Upcoming talk
Precise descriptions of bands of the Airy-Schrödinger operator on the real line
Tuesday, November 26th, 14h00-15h00, room 5448 AA
Joint work with Hakim Boumaza, LAGA, Université Paris 13
In this talk, we present recent results on band spectrum generated by a Schrödinger operator with a non C^1 potential for which one has eigenfunctions described by special functions. This generalizes a result Harrell (1979) and in particular we are able to have a precise estimate on the validity regime of the semi-classical behavior as well as the exact width of each band.
The ongoing work on a multiple-wells potential will be as well presented.
Past talks
Level sets of eigenfunctions and minimal surfaces
Tuesday, November 19th, 14h00-15h00, room 5448 AA
I will follow B. Klartag's recent paper "Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula". In it he explains how to think of nodal sets as minimal surfaces, and applies this idea to study the value distribution of eigenfunctions, and the shape of level sets of harmonic polynomials.
Spectrum of the Neumann-Poincare operator
Tuesday, November 12th, 14h00-15h00, room 5448 AA
I will discuss the spectrum of the Neumann-Poincare operator, which is the boundary version of the double layer potential operators. There operators arise when we are trying to solve the Dirichlet problem (given u on the boundary of a manifold M, find f on M such that f is harmonic and f restricted to the boundary is equal to u). However, this operator is not self-adjoint and therefore will not have the same properties as the Dirichlet-to-Neumann operator. I will also discuss some recent results by Yoshihisa Miyanishi, Takashi Suzuki and Grigori Rozenblum concerning the structure of the spectrum on somewhat smooth domains and results by Karl-Michael Perfekt on the spectrum when the boundary has corners.
Isoperimetric relations between Dirichlet and Neumann eigenvalues
Tuesday, October 29th, 14h00-15h00, room 5448 AA
Given a Dirichlet eigenvalue λk on a domain in Rn, one can ask how many Neumann eigenvalues exist which are less than or equal to λk, and how this number depends on the geometry of the domain. The case k=1 is interesting for observing nodal domains, because any eigenvalue λ is first Dirichlet eigenvalue on any nodal domain of its associated eigenfunction. In this talk, we follow the work of Cox, MacLachlan and Steeves, in which the authors study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. They also conjecture this number depends on the isoperimetric ratio, based on analytical and numerical results.
Bornes sur les nombres de Betti pour les fonctions propres du Laplacien
Tuesday, October 15st, 14h00-15h00, room 5448 AA
Dans cette présentation, nous allons étudier le sujet qui consiste à établir un lien entre l'homologie de l'ensemble des zéros d'une fonction propre du Laplacien et la valeur propre correspondante. En particulier, Je présenterai les résultats de ma maîtrise où je me concentre sur le tore plat et la sphère munie de sa métrique naturelle. Les résultats obtenus, d'une certaine façon, généralisent le théorème de Courant.
Complete spectral asymptotics for a parametric Dirichlet-to-Neumann operator on surfaces
Tuesday, October 8st, 14h00-15h00, room 5448 AA
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 oneterm 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 λ, maps functions on the boundary to the normal derivative of their extension satisfying Δu = λu 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.
Escobar's conjecture for the first Steklov eigenvalue
Tuesday, October 1st, 14h00-15h00, room 5448 AA
In 1990, Escobar conjectured a sharp lower bound for the first non-zero Steklov eigenvalue of a manifold with Ricci ≥ 0, and proved it for the 2 dimensional case. After a review of the conjecture and some related results, I will present the paper "Escobar's Conjecture on a Sharp Lower Bound for the First Non-Zero Steklov Eingenvalue" by Chao Xia and Changwei Xiong, in which the authors proved the conjectured results but with a stronger condition of non-negative sectional curvature. Their proof is based on two equalities: a weighted Reilly type formula and a generalized Pohozaev type identity. Their method also provides an inequality between the Steklov eigenvalues of a manifold and the Laplace eigenvalues of its boundary.
