## Portfolio item number 1

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(joint with Leonid Parnovski)

Published in *Journal of Spectral Theory 4:(6) pp. 859-879*, 2016

We propose a variation on the Gauss circle problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. This corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. ** Read more**

(joint with Alexandre Girouard, Iosif Polterovich and Alessandro Savo )

Published in *arXiv preprint*, 2017

We study the Steklov problem on cuboids of arbitrary dimension as a toy model for the Steklov spectrum on domains with non-smooth boundary. We derive two-term asymptotic formulae for the counting function of the eigenvalues, derive an isoperimetric inequality and study the concentration of eigenfunctions. ** Read more**

** Published:**

In this workshop talk, I present Gregory Berkolaiko and Andrew Comech’s paper *Symmetry and Dirac points in graphene spectrum* . I overview the results presented therein, with more attention given specifically to the geometric aspect of the proofs. ** Read more**

** Published:**

While the computation of the spectrum of the Laplacian on a hyperrectangle in Euclidean space is a standard exercise for PDE students, understanding the Steklov spectrum on such a domain is more subtle. I will present recent results, joint work with Alexandre Girouard, Iosif Polterovich and Alessandro Savo, about both high energy asymptotics and bottom of the spectrum characterisation. More precisely, I will give a complete characterisation of the eigenfunction and the eigenvalues then show that we have two term asymptotics for the counting function of the eigenvalues that determine the (d−1)-volume of the boundary of the hyperrectangle and the (d−2)-volume of 2-boundary. I will also show that the hypercube maximises the first eigenvalue when normalised either by d-volume of the hyperrectangle or (d−1)-volume of the boundary. ** Read more**

** Published:**

The spectral asymptotics of elliptic operators (e.g. the Laplacian) and analytic number theory are linked together via lattice point counting problems. As an example, counting eigenvalues of the Laplacian on a flat square torus corresponds to counting integer points in a disk of large radius, which is precisely the Gauss circle problem, a longstanding problem in analytic number theory. In this presentation, I will explain how lattice counting methods are applied to study spectral asymptotics for Schrödinger operators on waveguides and resonators, and also for the Steklov eigenvalue problem on a cube. These two settings will illustrate different flavors of lattice counting problems arising in spectral theory. The talk is based on joint works with L. Parnovski (UCL), as well as with A. Girouard (Laval), I. Polterovich (Montréal) and A. Savo (Rome). ** Read more**

** Published:**

Almost nothing is known in general about the Steklov spectrum of domains or manifolds with singularities on the boundary. In this joint work with A. Girouard, I. Polterovich and A. Savo, we use right cuboids as a model for such domains and obtain various spectral properties: two terms spectral asymptotics, characterisation of the eigenfunctions and scarring sequences, bottom of the spectrum behaviour and shape optimisation for the first eigenvalue. I will formulate more precisely those results and I will make some remarks as to how they would help us understand the general spectral properties of domains with singular boundaries. ** Read more**

Undergraduate course, teaching assistant, *Université de Montréal*, Winter 2013

This course is aimed at future high school teachers in mathematics at Université de Montréal. The students are introduced to the Peano axioms, the induction principle, Euclid’s algorithm, the construction of integers, rational, algebraic, real and complex numbers, cardinals and the fundamental theorem of algebra. My duties included conducting a weekly exercise session, holding office hours for student questions and marking exams. ** Read more**

Undegraduate course, teaching assistant, *Université de Montréal*, Fall 2014

This course is a first introduction to rigorous real analysis. The students learn the axioms of real numbers, properties and convergence of sequences and series of real numbers, properties of continuous functions and differential calculus. My duties included conducting a weekly exercise session, holding office hours and marking exams. ** Read more**

Undegraduate course, teaching assistant, *Université de Montréal*, Winter 2015

In this course, students put the theory of Riemann integration on a rigorous basis. The covered subjects include Riemann integration, the fundamental theorem of calculus and sequences and series of functions, including Taylor series, exponential and trigonometric functions and their inverses, as well as Fourier series. My duties included conducting a weekly exercise session, holding office hours and marking exams. ** Read more**

Undegraduate course, teaching assistant, *Université de Montréal*, Summer 2015

This course is a first introduction to rigorous real analysis. The students learn the axioms of real numbers, properties and convergence of sequences and series of real numbers, properties of continuous functions and differential calculus. My duties included conducting a weekly exercise session, holding office hours and marking exams. ** Read more**

Undegraduate course, teaching assistant, *Université de Montréal*, Fall 2015

In this course, students learn about some applications of analysis. The covered subjects include Fourier series, the Sturm-Liouville problem, Fourier transforms, separation of variables for partial differential equations, other sets of orthogonal functions and special functions. My duties included conducting a weekly exercise sessions and holding office hours. ** Read more**

Undegraduate course, teaching assistant, *Université de Montréal*, Fall 2016

In this course, students put the theory of Riemann integration on a rigorous basis. The covered subjects include Riemann integration, the fundamental theorem of calculus and sequences and series of functions, including Taylor series, exponential and trigonometric functions and their inverses, as well as Fourier series. My duties included conducting a weekly exercise session, holding office hours and marking exams. ** Read more**

Undegraduate course, Instructor of record, *Université de Montréal*, Fall 2017

This is the first course where students learn about multivariable calculus. The covered subjects start with sequence and series, before going into functions of more than one variables. We cover multivariable differential calculus (partial derivative, differentials, gradient, chain rule, level surfaces, optimisation, Lagrange multipliers) before introducing multivariable integral calculus (Iterated integrals, Fubini’s theorem, change of variables, Jacobians). My duties included designing and teaching the class, creating and marking exams, holding office hours and general administration of the course. The class consisted of two sections of 200 students. ** Read more**