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seminar

Barcodes of Morse functions on surfaces

Vukasin Stojisavljevic (Tel-Aviv University)

, Monday, January 22, 2018, Pavillon André-Aisenstadt, room AA-5183

We will quickly review the material presented in the first talk and focus on studying barcodes of Morse functions in greater depth. This will allow us to sketch the proof of the main theorem from the first talk and if time permits discuss additional applications and examples. The talk is based on a joint work with Iosif Polterovich and Leonid Polterovich. Read more

Higher-order Cheeger inequalities for Laplacian and Steklov eigenvalues

Antoine Métras (Université de Montréal)

14h, Monday, January 29, 2018, Pavillon André-Aisenstadt, room AA-5183

Cheeger’s inequality is a lower bound for the Laplacian’s first eigenvalue on a manifold depending only a geometric constant called Cheeger’s constant. I will present a higher-order Cheeger inequality for compact manifold without boundaries, giving a lower bound for higher eigenvalues using a natural generalisation of Cheeger’s constant. This result is adapted from the proof by Lee, Gharan and Trevissan [1] of this inequality for the discrete Laplacian on graphs. I will also apply their methods to prove a similar inequality for Steklov eigenvalues. Read more

Shape optimization for Neumann eigenvalues of euclidean domains

Alexandre Girouard (Université Laval)

14h, Monday, February 5, 2018, Pavillon André-Aisenstadt, room AA-5183

In this talk I will review known bounds for the eigenvalues of the Neumann problem on Euclidean domains. My main goal will be to prove an optimal upper bound for the second nonzero eigenvalue mu_2 of planar simply-connected domains of prescribed area. This is joint work from 2009 with Iosif Polterovich and Nikolai Nadirashvili. This talk will be a serve as a preparation for the very recent work of Bucur and Henrot who solved the maximization problem for mu_2 on arbitrary bounded euclidean domains of fixed measure (that is, there is no connectedness assumption, and no constraint on the dimension). Read more

Maximization of the second Neumann eigenvalue: the general case

Antoine Henrot (Institut Élie Cartan)

14h, Monday, February 12, 2018, Pavillon André-Aisenstadt, room AA-5183

In this talk we consider the second (non-trivial) eigenvalue $\mu_2(\Omega)$ of the Laplacian with Neumann boundary conditions. Using suitable test functions and a topological argument, we prove that $\mu_2(\Omega)$ is always less than $\mu_2(\Omega^)$ where $\Omega^$ is the ball of same volume as $\Omega$. Read more

Fractal Weyl laws and wave decay for general trapping

Jeff Galkowski (Stanford University)

14h, Wednesday, February 28, 2018, Pavillon André-Aisenstadt, room AA-5183

We prove a Weyl upper bound on the number of scattering resonances in strips for manifolds with Euclidean infinite ends. In contrast with previous results, we do not make any strong structural assumptions on the geodesic flow on the trapped set (such as hyperbolicity) and instead use propagation statements up to the Ehrenfest time. By a similar method we prove a decay statement with high probability for linear waves with random initial data. The latter statement is related heuristically to the Weyl upper bound. For geodesic flows with positive escape rate, we obtain a power improvement over the trivial Weyl bound and exponential decay up to twice the Ehrenfest time. (Joint with S. Dyatlov) Read more

Pleijel’s theorem for Schrödinger operators with radial potentials

Philippe Charron (Université de Montréal)

14h, Monday, March 12, 2018, Pavillon André-Aisenstadt, room AA-5183

The study of nodal domains for eigenfunctions of the Laplacian has garnered some attention over the years. First, Courant proved in 1923 that the $n$-th eigenfunction cannot have more than $n$ nodal domains. Then, Pleijel proved in 1956 that the $n$-th eigenfunction has at most $C(d) n + o(n)$ nodal domains with $C(d) < 1$ depending only on the dimension. This holds for the Laplacian with Dirichlet boundary condition. This was extended to smooth manifolds with or without bondary par Berard and Meyer in 1982, and to the Laplacian with Neumann boundary condition by Polterovich in 2009 and Léna in 2016. In this talk we will generalize these results for a large class of Schrödinger operators with radial potentials in $R^d$. This is based on joint work with Bernard Helffer and Thomas Hoffmann-Ostenhof. Read more

Spectral Properties of Potential Type Operators

Broderick Causley (McGill)

14h, Monday, March 26, 2018, Pavillon André-Aisenstadt, room AA-5183

The Single Layer, Double Layer, and Steklov operators share many similarities. For example, the boundary plays a main role in all three. However, there are already striking differences with how their eigenvalues behave when we look at a circle. In this survey talk, we discuss each operator and it’s spectral properties. We start with relating the operators to each-other via decomposition and jump relations. Next, we discuss eigenvalue asymptotics, comparing with some of the few known explicitly computed examples, and state other known geometric facts. Read more

Uniform spectral asymptotics on flat tori

Jean Lagacé (Université de Montréal)

14h, Monday, April 9, 2018, Pavillon André-Aisenstadt, room AA-5183

Spectral asymptotics for the eigenvalue of the Laplacian on a manifold are governed by the well known Weyl’s law. However, the implicit constant in the error term in Weyl’s law is not uniform when one takes large families of manifolds. In this presentation, I will show how one can get a uniform estimate on the remainder under a genericity condition in the family of flat tori. We will also see that this condition is necessary. Read more

talks

Symmetry and Dirac points in graphene spectrum

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

The Steklov spectrum of hyperrectangles

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

Lattice point counting in spectral theory

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

The Steklov spectrum of cuboids

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

teaching

MAT1101 - Mathématiques fondamentales (Fundamental Mathematics)

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

MAT1000 - Analyse 1 (Analysis 1)

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

MAT2050 - Analyse 2 (Analysis 2)

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

MAT1000 - Analyse 1 (Analysis 1)

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

MAT2466 - Analyse appliquée (Applied Analysis)

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

MAT2050 - Analyse 2 (Analysis 2)

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

MAT1400 - Calcul 1 (Multivariable calculus 1)

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