Critical points for eigenfunctions of the Laplacian
Tuesday, September 17, 14h00-15h00, room 5448 AA
In this talk, we will discuss a series of recent constructions of smooth manifolds with many critical points. I will show that there exists a sequence of analytic metrics on the 2-torus converging to the flat metric such that an eigenfunction of finite index has a growing number of critical points. I will also create a sequence of smooth metrics on the sphere such that each metric has a eigenfunction of finite index which has infinitely many critical points. I will then show that each of those constructions give a very strong refutation of the "Extended Courant Property", but does not seem to contradict recent results on barcodes by I. Polterovich, L. Polterovich and V. Stojisavljević. This is based on joint work with Bernard Helffer and Pierre Bérard.
Zero and negative eigenvalues of conformally covariant operators, and nodal sets in conformal geometry.
Tuesday, September 10, 14h00-15h00, room 5448 AA
We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n ≥ 3. We show that 0 is generically not an eigenvalue of the conformal Laplacian. If time permits, we shall discuss related results for manifolds with boundary, and for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire.
Index of minimal surfaces in spheres and eigenvalues of the Laplacian.
Tuesday, July 2nd, 14h00-15h00, room 5183 AA
The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of upper bounds for its eigenvalues is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. It turns out that the optimal isoperimetric inequalities for Laplace eigenvalues are closely related to minimal surfaces in spheres. At the same time, the index of a minimal surface is defined as a number of negative eigenvalues of a different second order elliptic operator. It measures the instability of the surface as a critical point of the area functional. In the present talk we will discuss the interplay between index and Laplace eigenvalues, and present some recent applications, including a new bound on the index of minimal spheres as well as the optimal isoperimetric inequality for Laplace eigenvalues on the projective plane.
Deux disques maximisent la troisième valeur propre du problème de Robin d'un domaine du plan.
Tuesday, April 30th, 13h30-14h30, room 5183 AA
Étant donné un domaine simplement connexe du plan, on montre que la troisième valeur propre d'un problème de Robin convenablement normalisé est bornée supérieurement par la troisième valeur propre du domaine constitué de deux disques disjoints identiques. Cette borne est optimale: elle est atteinte à la limite par une suite de domaines qui tend vers deux disques disjoints de même rayon. La preuve repose sur l'utilisation de fonctions tests appropriées dont l'existence découle d'un argument homotopique. Il s'agit d'un travail conjoint avec Richard Laugesen de l'Université d'Illinois.
TBA
Tuesday, April 23th, 14h30-15h30, room 5183 AA
Savoir que les timbales sont presque harmoniques c'est bien, mais le prouver c'est mieux!
Tuesday, April 9th, 13h30-14h30, room 5183 AA
Note that this time, I will give my talk in french! #SciencesInToutesLesSprachen
Nevertheless, a little english abstract will follow.
Pour la petite histoire, en juin 2017 je m'inscrivais au programme de musicologie de l'Université de Montréal afin d'étudier la théorie derrière les grands classiques de la musique. Pourtant, en septembre je me joignais à ce groupe de recherche de Iosif... Ces deux démarches furent finalement relativement similaires puisque j'ai été amenée à travailler un vieux classique en théorie spectrale: faire l'étude harmonique d'un instrument de percussion, la timbale.
Pour ce séminaire, je présenterai mon travail de justification mathématique de la démarche de l'ingénieur physicien Robert E. Davis (Purdue University, 1988), qui a montré que certains des modes normaux de la timbale qui perdurent le plus dans le temps sont dans un rapport presque harmonique 2:3:4:5. Ceci explique le fait que cet instrument, au contraire du tambour régulier, puisse s'accorder au reste de l'orchestre, qu'il fasse entendre des notes plutôt que des bruits. La preuve constitue un bon entraînement pour revoir les notions du cours d'EDP et s'appuie principalement sur la méthode des fonctions de Green et sur la complétude de certains ensembles de fonctions, notamment des fonctions de Bessel de premier ordre.
This week, I will share my master's project: a mathematical justification of the phD work (1988) of the engineering physicist Robert E. Davis, who showed that the timpani is an almost harmonic percussion instrument. It is almost harmonic in the sense that some of the eigenmodes that last the longest are in a ratio that is very near 2:3:4:5. This proof is a good training for using the notions of a PDE course and it lies mostly on the Green's functions method and on the completeness of some sets of functions, like Bessel ones.
