years of research on the subject. A complete and precise theory of turbulence would be useful in areas as diverse as aerodynamics, combustion, urban pollution modelling, weather prediction and climate modelling. Although we are still far from being able to formulate such a theory, much progress has been made in the last few decades. The aim of my research is to combine several recent discoveries in order to develop a new approach to turbulence modelling. These discoveries include wavelet transforms (which are used to compress data and solve partial differential equations), penalisation methods (which can be used with any numerical method to simulate complex geometries, such as an airplane), and coherent vortices (flow structures that control turbulence dynamics). Current research projects include adaptive multiscale climate modelling, fluid-structure interaction, compressive sampling and the role of turbulence in star formation.

### Nicholas Kevlahan

Professor

**Research Area:** applied-mathematics, Fluids & Turbulence

**Research Profile:** *
applied-mathematics, Turbulent fluid flow, wavelet methods *My research focuses on the theory and computation of fluid turbulence, with a special interest in dynamically adaptive numerical methods based on the wavelet transform. There are numerous problems that remain unresolved in the theory of turbulence, despite more than 100 years of research on the subject. A complete and precise theory of turbulence would be useful in areas as diverse as aerodynamics, combustion, urban pollution modelling, weather prediction and climate modelling. Although we are still far from being able to formulate such a theory, much progress has been made in the last few decades. The aim of my research is to combine several recent discoveries in order to develop a new approach to turbulence modelling. These discoveries include wavelet transforms (which are used to compress data and solve partial differential equations), penalisation methods (which can be used with any numerical method to simulate complex geometries, such as an airplane), and coherent vortices (flow structures that control turbulence dynamics). Current research projects include adaptive multiscale climate modelling, fluid-structure interaction, compressive sampling and the role of turbulence in star formation.

### Dmitry Pelinovsky

Professor

**Research Area:** applied-mathematics

**Research Profile:** *Partial differential equations, solitons*

My research concentrates in the areas of applied nonlinear partial differential equations, spectral analysis, nonlinear waves (solitons), mathematical modeling in nonlinear optics, photonic crystals and Bose-Einstein condensates.

### Dmitry Pelinovsky

Professor

**Research Area:** applied-mathematics

**Research Profile:** *Partial differential equations, solitons*

My research concentrates in the areas of applied nonlinear partial differential equations, spectral analysis, nonlinear waves (solitons), mathematical modeling in nonlinear optics, photonic crystals and Bose-Einstein condensates.

### Bartosz Protas

Professor and Department Chair

**Research Area:** applied-mathematics, Fluids & Turbulence

**Research Profile:** *applied-mathematics, Fluids & Turbluence*

applied-mathematics, Fluids & Turbluence I am an applied mathematician interested primarily in problems combining the theoretical and computational fluid dynamics with the theory of optimization and control. In addition, I also have a broad interest in applied-mathematics and scientific computing. My research is a blend of fundamental investigations in core problems of hydrodynamics and more applied studies relevant for industrial fluid mechanics. I focus on the use of advanced mathematical methods to develop efficient computational algorithms and models useful for studying problems in fluid mechanics. My administrative responsibilities include directing McMaster’s School of Computational Science and Engineering.

### Bartosz Protas

Professor and Department Chair

**Research Area:** applied-mathematics, Fluids & Turbulence

**Research Profile:** *applied-mathematics, Fluids & Turbluence*

applied-mathematics, Fluids & Turbluence I am an applied mathematician interested primarily in problems combining the theoretical and computational fluid dynamics with the theory of optimization and control. In addition, I also have a broad interest in applied-mathematics and scientific computing. My research is a blend of fundamental investigations in core problems of hydrodynamics and more applied studies relevant for industrial fluid mechanics. I focus on the use of advanced mathematical methods to develop efficient computational algorithms and models useful for studying problems in fluid mechanics. My administrative responsibilities include directing McMaster’s School of Computational Science and Engineering.

### Gail S.K. Wolkowicz

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology*

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.

### Gail S.K. Wolkowicz

Professor

**Research Area:** applied-mathematics, mathematical-biology

**Research Profile:** *Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology*

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. For example, one goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of micro-organisms most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, the prevention of species’ extinction or the control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures, and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, integro- and functional differential equations, and bifurcation theory.