Spectral Geometry in the Clouds
Intergalactic seminar
on spectral theory and geometric analysis
on spectral theory and geometric analysis
| Home | Program | Useful Links |
| September 11 | Cyril Letrouit (Université Paris-Saclay) |
| Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces | |
View AbstractThe problem of finding the maximal possible multiplicity of the first Laplacian eigenvalues has been studied at least since the 1970’s. Colin de Verdière (1986) showed that on any closed manifold of dimension at least 3 the multiplicity of the first non-trivial Laplacian eigenvalue can be made arbitrarily large by an appropriate choice of metric. On surfaces, finding upper bounds on the multiplicity is a difficult problem; Cheng (1975) and then Besson (1980) proved bounds which are linear in the genus \(g\) of the surface.I will present a recent work in collaboration with Simon Machado (ETH) in which we proved, for negatively curved surfaces, the first upper bound which is sublinear in the genus \(g\). Our method is robust enough to also yield an upper bound on the “approximate multiplicity” of eigenvalues, i.e., the number of eigenvalues in windows of size \(1/\log^\kappa(g)\), \(\kappa>0\), and this upper bound is shown to be nearly sharp. Our proof combines a trace argument for the heat kernel and the idea of leveraging an \(r\)-net in the surface to control this trace. The latter idea has been introduced in the context of graphs of bounded degree in a paper of Jiang-Tidor-Yao-Zhang-Zhao (2021). Our work provides new insights on a conjecture by Colin de Verdière and a new way to transfer spectral results from graphs to surfaces. |
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| September 18 | Michela Egidi (Ruhr-Universität Bochum) |
| Complex analysis, thick sets, and spectral inequalities | |
View AbstractIt is a well-known fact in control theory, that null-controllability in time \(T\) of a control system given by an inhomogeneous Cauchy problem is equivalent to final-state-observability of the associated dual system. A key ingredient to show the latter property is the availability of a spectral inequality for elements in the range of the spectral projection of an appropriate operator up to a certain energy value. In this talk we explain what a spectral inequality is and we present a general framework based on complex analytical technique to obtain them for a certain class of domains and operators. Moreover, we will expand our results to the realm of graphs. The main ideas are a generalization of techniques first exploited by Kovrijkine to study functions with compactly supported Fourier Transform. This talk is based on a joint work with Ivan Veselic (TU Dortmund), Albrecht Seelmann (TU Dortmund) and Delio Mugnolo (FernUni Hagen). |
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| September 25 | Yves Colin de Verdière (Institut Fourier) |
| On the spectrum of the Poincaré operator in ellipsoids | |
View AbstractThe Poincaré equation describes the motion of an incompressible fluid in a domain submitted to a rotation. The associated wave operator is called the "Poincaré operator". If the domain is an ellipsoid, it was observed by several physicists that the spectrum is pure point with polynomial eigenfields. I will give a conceptual proof of this fact and an asymptotic result on the eigenvalues. |
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| October 2 | Sam Farrington (Durham University) |
| On the geometric stability of Weyl's law and some applications to asymptotic spectral shape optimisation | |
View AbstractDenoting the Dirichlet and Neumann eigenvalues of a bounded Lipschitz domain \(\Omega \subset \mathbb{R}^{d}\) by \(0<\lambda_{1}(\Omega) \leq \lambda_{2}(\Omega) \leq \lambda_{3}(\Omega) \leq \cdots \) and \(0=\mu_{1}(\Omega) \leq \mu_{2}(\Omega) \leq \mu_{3}(\Omega) \leq \cdots \) respectively, the well-known Weyl asymptotic formula asserts that \begin{equation} \lambda_{k}(\Omega) \sim \mu_{k}(\Omega) \sim W_{d}k^{2/d}|\Omega|^{-2/d}, \enspace \text{as} \enspace k\to +\infty, \end{equation} where \(W_{d}>0\) is the Weyl constant depending only on the dimension \(d\) and \(|\Omega|\) is the volume of \(\Omega\).In spectral shape optimisation problems, it is often desirable to have stability of the Weyl asymptotic formula for sequences of domains under some prescribed geometric conditions. For example, if one has a sequence of bounded Lipschitz domains \(\Omega_{k} \subset \mathbb{R}^{d}\) of unit volume, under which further geometric conditions on the \(\Omega_{k}\) is it true that \begin{equation} \lambda_{k}(\Omega_{k}) \sim \mu_{k}(\Omega_{k}) \sim W_{d}k^{2/d}, \enspace \text{as} \enspace k\to +\infty? \end{equation} One can ask similar questions more generally regarding the asymptotic behaviour of suprema/infima of Dirichlet/Neumann eigenvalues over collections of bounded Lipschitz domains. We will discuss proofs of results in this spirit for bounded convex domains and consider some applications of these ideas in obtaining asymptotic results for spectral shape optimisation problems. In particular, we will consider minimising Dirichlet, Neumann and mixed Dirichlet-Neumann eigenvalues over collections of convex domains of a prescribed perimeter or diameter. |
