Previous Talks
Date: 04/05/23
Speaker: Nicolas Frantz (Université de Lorraine)
Title: Introduction to scattering theory on Hilbert spaces.
Abstract: The goal of scattering theory is to write the asymptotic of solutions of Schrödinger equation associated to a complex Hamiltonian in term of solutions of Schrödinger equation associated to a simpler Hamiltonian. After describing how theory of Hilbert space can describe quantum system, I will introduce the main ideas of scattering theory for self-adjoint operator. If I have enough time, I will explain how to extend this theory to non-self-adjoint Hamiltonian.
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Date: 20/04/23
Speaker: Yuveshen Mooroogen (University of British Columbia)
Title: Large subsets of Euclidean space avoiding infinite arithmetic progressions
Abstract: An arithmetic progression (AP) is a collection of equally-spaced real numbers. It may be finite or countably infinite. It is known that if a subset of the real line has positive Lebesgue measure, then it contains a k-term AP for every natural number k. In joint work with Laurestine Bradford (McGill, Linguistics) and Hannah Kohut (UBC, Mathematics), we prove that this result does not extend to infinite APs in the following sense: for each real number p in [0,1), we construct a subset of the real line that intersects every interval of unit length in a set of measure at least p, but that does not contain any infinite AP. In this presentation, I will explain the geometric features of our set that allow it to avoid such progressions. I will also briefly discuss two recent preprints, due to Kolountzakis-Papageorgiou and Burgin-Goldberg-Keleti-MacMahon-Wang, that were inspired by our work. These respectively employ probabilistic and topological methods, in contrast to our argument, which relies on measure theory and equidistribution of sequences mod 1.
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Date: 06/04/23
Speaker: Pierre-Olivier Parisé (University of Hawaii at Manoa)
Title: Involutions of multicomplex numbers
Abstract: Given a real algebra $A$, a function $f : A \to A$ is called a (real)-linear involution if $f$ is (real)-linear and $f(f(a)) = a$ for any element $a \in A$. A natural question, at least when $\dim A < \infty$, is: How many (real)-linear involutions are there for a given complex algebra?
We will answer this question in the first part of the talk for the commutative real algebra $\mathbb{M}\mathbb{C}(n) (n \geq 1)$ of multicomplex numbers, a commutative generalization of the complex numbers. In the second part of the talk, I will show how to define different Laplacians using the (real)-linear involutions of the multicomplex numbers.
The first part of this talk is a joint work with Nicolas Doyon and William Verreault.
Date: 09/03/23
Speaker: Antoine Métras (University of Bristol)
Title: Eigenvalue optimisation and n-harmonic maps
Abstract: On a surface, eigenvalue optimisation with respect to the metric leads to minimal surfaces (in a sphere for Laplace eigenvalue, free boundary minimal in a ball for Steklov ones). When we restrict the optimisation problem to a conformal class, the corresponding object we obtain are harmonic maps. I will discuss generalisation to higher dimension of these results and how n-harmonic maps play a crucial role in it.
Link to Recording
Date: 23/02/23
Speaker: Alain Didier Noutchegueme (Université de Montréal)
Title: On minimal surface and eigenvalues isoperimetric inequalities.
Abstract: In the same way that geodesics are critical curves for the length fonctional in a Riemanniann Manifold, Minimal Surfaces are critical hypersurfaces for the area functional.
In 1996, a passionating connection have been made between Minimal Surfaces in low dimensional spheres, and extremal riemannian metrics for eigenvalues of the Laplace-Beltrami operator on Compact Riemannian Surfaces. The aim of this talk is to present such a connection, and some more recent extensions to more general eigenvalues problems.
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Date: 09/02/23
Speaker: Mahishanka Withanachchi (Université Laval)
Title: Polynomial approximation in local Dirichlet spaces
Abstract: The partial Taylor sums $S_n$, $n \geq 0$, are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra, the precise value of the norm of $S_n$ is not known and thus in the literature they are referred as the Lebesgue constants. In this setting, we just know that they grow like $\log n$, modulo a multiplicative constant, as $n$ tends to infinity. However, on the weighted Dirichlet spaces $\Di_w$, we precisely evaluate the norm of $S_n$. As a matter of fact, there are different ways to put a norm on $\Di_w$. Even though these norms are equivalent, they lead to different values for the norm of $S_n$, as an operator on $\Di_w$. We present three different norms on $\Di_w$, and in each case we try to obtain the precise value of the operator norm of $S_n$. These results are in sharp contrast to the classical setting of the disc algebra. We also consider the problem for the cesaro means $\sigma_n$ on local Dirichlet spaces and try to find the norm of $\sigma_n$ precisely for the three different norms that we introduced.
Link to Recording
Date: 26/01/23
Speaker: Thomas Tendron (University of Oxford)
Title: A central limit theorem for a spatial logistic branching process in the slow coalescence regime
Abstract: We study the scaling limits of a spatial population dynamics model which describes the sizes of colonies located on the integer lattice, and allows for branching, coalescence in the form of local pairwise competition, and migration. When started near the local equilibrium, the rates of branching and coalescence in the particle system are both linear in the local population size - we say that the coalescence is slow. We identify a rescaling of the equilibrium fluctuations process under which it converges to an infinite dimensional Ornstein-Uhlenbeck process with alpha-stable driving noise if the offspring distribution lies in the domain of attraction of an alpha-stable law with alpha between one and two.
Date: 14/12/22
Speaker: Billel Guelmame (ENS de Lyon)
Title: On some regularized nonlinear hyperbolic equations
Abstract: It is known that the solutions of nonlinear hyperbolic partial differential equations develop discontinuous shocks in finite time even with smooth initial data. Those shock are problematic in the theoretical study and in the numerical computations. To avoid these shocks, many regularizations have been studied in the literature. For example, adding diffusion and/or dispersion to the equation. In this talk, we present and study some non-diffusive and non-dispersive regularizations of the Burgers equation and the barotropic Euler equations that have similar properties as the classical equations.
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Date: 30/11/22
Speaker: Reihaneh Vafadar (Université Laval)
Title: On divergence-free (form-bounded type) drifts
Abstract: We develop regularity theory for elliptic Kolmogorov operator with divergence-free drift in a large class (or, more generally, drift having singular divergence). A key step in our proofs is "Caccioppoli's iterations", used in addition to the classical De Giorgi's iterations and Moser's method.
This talk is based on joint work with Damir Kinzebulatov (Université Laval)
Link to Recording
Date: 16/11/22
Speaker: Marcu-Antone Orsoni (University of Toronto)
Title: Separation of singularities for the Bergman space and reachable space of the heat equation
Abstract: Let $\Omega_1$ and $\Omega_2$ be two open sets of the complex plane with non empty intersection. The separation of singularities problem can be stated as follows: if $f$ belongs to the Bergman space of $\Omega_1 \cap \Omega_2$, can we find $f_1$ and $f_2$ belonging respectively to the Bergman spaces of $\Omega_1$ and $\Omega_2$, such that $f= f_1 + f_2$?
In this talk, we will see general settings in which the previous question has a positive answer and we will apply these results to the description of the reachable space of the heat equation. Joint work with Andreas Hartmann.
Link to Recording
Date: 02/11/22
Speaker: Jade Brisson (Université Laval)
Title: Looking for eigenvalues
Abstract: In this talk, you will be introduced to spectral geometry. We will briefly talk about what it is before focusing on one of the problem that is studied in this field: the Steklov problem. After a historic review of this problem, we will focus on the following question: Can we calculate its eigenvalues?
Link to Recording