Computations of eigenvalues and resonances on perturbed hyperbolic surfaces with cusps
Tuesday, April 2nd, 13h30-14h30, room 5183 AA
I will describe a simple method (joint work with Alex Strohmaier) that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. I will give some interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichm\"uller space. For instance, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible, in essence, to do number theory using the finite element method.
Past talks
On the Yang-Yau inequality for the first Laplace eigenvalue
Tuesday, March 26th, 13h30-14h30, room 5183 AA
In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus \gamma and the area. It is known to be sharp for \gamma=0,2. In the present talk we outline the main steps in the proof of the fact that the Yang-Yau's inequality is strict for all genera \gamma>2. The talk could be viewed as the second part of the Friday's talk on the Analysis seminar, but the main concepts will be reviewed one more time.
The conformal eigenvalue problem on the standard sphere
Tuesday, March 19th, 13h30-14h30, room 5183 AA
On a n-dimensional Riemannian manifold, one can consider the supremum \Lambda_k(M,[g]) of a Laplace-Beltrami eigenvalue when the metric varies in some conformal class [g], which was shown by Korevaar to be always finite. This problem on the n-sphere is of particular interest since it is known that for any manifold M, \Lambda_k(M,[g]) is greater or equal than \Lambda_k(S^n, [h]) where h is the round metric. I will discuss a result by Druet proving that on a n-sphere (n > 2), the second normalized eigenvalue in the conformal class of the round metric on S^n is NOT maximized by two disjoint spheres. This is in contrast to the two dimensional case and also a known upper bound by Girouard, Nadirashvili, Polterovich (for odd dimensions) and Petrides (all dimensions).
On the Friedlander-Nadirashvili invariants of surfaces
Tuesday, March 12th, 13h30-14h30, room 5183 AA
Let M be a closed smooth manifold. In 1999, L. Friedlander and N. Nadirashvili introduced a new differential invariant I_1(M) using the first normalized nonzero eigenvalue of the Laplace-Beltrami operator \Delta_g over a Riemannian metric g. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use the k-th eigenvalues of g to define the invariants I_k(M) indexed by positive integers k. In a joint paper with M. Karpukhin we show that I_k(M) = I_k(\mathbb S^2) unless M is a nonorientable surface of even genus. For orientable surfaces and k = 1 this was earlier shown by R. Petrides. In fact L. Friedlander and N. Nadirashvili suggested that I_1(M) = I_1(\mathbb S^2) for any surface M different from \mathbb{RP}^2. We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has I_k(M) > I_k(\mathbb S^2). We also discuss the connection between the Friedlander-Nadirashvili invariants and the theory of cobordisms, and conjecture that I_k(M) is a cobordism invariant. I will try to explain these results and give the main idea of proofs.
Eigenfunctions of the Laplacian with infinitely many critical points
Tuesday, February 19th, 13h30-14h30, room 5183 AA
In 1997, Jakobson and Nadirashvili constructed a sequence of eigenfunctions on the 2-torus such that all of them have exactly 16 critical points. In a recent paper (november 2018), Buhovsky, Logunov and Sodin constructed a metric on the 2-torus such that there exists an infinite sequence of eigenfunctions such that all of them have infinitely many critical points. In this talk, I will sketch the construction of this metric and highlight the key features that allow the eigenfunctions to oscillate very rapidly.
Stability results on the inverse Steklov problem for Riemannian warped products
Tuesday, February 12th, 13h30-14h30, room 5183 AA
We consider the inverse Steklov problem on a class of Riemannian warped products which can be thought of as deformations of the closed ball in R^d. We first prove that the Steklov spectrum determines uniquely the warping function of the metric. Next, we show that the approximate knowledge of the Steklov spectrum -in a given technical sense- is enough to determine uniquely the warping function in a neighbourhood of the boundary. Finally, we provide stability estimates of log- type on the warping function. This is joint work with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes), see arXiv 1812.07235.
Spectral asymptotics for the Steklov problem on curvilinear polygons (Part III)
Tuesday, February 5th, 13h30-14h30, room 5183 AA
The asymptotic behavior of the Steklov eigenvalues and eigenfunctions on non-smooth domains is significantly more complex and more intriguing than on domains with smooth boundaries. In the talk we will explain how to compute precise spectral asymptotics for the Steklov problem on curvilinear polygons. In particular, it turns out that the behavior of the Steklov eigenvalues and eigenfunctions depends in a surprising way on the arithmetic properties of the angles at the corner points. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.