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| October 9 | Jonathan Rohleder (Stockholms universitet) |
| Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains | |
View AbstractWe generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the \(k\)-th eigenvalue of the Dirichlet Laplacian there exist at least \(k+2\) eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains. The proof relies on a novel variational principle. |
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| October 16 | Antoine Métras (University of Bristol) |
| Dirac eigenvalue optimization | |
View AbstractThe optimization of Laplace eigenvalues with respect to metrics in a given conformal class naturally leads to a correspondence between critical metrics and harmonic maps to spheres, the maps being given component wise by the corresponding eigenfunctions. In my talk, I will look at the optimization of Dirac eigenvalues and discuss the similarities and differences of this problem with the Laplace one. Indeed, in this case we still obtain a correspondence of critical metric with harmonic maps, but this time the maps are to complex projective spaces and need to satisfy additional conditions. I will also give some applications to the minimization of Dirac eigenvalues on the sphere and the torus. This talk is based on joint work with Mikhail Karpukhin and Iosif Polterovich. |
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| October 23 | Wadim Gerner (Sorbonne Université) |
| Optimal Metrics and Domains for the first Curl eigenvalue | |
View AbstractThis talk surveys recent developments regarding optimisation problems for the first curl eigenvalue. In a first part I will discuss results concerning the existence of optimal metrics for the conformal curl eigenvalue and their relation to conformal eigenvalues of the Laplace-operator. In a second part I will discuss the question of existence of optimal domains for the first curl eigenvalue under a volume constraint. I will present existence results within certain subclasses of domains (e.g. convex domains) and also discuss restrictions on possible optimal shapes in the general setting in which the existence of optimal domains is still an open problem. |
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| October 30 | Ana Menezes (Princeton University) |
| Eigenvalue problems and free boundary minimal surfaces in spherical caps | |
View AbstractIn a recent work with Vanderson Lima (UFRGS, Brazil), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk we will prove that each such functional is uniformly bounded from above, and we will characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we will determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we will prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions. |
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| November 6 - 10 | GEMSTONE mini-course |
| Dorin Bucur (Université de Savoie) | |
| Shape optimization of spectral functionals | |
| November 13 | Sara Farinelli (Lagrange Mathematics and Computation Research Center) |
| Peijel's nodal domain theorem in spaces with synthetic Ricci curvature lower bounds | |
View AbstractIn the last decades the study of geometric and analytical properties of metric measure spaces satisfying synthetic Ricci curvature lower bounds has received a huge attention. However not much is known about the classical topics of nodal sets and nodal domains of eigenfunctions of the Laplace operator in this setting. In this seminar we will show how the Peijel's nodal domain theorem can be proved in this possibly singular setting where the main difficulty relies on the possibly non locally Riemannian structure of the space. The techniques used to face the problem in the case of Dirichlet and Neumann Laplacian on domains, apply in the particular case of domains of the Euclidean space, allowing us to extend existing results on the validity of the Pleijel's nodal domain theorem to a bigger class of sets.Based on a joint work with Nicoló de Ponti and Ivan Yuri Violo |
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| November 20 | Samuel Lin (University of Oklahoma) |
| Spectral Multiplicity and Nodal Domains of Torus-invariant Metrics | |
