Simon St-Amant

The principle behind X-ray imaging is quite simple. We send a ray along a straight line through an object and we measure the difference between its initial intensity and its intensity as it exits the object. The denser the object, the more energy it will absorb and the starker this difference will be. This then allows us to probe the inside of the object in a non-destructive fashion. Suppose now that we do not know the nature of the object, but that we know its X-ray data along every line in space. Would we be able to reconstruct the object? This type of problem is often called an "inverse problem" and leads to fascinating mathematics. We will see other examples of inverse problems, such as the denoising of an image or electrical impedance tomography, and will explore analytical and statistical ways to solve them.

From the analysis of signals to number theory and crystallography, the applications of Fourier analysis are was too numerous to name them all. It so happens that Joseph Fourier's idea to decompose a function as a sum of trigonometric functions has other uses than solving the heat equation. We will present from a mathematical standpoint a modern application of Fourier theory, the analysis of images and JPEG compression. Using elements of spectral graph theory, we will generalize the ideas of JPEG compression to images defined on a torus or any other surface. Turns out that our friend Joseph often hides where we least expect it.

If \(f\) is a sufficiently nice function, the Fourier transform of its derivative \(f'\) can be obtained from that of \(f\) simply by multiplying it by \(i\xi\). The generalization of this idea leads to the introduction of pseudodifferential operators. They Si \(f\) est une fonction suffisamment gentille, la transformée de Fourier de sa dérivée \(f’\) peut être obtenue de celle de \(f\) tout simplement en la multipliant par \(i\xi\). La généralisation de cette idée mène à l’introduction des opérateurs pseudo-différentiels. Ces derniers forment une classe plus grande que celle des opérateurs différentiels habituels, mais jouissent tout de même de plusieurs propriétés fascinantes. Après avoir exploré plusieurs facettes de ces opérateurs, nous étudierons leur spectre et leurs applications au problème de valeur propre de Steklov.

Consider a simply connected surface \(M\) with smooth boundary. Given a function on the boundary of \(M\), the usual Dirichlet-to-Neumann operator maps it to the normal derivative of its harmonic extension. It is known that the spectrum of this operator is asymptotically double and follows a one term asymptotic expansion that is superpolynomially precise. Furthermore, this expansion depends only on the length of the boundary of \(M\). We will consider a parametric Dirichlet-to-Neumann operator which, given a real number \(\lambda\), maps functions on the boundary to the normal derivative of their extension satisfying \(\Delta u = \lambda u\) in \(M\), rather than being harmonic. Using pseudo-differential techniques, we will show that this operator's spectrum admits a complete asymptotic expansion and from it deduce two new spectral invariants. This talk is based on joint work in progress with Jean Lagacé.

If you spend a lot of time on the fifth floor of André-Aisenstadt, you have probably seen the fake blackboard glued on the well next to office 5247 on which can be read the evocative title "Problème du Trio (+Théo) (+Florence?)". Many problems have been displayed on the blackboard in the past few months. Whether they are analysis, geometry, algebra or combinatorics problems, most of them were solved within days of their display. Last week, I put up the problem of finding the number of covering trees in the cubic graph. Seeing that no one found the answer after one week, I decided to solve the problem in this week's Clubmath by presenting a clever theorem that uses linear algebra to come to the rescue of combinatorics.

Up until quite recently, very few (maybe none) nontrivial examples of Steklov isospectral non-isometric manifolds had been discovered. In this talk, we will cover a recent paper by C. Gordon, P. Herbrich and D. Webb which uses the Sunada and the torus action methods to construct manifolds which share the same Steklov spectrum. These methods were previously known but were adapted to the Steklov setting. We will provide explicit examples of such pairs of manifolds and compare them to previously known Dirichlet or Neumann (or mixed) isospectral manifolds.

Nowadays, Euclidean geometry seems trivial when compared to the more complex domains of modern mathematics. For many students in mathematics, their exposure to Euclidean geometry is limited to a few courses in secondary 4. Therefore, why is it that, for thousands of years, the most brilliant minds of humanity struggled to get a B+ on their second semester? And why, despite their genius, were they unable to trisect an angle, duplicate a cube, find the quadrature of a circle, or construct a perfect regular heptagon? We will try to answer those questions by introducing constructible numbers and a hint of Galois theory.

Dating back from 1885, Runge's theorem is a founding theorem in the theory of complex approximations. Given a holomorphic function on compact subset of the complex plane, it guarantees the existence of a sequence of rational functions approximating it uniformly. We will try to show its proof in a relatively direct way. Do not be afraid if your exposure to complex numbers consists at most of Euler's formula, the talk will be preceded by a brief introduction to the necessary concepts of complex analysis. Come and join us in the fabulous world of complex analysis!

At first sight, the approximation theorem of Weierstraß (1885) and its extension found by Stone (1937) seemed to solve any problem of polynomial approximation possible. However, both theorems fail when applied to complex numbers and to holomorphic functions. But then under what conditions can we get a similar theorem in \(\mathbb{C}\) and can we extend it to \(\mathbb{C^n}\)? We will try to answer this fundamental question all while introducing wonderful concepts of complex analysis.

Can one hear the shape of a drum? This question asked by Mark Kac in the '60s is directly linked with the study of the eigenvalues of the Laplacian on different domains and is central to spectral geometry. Following an introduction to spectral geometry, our goal will be to find counter-examples to Kac's question and an efficient method to construct them by allowing different boundary conditions on our domains.