# Spectral Geometry Seminar

## 2018

### A summary on recent results on bounds for complex eigenvalues arising in non self-adjoint problems

Wednesday, April 25th, 2018, room AA-5448

In this talk I will present the progress made in the recent years towards the formulation of Keller and Lieb-Thirring type of bounds for complex eigenvalues arising in non-selfadjoint problems. In particular, I will consider the case of the discrete spectrum for operators obtained introducing pertubations in form of complex potentials. I will present at first the result that hold on single and sum of eigenvalues for the illustrative case of the Laplacian and a glimpse on the main ideas which are underlying the proofs will be given. I will present the results that holds for different types of operators, focusing in particular on the Dirac and the two layer graphene's perturbed operator. If time will allow, I will also briefly mention some results on the stability of the spectrum for complex perturbed operators.

### Uniform spectral asymptotics on flat tori

Monday, April 9th, 2018, 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.

### Spectral properties of potential type operators

Monday, March 26th, 2018, 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.

### Pleijel's theorem for Schrödinger operators with radial potentials

Monday, March 12th, 2018, 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

### Fractal Weyl laws and wave decay for general trapping

Wednesday, February 28th, 2018, 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)

### Maximization of the second Neumann eigenvalue: the general case

Monday, February 12th, 2018, 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 $\Omega$. This is a joint work with Dorin Bucur (Chambery, France).

### Shape optimization for Neumann eigenvalues of euclidean domains

Monday, February 5, 2018, 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)

### Higher-order Cheeger inequalities for Laplacian and Steklov eigenvalues

Monday, January 29th, 2018, 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 of this inequality for the discrete Laplacian on graphs. I will also apply their methods to prove a similar inequality for Steklov eigenvalues.

### Barcodes of Morse functions on surface

Monday, January 22th, 2018, 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.

## 2017

### About a rigidity result of R. Petrides

Monday, December 4th, 2018, room AA-5340

I will proof the following result of R. Petrides: Let (\Sigma, g) be a compact Riemannian surface without boundary not diffeomorphic to the sphere \mathbb{S}^2, then \Lambda_1(\Sigma,[g])>\Lambda_1(\mathbb{S}^2,[can])=8\pi, where \Lambda_1(\Sigma,[g]) is the supremum of the functional \Lambda_1(\Sigma, g) in the conformal class [g] of the metric g and can is the canonical metric on the sphere.

### Bourguignon--Li--Yau upper bound for toric manifolds

Monday, November 27th, 2018, room AA-5340

In a joint work with Rosa Sena-Dias, we study the variations of the first eigenvalue corresponding to variation of the torus invariant Kähler metric on a fixed symplectic toric manifold. We prove that the first torus invariant eigenvalue is unbounded, extending work of Abreu and Freitas on the 2 sphere. Using the work of Bourguignon Li Yau we give an explicit upper bound for the first eigenvalue of compact toric Kähler manifolds in terms of the moment polytope.

### Steklov eigenvalues of submanifolds with prescribed boudary in Euclidean space

Monday, November 20th, 2018, room AA-5340

Let S be a fixed closed (n-1)-dimensional submanifold of Euclidean space R^{n+1}. I will discuss upper and lower bounds for the Steklov eigenvalues sigma_k(M), where M is any compact manifold with boundary S. An upper bound will be given, in term of the volume of M. This is based on methods from metric geometry. In the particular situation where S is the unit sphere S^{n-1} (lying in a coordinate hyperplane) and M is an hypersurface of revolution, I will prove that sigma_k(M)\geq sigma_k(B^n) with equality if and only M is the n-dimensional ball B^n. This based on joint work with Bruno Colbois (Neuchâtel) and Katie Gittins (MPIM Bonn).

### Spectral matching & learning of surface date - Example on brain surfaces

Monday, November 13th, 2018, room AA-5340

How to analyze complex shapes, such as of the highly folded surface of the brain? In this talk, I will show how spectral representations of shapes can benefit neuroimaging and, more generally, problems where data fundamentally lives on surfaces. Key operations, such as segmentation and registration, typically need a common mapping of surfaces, often obtained via slow and complex mesh deformations in a Euclidean space. Here, we exploit spectral coordinates derived from the Laplacian eigenfunctions of shapes and also address the inherent instability of spectral shape decompositions. Spectral coordinates have the advantage over Euclidean coordinates, to be geometry aware and to parameterize surfaces explicitly. This change of paradigm, from Euclidean to spectral representations, enables a classifier to be applied *directly* on surface data, via spectral coordinates. The talk will focus, first, on spectral representations of shapes, with an example on brain surface matching, and second, on the learning of surface data, with an example on automatic brain surface parcellation.

### Can we hear how a drum is attached?

Monday, November 6th, 2018, room AA-5340

In this talk, we will wonder whether isospectrality is possible by only changing boundary conditions. Specifically, we will try to achieve this by decomposing the boundary in two, then interchanging conditions. My talk will be mainly based on "Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond" by multiple authors, including Iosif and Dmitry.

### Operator-based techniques for frame field design

Monday, October 30th, 2018, room AA-5340

### Formulas of Szegö type for the periodic Schrödinger operator

Monday, October 23th, 2018, room AA-5340

In the 1980's H.J. Landau and H. Widom proved trace asymptotics for truncated Wiener-Hopf operators with piecewise continuous symbol, as the truncation parameter grows. We present an asymptotic trace formula for the periodic Schrödinger operator in dimension one, which may be seen as an extension of Landau and Widom's results. The subleading order of the asymptotics identifies the spectrum of the periodic Schrödinger operator. This is joint work with Alexander V. Sobolev.

### Small eigenvalues and multiplicity

Monday, October 2nd, 2018, room AA-5340

Small eigenvalue of a Riemannian manifold is any Laplacian eigenvalue less than the fundamental tone of its universal cover. In the case of hyperbolic surfaces they are of particular interest and have been studied in the works of Buser, Schoen, Yau and Wolpert. In the present talk we will discuss the sharp upper bound on the number of small eigenvalues proved by Otal and Rosas. The proof uses a smart application of elementary algebraic topology initially introduced by Sevennec to study multiplicity of the first eigenvalue.

### Distribution of coefficients of rank polynomials for random sparse graphs

Monday, September 25th, 2018, room AA-5340

We study the distribution of coefficients of rank polynomials of random sparse graphs. We rst discuss the limiting distribution for general graph sequences that converge in the sense of Benjamini-Schramm. Then we compute the limiting distribution and Newton polygons of the coeffcients of the rank polynomial of random d-regular graphs.

### On the Steklov spectrum of cuboids

Monday, September 11th, 2018

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.