View AbstractA classical result of Uhlenbeck states that for a generic Riemannian metric, the Laplace spectrum is simple, i.e., each eigenspace is real one-dimensional. On the other hand, manifolds with symmetries do not typically have simple spectra. If a compact Lie group \(G\) acts on a manifold as isometries, then each eigenspace is a representation of \(G\), and hence, the spectrum cannot be simple in general. That each eigenspace is an irreducible representation for a generic \(G\)-invariant metric is a conjecture originating from quantum mechanics and atomic physics. In this work, we prove this conjecture for torus actions.We also prove that for a generic torus-invariant metric, if \(u\) is a real-valued, non-invariant eigenfunction that vanishes on an orbit of the torus action, then the nodal set of \(u\) is a smooth hypersurface. This result provides a large class of Riemannian manifolds such that almost every eigenfunction has precisely two nodal domains. This starkly contrasts with previous results on the number of nodal domains for surfaces with ergodic geodesic flows. This is a joint project with Donato Cianci, Chris Judge, and Craig Sutton. |
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| November 27 | Dan Mangoubi (The Hebrew University) |
| On the inner radius of nodal domains | |
View AbstractConsider a closed Riemannian manifold of dimension \(d\). Let \(u\) be an eigenfunction of the Laplace-Beltrami operator with eigenvalue \(\lambda\). Every connected component \(\Omega\) of \(u\neq 0\) is called a nodal domain of \(u\). It follows from the Faber-Krahn inequality that \(\text{Vol}(\Omega)\geq C \lambda^{-d/2}\), where \(C=C(M, g)>0\). A refined question due to Leonid Polterovich is whether one can inscribe in \(\Omega\) a ball of radius \(C\lambda^{-1/2}\). The answer is positive in dimension two (M., 2006). In higher dimensions we show that this is true up to a logarithmic power factor: One can inscribe in \(\Omega\) a ball of radius \(C\lambda^{-1/2}(\log\lambda)^{-a_d}\). where \(a_d\) is a positive constant depending on dimension only. I will explain several ideas which go into the proof.The talk is based on joint work with Philippe Charron. |
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| December 4 | Matthew Kwan (Institute of Science and Technology Austria) |
| Exponentially many graphs are determined by their spectrum | |
View AbstractAs a discrete analogue of Kac's celebrated question on “hearing the shape of a drum”, and towards a practical graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by their spectrum (of their adjacency matrix). In this talk I'll introduce the topic and discuss some recent work with Ilya Koval. |
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| December 11 | Denis Grebenkov (CNRS — Ecole Polytechnique) |
| Probabilistic and numerical insights onto the spectral properties of the Dirichlet-to-Neumann operator | |
View AbstractIn this talk, I show how the spectral properties of the Dirichlet-to-Neumann operator can be used to describe various characteristics of diffusion-controlled reactions, i.e., of reflected Brownian motion in an Euclidean domain with appropriate stopping conditions. For instance, one can derive a spectral expansion for the probability flux density that determines the joint probability law for the stopping time and location of the stochastic process. I discuss the advantages of this approach as compared to conventional spectral expansions based on the eigenfunctions of the Laplace operator. In particular, the Dirichlet-to-Neumann operator allows one to disentangle the diffusive dynamics in an Euclidean domain from surface reactions on its boundary. Various numerical results in the case of planar Euclidean domains with smooth and nonsmooth boundaries will be given. Several conjectures on spectral properties of this operator and related open problems will be presented. |
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| January 15 | Ram Band (Technion) |
| The dry ten Martini problem for Sturmian Schrödinger operators | |
View Abstract"Are all gaps there?", asked Mark Kac in 1981 during a talk at the AMS annual meeting, and offered ten Martinis for the answer. This led Barry Simon to coin the names the Ten Martini Problem (TMP) and the Dry Ten Martini Problem for two related problems concerning the Almost-Mathieu operator. The TMP is about showing that the spectrum of the Almost-Mathieu operator is a Cantor set. The Dry TMP is about the values that the integrated density of states (IDS) attains at the spectral gaps. The gap labelling theorem predicts the possible set of values which the IDS may attain at the spectral gaps. The Dry TMP is whether or not all these values are attained, or equivalently, "are all gaps there?".The TMP was fully solved by Artur Avila and Svetlana Jitomirskaya in 2005. Artur Avila, Jiangong You and Qi Zhou posted this year a preprint with the solution of the Dry TMP for the non-critical case (the coupling constant of the potential differs from one). This talk is about the Dry TMP for Sturmian Schrödinger operators. These are one-dimensional Schrödinger operators with aperiodic potentials which are Sturmian sequences. The potential is determined in terms of two parameters: the frequency and the potential strength (a.k.a coupling constant). As for the Almost-Mathieu operator the Dry TMP is whether all the possible spectral gaps are there for all irrational frequencies and all values of the coupling constant. For large values of the coupling constant, the Sturmian Dry TMP was solved by Laurent Raymond in 1995. In 2016, David Damanik, Anton Gorodetski and William Yessen provided a solution if the frequency is the golden mean and for all couplings. In our current work, jointly with Siegfried Beckus and Raphael Loewy, we solve the Sturmian Dry TMP for all irrational frequencies and all couplings. |