Spectral asymptotics for the Steklov problem on curvilinear polygons (Part II)
Tuesday, January 29th, 13h30-14h30, room 5183 AA
The asymptotic behavior of the Steklov eigenvalues and eigenfunctions on non-smooth domains is significantly more complex and more intriguing than on domains with smooth boundaries. In the talk we will explain how to compute precise spectral asymptotics for the Steklov problem on curvilinear polygons. In particular, it turns out that the behavior of the Steklov eigenvalues and eigenfunctions depends in a surprising way on the arithmetic properties of the angles at the corner points. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.
Spectral asymptotics for the Steklov problem on curvilinear polygons (Part I)
Tuesday, January 22th, 13h30-14h30, room 5183 AA
The asymptotic behavior of the Steklov eigenvalues and eigenfunctions on non-smooth domains is significantly more complex and more intriguing than on domains with smooth boundaries. In the talk we will explain how to compute precise spectral asymptotics for the Steklov problem on curvilinear polygons. In particular, it turns out that the behavior of the Steklov eigenvalues and eigenfunctions depends in a surprising way on the arithmetic properties of the angles at the corner points. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.
The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Tuesday, January 15th, 13h30-14h30, room 5183 AA
The Dirichlet-to-Neumann map is a first order pseudodifferential operator acting on the smooth functions of the boundary of a compact Riemannian manifold M. Its spectrum is known as the Steklov spectrum of M. The asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues s_j is determined by the Riemannian metric on the boundary of M. Neverthless, each individual eigenvalue can become arbitrarily big if the Riemannian metric is perturbed adequately. This can be achieved while keeping the geometry of the boundary unchanged, but it requires modifications of the metric in arbitrarily small neighborhoods of the boundary. In our recent work with Bruno Colbois and Asma Hassannezhad, we impose constraints on the geometry of M on and near its boundary. This allows the comparison of each Steklov eigenvalue s_j with the eigenvalues l_j of the Laplace operator acting on the boundary. This control is uniform in the index j. In this talk I will discuss the proof of this result, which is based on the Pohozaev identity and on comparison results for the principal curvatures of hypersurfaces that are parallel to the boundary.
Optimizing the second Robin eigenvalue for planar domains
Thursday, December 13th, 14h-15h, room 5448 AA
I will present the recent article "From Steklov to Neumann and beyond, via Robin : the Szego way" by P. Freitas and R. Laugesen, in which it is shown that the disk maximizes the second Robin eigenvalue for simply-connected planar domains of fixed area, with the Robin parameter scaled by the perimeter and between -2\pi and 2\pi. The proof is based on Szego's method and the new idea of scaling the Robin parameter by the perimeter in order to obtain a scale-invariant isoperimetric inequality. Their isoperimetric inequality unifies Weinstock's and Szego's inequalities for Steklov (for a given perimeter) and Neumann (for a given area) eigenvalues as a single inequality for the scaled Robin problem.
Equidistribution of the conormal cycle of random nodal sets (after N. V. Dang and G. Riviere)
Thursday, November 29th, 14h-15h, room 5448 AA
One direction in the study of nodal domains is that of their average properties, as a function with bounded spectrum is chosen randomly. In this work of N. V. Dang and G. Riviere, (the asymptotics of) the average of the conormal cycle (considered as a current) of a random nodal set is computed. Valuation theory then allows to deduce the asymptotics of the averages of various topological/metric characteristics of the nodal domains, such as their Euler characteristic, surface area and intrinsic volumes. A nice tool guiding the computation is the Berezin integral.
An alternative proof for Sturm's theorem on the number of zeros of linear combinations of eigenfunctions
Thursday, November 22th, 14h-15h, room 5448 AA
Gelfand had this idea of replacing the analysis of linear combinations of the n fisrt eigenfunctions by the analysis of their Slater determinant, but he never rigorously implemented his strategy to the proof of Sturm's theorem. In this talk we will follow the proof of Bérard and Helffer who took on them to finish the job. The Slater determinant will lead us to consider the Dirichlet Sturm-Liouville operator acting on Fermions and to use simple tools from linear algebra.
