# Maxime Fortier Bourque

**
Coxeter Assistant Professor |
University of Toronto, Department of Mathematics
**

Toronto, ON, CA

Maxime Fortier Bourque's research focuses on Teichmüller theory

## Industry Applications (2)

## Research Interests (3)

## Accomplishments (7)

#### Postdoctoral Fellowship (professional)

Awarded by the Fonds de recherche du Québec - Nature et technologies.

#### Postgraduate Scholarship (professional)

Awarded by the Natural Sciences and Engineering Research Council of Canada.

#### Masters Research Scholarship (professional)

Awarded by the Fonds de recherche du Québec - Nature et technologies.

#### Alexander Graham Bell Canada Graduate Scholarship (professional)

Awarded by the Natural Sciences and Engineering Research Council of Canada.

#### Math in Moscow Scholarship (professional)

Awarded by the Natural Sciences and Engineering Research Council of Canada and the Canadian Mathematics Society.

#### Undergraduate Student Research Award (professional)

Awarded by the Natural Sciences and Engineering Research Council of Canada.

#### Governor General’s Academic Medal (professional)

Awarded to the student graduating with the highest average from a high school, as well as from approved college or university programs.

## Education (4)

#### City University of New York (CUNY): Ph.D., Mathematics 2015

Advisor: Jeremy Kahn

#### Brown University: M.Sc, Mathematics 2013

#### Université Laval: M.Sc, Mathematics 2010

#### Université Laval: B.Sc, Mathematics 2009

## Links (1)

## Languages (2)

- English
- French

## Articles (6)

**Non-convex balls in the Teichmüller metric**

*arXiv*

2016

We prove that the Teichmuller space of surfaces of genus g with p punctures contains balls which are not convex in the Teichmuller metric whenever 3g−3+p>1.

**The holomorphic couch theorem**

*arXiv*

2015

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.

**The converse of the Schwarz Lemma is false**

*Annales Academiæ Scientiarum Fennicæ Mathematica*

2016

Let h : X → Y be a homeomorphism between hyperbolic surfaces with finite topology. If h is homotopic to a holomorphic map, then every closed geodesic in X is at least as long as the corresponding geodesic in Y, by the Schwarz Lemma. The converse holds trivially when X and Y are disks or annuli, and it holds when X and Y are closed surfaces by a theorem of Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto.

**Conformal grafting and convergence of Fenchel-Nielsen twist coordinates**

*Conformal Geometry and Dynamics*

2015

We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.

**Rational Ahlfors Functions**

*Constructive Approximation*

2015

We study a problem of Jeong and Taniguchi to find all rational maps which are Ahlfors functions. We prove that the rational Ahlfors functions of degree two are characterized by having positive residues at their poles. We then show that this characterization does not generalize to higher degrees, with the help of a numerical method for the computation of analytic capacity. We also provide examples of rational Ahlfors functions in all degrees.

**Super-identical pseudospectra**

*Journal of The London Mathematical Society*

2009

The complex N × N matrices A and B are said to have super-identical pseudospectra if, for each z ∈ ℂ, the singular values of A − zI are the same as those of B − zI. We explore this condition and its consequences. On the positive side, drawing on ideas from invariant theory, we prove that there exists an integer m = m(N) such that ‘almost every’ m-tuple of N × N matrices with super-identical pseudospectra contains a pair that are unitarily equivalent. On the negative side, we present an example of a pair of non-derogatory 4 × 4 matrices A and B with super-identical pseudospectra such that ||A^2|| ≠ ||B^2||.