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| January 22 | Carolyn Gordon (Dartmouth College) |
| The Steklov spectrum of convex Euclidean polygons | |
View AbstractWe will first address upper bounds for the Steklov eigenvalues, normalized by perimeter, of convex polygons. We then turn to the inverse problem for the Steklov spectrum, establishing that generic convex n-gons are determined up to finite many possibilities among all n-gons by their Steklov spectra and seeking bounds on the size of isospectral sets. A sample result: Every convex \(n\)-gon all of whose interior angles are obtuse and whose sides lengths are incommensurable over \(\{-1,0,1\}\) is uniquely determined by its Steklov spectrum among all \(n\)-gons.Our main tools in addressing the inverse spectral problem are the very powerful Steklov spectral invariants found by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich and David Sher and the new eigenvalue bounds mentioned above. This work is joint with Emily Dryden, Javier Moreno, Julie Rowlett, and Carlos Villegas Blas. |
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| January 29 | Vladimir Medvedev (Higher School of Economics) |
| Free boundary minimal surfaces in geodesic balls in the hyperbolic space and the upper half-sphere | |
View AbstractRecently, in [1] Lima and Menezes have found a connection between free boundary minimal immersions in geodesic balls in the spherical caps and maximal metrics for a functional on the set of Riemannian metrics on a given surface with boundary. In this talk I will explain how to extend their result to the case of free boundary minimal immersions in geodesic balls in the hyperbolic space and critical metrics for the high order generalization of the Lima-Menezes functional. Also, I plan to consider some applications to the area index of free boundary minimal surfaces in geodesic balls in the hyperbolic space and the upper half-sphere. This talk is based on my recent paper [2].[1] V. Lima and A. Menezes. Eigenvalue problems and free boundary minimal surfaces in spherical caps. arXiv preprint arXiv:2307.13556, 2023. [2] V. Medvedev. On free boundary minimal submanifolds in geodesic balls in \(\mathbb{H}^n\) and \(\mathbb{S}^n_+\). arXiv preprint arXiv:2311.02409, 2023. |
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| February 5 | Yevgeny Liokumovich (University of Toronto) |
| Parametric geometric inequalities and Weyl law for the volume spectrum | |
View AbstractThe isoperimetric inequality and coarea inequality are basic tools in Geometric Analysis. But what if instead of applying them to a fixed submanifold we try to apply them to a continuous family of submanifolds? These "parametric" versions of classical inequalities are open problems. It turns out that they are closely related to the properties of the "volume spectrum" - volumes of minimal submanifolds that arise from Morse theory on the space of flat cycles. I will discuss proofs of these inequalities in low dimensions, their applications to Weyl law for the volume spectrum in higher codimension and existence results for minimal surfaces and stationary geodesic nets that follow from the Weyl law. The talk will be based on joint works with Marques and Neves, Larry Guth and Bruno Staffa. |
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| February 12 | Grigori Rozenblum (Chalmers University of Technology) |
| Weyl asymptotics for Poincare-Steklov eigenvalues in a domain with Lipschitz boundary | |
View AbstractWe discuss the recent result on establishing the Neumann-to-Dirichlet eigenvalue asymptotics formula for an elliptic operator with very weak regularity of coefficients in a domain with Lipschitz boundary. The approach is based upon an approximation of this, rather singular, problem by the ones with smooth boundary and smooth coefficients.Reference: J. Spectr. Theory 13 (2023), no. 3, 755—803 |
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| February 19 | James Kennedy (University of Lisbon) |
| Spectral minimal partitions of metric graphs: what and why? | |