Study of nodal domains on Kaluza-Klein manifolds
Thursday, November 15th, 14h-15h, room 5448 AA
The study of nodal domains of eigenfunctions of the Laplacian on Riemannian manifolds helps to highlight some of their topological properties. The celebrated Courant and Pleijel theorems give upper bounds on the number of nodal domains for a any manifold. In most cases, lower bounds are not known. In this talk, we will discuss recent results by Zelditch and Jung on lower bounds for a certain type of S^1 bundles over Riemannian surfaces equipped with a Kaluza-Klein metric.
Isoperimetric inequalities for integral operators in potential theory
Thursday, November 8th, 14h-15h, room 5448 AA
The logarithmic and Newton potential operators are integral operators which act on a bounded domain in R^n. In this talk we review isoperimetric results for these operators, including an analogues of the Luttinger inequality and the Rayleigh-Faber-Krahn and Pólya inequalities. We will discuss these bounds as spectral invariants on arbitrary domains and compare them with some explicit examples. This talk is based off of work by M. Ruzhansky and D. Suragan.
A one point non-concentration estimate for Laplace eigenfunctions on polygons
Thursday, November 1st, 14h-15h, room 5448 AA
Non-concentration-type results for eigenfunctions have a rich history, with one of the most well known being the Quantum Ergodicity Theorem. In this talk, we will consider eigenfunctions of the Laplacian on a planar domain with polygonal boundary and Dirichlet conditions. In particular, I will present a recent result of Hans Christianson which gives a quantitative estimate on the L^2 mass of these eigenfunctions near a point in terms of the distance to the nearest non-adjacent boundary face.
The aim for the first 20 minutes is to present background on non-concentration which places Christianson's result in context. The remaining 30 minutes will be focused on presenting the explicit computation and estimates which give his main result; I will also try to explain how his method is inspired by energy estimates via positive commutators, the latter being the heart of his proof.
Robin and Steklov isospectral manifolds
Thursday, October 18th, 14h-15h, room 5448 AA
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.
About the Friedlander-Nadirashvili Invariant of Closed Surfaces
Thursday, October 4th, 14h-15h, room 5448 AA
Let \Sigma be a closed surface. The Friedlander-Nadirashvili invariant I_k(\Sigma) is a little brother of the invariant \Lambda_k(\Sigma). It was introduced by Friedlander and Nadirashvili in 1999. Unlike the invariant \Lambda_k the invariant I_k is well-defined in any dimension. Precisely, let M be a closed n-dimensional manifold and g a Riemannian metric on it then I_k(M)=\inf_{[g]}\sup_{\tilde{g} \in [g]}\lambda_k(\tilde{g})Vol(M, \tilde{g})^{2/n}, where [g] is the conformal class of the metric g, \lambda_k(\tilde{g}) is the k-th eigenvalue of the metric \tilde{g} and Vol(M, \tilde{g}) is the volume of M with this metric. In this talk we will discuss some basic properties of the Friedlander-Nadirashvili invariant of surfaces including some new results obtained in our work (joint with M. Karpukhin) in progress.
Minimization of Steklov eigenvalues for submanifolds with prescribed boundary in R^{n+1}
Thursday, Septembre 27th, 14h-15h, room 5448 AA
In this talk, I will show that for a given n dimensional manifold embedded in R^{n+1}, one can deform it locally while keeping its boundary fixed in R^{n+1} to make the k-th Steklov eigenvalue arbitrarily small. Furthermore, these deformations can be done such that the change in volume and diameter of the new manifold and also the curvature of its boundary is insignificant. This is based on a joint work with Bruno Colbois and Alexandre Girouard.
Free boundary minimal surfaces and overdetermined boundary values problems
Thursday, Septembre 20th, 14h-15h, room 5448 AA
In this talk we prove that a doubly connected minimal surface with free boundary in a ball is a catenoid. We also discuss a connection between free boundary minimal surfaces in a ball and free boundary cones arising in a one-phase problem. Based on a joint paper with Nikolai Nadirashvili.