# Subjects by keyword

Our faculty members and researchers have vast expertise in many advanced aeras, as can be seen in the list below.

For the complete list of our experts, see the Departmental directory.

### Axes

### Fortier Bourque, Maxime

Professeur adjoint

- Analysis and probability
- Complex analysis
- Espaces de modules
- Extreme problems
- Geometric invariants
- Geometry
- Riemann surfaces
- Teichmüller's theory

### Fournier, Richard

Professeur associé

- Analysis and probability
- Approximation theory
- Bernstein-Markov inequalities for algebraic and trigonometric polynomials
- Complex analysis
- Univalent functions
- Universality

### Gauthier, Paul M

Professeur associé

My research has been in complex analysis and potential theory - mostly approximation by holomorphic or harmonic functions. In recent years, I have been trying to approach the Riemann Hypothesis, using methods of complex approximation rather than number theory. Below, I give a very small sample of some recent articles.

For all my publications, see mathscinet

### Perron, François

Professeur titulaire

- Analysis and probability
- Asymptotic methods
- Bayesian statistics
- Computational statistics
- Copula
- Decision theory
- Markov chain
- Monte Carlo methods

### Rousseau, Christiane

Professeure associée

- Analysis and probability
- Dynamical systems
- Hilbert's sixth problem
- Limit cycle
- Linear differential systems
- Normal forms
- Parabolic point
- Predator-prey system
- Singularity
- Unfoldings of singularities

My research interests are around dynamical systems in small dimension, either ODE or difference equations.

In the case of ODE, I am interested in the qualitative theory of ODE and the development of methods allowing understanding the geometric organization of the solutions, often summarized in the phase portrait. I am especially interested to parameter dependent ODE and bifurcation analysis: bifurcations correspond to qualitative changes on the phase portraits occurring for particular values of the parameters. I am interested in applications to Hilbert 19s 16th problem on one side and, occasionally, to some predator-prey models in mathematical biology.

The main part of my recent research deals with the study of equilibrium positions of analytic dynamical systems depending on parameters, more precisely with the problem

of analytic classification of singularities of families of dynamical systems depending on parameters: when are two analytic families of dynamical systems equivalent modulo an analytic change of parameters and possibly a reparameterization of time? There are many obstructions to such equivalences and I am interesting in understanding their geometric meaning.

I am also very involved in popularization of mathematics and the training of future high school teachers.

I was the instigator and international coordinator of the international year Mathematics of Planet Earth 2013 (MPE2013).

### Rousseau, Christiane

Professeure émérite

- Analysis and probability
- Dynamical systems
- Hilbert's sixth problem
- Limit cycle
- Linear differential systems
- Normal forms
- Parabolic point
- Predator-prey system
- Singularity
- Unfoldings of singularities

My research interests are around dynamical systems in small dimension, either ODE or difference equations.

In the case of ODE, I am interested in the qualitative theory of ODE and the development of methods allowing understanding the geometric organization of the solutions, often summarized in the phase portrait. I am especially interested to parameter dependent ODE and bifurcation analysis: bifurcations correspond to qualitative changes on the phase portraits occurring for particular values of the parameters. I am interested in applications to Hilbert 19s 16th problem on one side and, occasionally, to some predator-prey models in mathematical biology.

The main part of my recent research deals with the study of equilibrium positions of analytic dynamical systems depending on parameters, more precisely with the problem

of analytic classification of singularities of families of dynamical systems depending on parameters: when are two analytic families of dynamical systems equivalent modulo an analytic change of parameters and possibly a reparameterization of time? There are many obstructions to such equivalences and I am interesting in understanding their geometric meaning.

I am also very involved in popularization of mathematics and the training of future high school teachers.

I was the instigator and international coordinator of the international year Mathematics of Planet Earth 2013 (MPE2013).

### Schlomiuk, Dana

Professeure associée

- Analysis and probability
- Bifurcation theory
- Center singularity
- Darboux and Liouville integrability
- Dynamical systems
- Global analysis of quadratic vector fields
- Group action
- Invariant algebraic curves
- Limit cycle
- Polynomial invariants
- Polynomial vector fields