View AbstractSMPs offer a way of dividing a given object (domain, manifold or graph) into a given number of pieces in an "analytically optimal" way: typically, one attempts to minimise an energy functional defined on all \(k\)-partitions built out of some norm of (say) the first Dirichlet Laplacian eigenvalue on each piece. They have deep connections both to fine spectral properties of the Laplacian defined on the whole object, and also to purely geometric functionals/partitions such as so-called Cheeger cuts.We will start by giving a brief overview of SMPs in the particular context of metric graphs. These are a useful sandbox since the existence theory is far easier than on domains or manifolds, but SMPs on metric graphs still enjoy the same (or even more) connections to spectral theory as their higher-dimensional counterparts. We will attempt to illustrate this principle with two new results for metric graphs that should also hold, mutatis mutandis, on domains (with more difficult proofs). First, we will discuss the problem of partitioning an unbounded graph, possibly equipped with an underlying potential. The existence or non-existence of a minimising \(k\)-partition, for given \(k\), is closely related to the infimum of the essential spectrum of the operator on the whole graph, and in particular whether there exists a "test" \(k\)-partition of energy below this infimum. This directly parallels min-max-type principles for discrete eigenvalues of the operator at energies below its essential spectrum. Second, we will introduce partitions of compact graphs based on Robin Laplacian-type first eigenvalues, where a Robin parameter \(\alpha > 0\) is imposed at all boundary points between partition pieces; \(\alpha = \infty\) corresponds, formally, to the Dirichlet case. But as \(\alpha \to 0\), the minimal partition energies, suitably normalised, converge to the purely geometric \(k\)-Cheeger constant of the graph; and, up to a subsequence, the minimising partitions also converge in a natural Hausdorff sense to a \(k\)-Cheeger cut of the graph. This talk will be based on the results of several projects with multiple different co-authors: Pavel Kurasov, Corentin Léna and Delio Mugnolo; Matthias Hofmann and Andrea Serio; and João Ribeiro. |
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| February 26 | Delio Mugnolo (FernUniversität in Hagen) |
| Non-canonical connectivity measures on metric graphs | |
View AbstractFiedler proposed the "algebraic connectivity" — i.e., the spectral gap of the graph Laplacian — as a measure of connectivity of discrete graphs; an analogous idea was substantiated by a two-sided bound on the spectral gap of the metric graph Laplacian obtained by Nicaise (1987) and Kennedy-Kurasov-Malenová-Mugnolo (2016).Further, possibly "more geometric" quantities can be shown to describe the connectivity of a metric graph, too. I will focus on the mean distance, a rather natural quantity that can be defined on each compact metric measure space. After presenting geometric bounds on this quantity, I will show its interplay with the spectral gap of the metric graph Laplacian and outline some similarities with the Kohler-Jobin inequality for metric graphs. This is joint work with Luis Baptista and James B. Kennedy. |
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| March 4 | Alexander Pushnitski (King's College London) |
| The spectra of some arithmetical matrices | |
View AbstractA GCD matrix is a (finite or semi-infinite) matrix whose \((n,m)\)'th matrix entry involves the greatest common divisor of \(n\) and \(m\). I will discuss what is known about the spectral analysis of such matrices. This question has a classical nature but it came to the attention of analysts in recent years in connection with some questions of analytic number theory. |
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| March 11 | Jade Brisson (Université de Neuchâtel) |
| Steklov eigenvalues in negatively curved manifolds | |
View AbstractIn the setting of negatively curved manifolds of dimension \(n\geq3\), we consider the Steklov eigenvalue problem on compact pinched negatively curved manifolds with totally geodesic boundaries. We show that the first nonzero Steklov eigenvalue is bounded below in terms of the total volume and boundary area when the dimension is at least three. In particular, it shows that Steklov eigenvalues can only tend to zero when the total volume and/or boundary area go to infinity. It can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalues on closed pinched negatively curved manifolds of dimension at least three as proved by Schoen in 1982. This is joint work with Ara Basmajian, Asma Hassannezhad and Antoine Métras. |
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| March 18 | Changwei Xiong (Sichuan University) |
| A weighted Reilly formula for differential forms and sharp Steklov eigenvalue estimates | |
View AbstractIn the talk first we will present how to establish a weighted Reilly formula for differential forms on a compact Riemannian manifold with boundary. Then we give some applications of this formula. One is a sharp lower bound for the first positive eigenvalue of the Steklov eigenvalue problem on differential forms investigated by Belishev and Sharafutdinov (2008) and Karpukhin (2019). A second one is a comparison result between the spectrum of this Steklov eigenvalue problem and the spectrum of the Hodge Laplacian on the boundary of the manifold. At the end we discuss an open problem for differential forms analogous to Escobar's conjecture (1999) for functions. The talk will be mainly based on the preprint arXiv:2312.16780. |
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| March 25-29 | GEMSTONE mini-course |
| Maxime Fortier Bourque (Université de Montréal) and Bram Petri (Sorbonne Université) | |
| Extremal problems on hyperbolic surfaces | |
| April 1 | Daniel Stern (Cornell University) |
| New minimal surfaces via equivariant eigenvalue optimization (part I) | |
View AbstractI'll discuss recent work with Karpukhin, Kusner, and McGrath, in which we construct many new examples of embedded minimal surfaces in \(S^3\) and free boundary minimal surfaces in \(B^3\) with prescribed topology via intrinsic shape optimization problems for Laplace and Steklov eigenvalues in the presence of symmetries. I'll describe some key analytic challenges in the existence theory for metrics maximizing Laplace and Steklov eigenvalues on surfaces — with or without prescribed symmetry — and discuss new techniques for overcoming these, as well as some lingering open questions.(To be continued in Peter McGrath's talk.) |
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| April 8 | Peter McGrath (North Carolina State University) |
| New minimal surfaces via equivariant eigenvalue optimization (part II) | |
View AbstractIn this follow-up to Daniel Stern's 4/1 talk, I discuss joint work with Karpukhin, Kusner, and Stern on the application of equivariant optimization of Laplace and Steklov eigenvalues on surfaces to constructions of embedded minimal surfaces in the 3-sphere and 3-ball. Riemannian metrics which maximize such normalized eigenvalues are known to give rise to branched minimal immersions by first eigenfunctions into spheres and balls, with codimension in general expected to grow with the topology of the surface. Nonetheless, there is a class ("Basic Reflection Surfaces", or BRS), of group actions on surfaces which allow each topological type and satisfy optimal eigenvalue bounds, ensuring that any branched minimal immersion by first eigenfunctions is a codimension-1 embedding, which doubles a minimal 2-sphere (or 2-disk) and has area less than \(8\pi\) (or \(2\pi\)). We show maximizing metrics can be found on each BRS, leading in particular to the existence of orientable, embedded minimal surfaces with free boundary in the 3-ball with arbitrary topological type, answering a question of Fraser and Li. |
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| April 22 | Luigi Provenzano (Sapienza Università di Roma) |
| On the critical points of Steklov eigenfunctions | |
View AbstractWe consider the critical points of Steklov eigenfunctions on a compact, smooth \(n\)-dimensional Riemannian manifold \(M\) with boundary \(\partial M\). For generic metrics on \(M\) we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to \(\partial M\), and the Euler characteristic of \(M\). In dimension 2 this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of \(M\) and of the number of sign changes of \(u\) on \(\partial M\).Based on a joint work with Luca Battaglia (Università degli Studi Roma Tre) and Angela Pistoia (Sapienza Università di Roma). |
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| April 29 | José Espinar (University of Granada) |
| An overdetermined eigenvalue problem and the Critical Catenoid conjecture | |
View AbstractWe consider the eigenvalue problem \(\Delta^{\mathbb{S}^2} \xi + 2 \xi=0 \) in \( \Omega \) and \(\xi = 0 \) along \( \partial \Omega \), being \(\Omega\) the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere \(\mathbb{S}^2\). Imposing that \(\|\nabla \xi\|\) is locally constant along \(\partial \Omega\) and that \(\xi\) has infinitely many maximum points, we are able to classify positive solutions as the rotationally symmetric ones. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points. |
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| May 6 | Panagiotis Polymerakis (University of Thessaly) |
| Large Steklov eigenvalues under geometric constraints | |
View AbstractIn this talk, we will discuss two recent constructions of compact Riemannian manifolds with boundary which satisfy certain geometric conditions and have arbitrarily large first non-zero Steklov eigenvalue. In the first part of the talk, under some assumptions, we will construct Riemannian metrics on a given manifold which coincide on the boundary, have fixed volume and arbitrarily large first non-zero Steklov eigenvalue. In particular, this provides the first examples of Riemannian metrics with these properties on three-dimensional manifolds. In the second part, we will construct compact submanifolds of the Euclidean space with fixed boundary and arbitrarily large first non-zero Steklov eigenvalue. This is a joint work with Alexandre Girouard. |
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| May 13 | David Tewodrose (Vrije Universiteit Brussel) |
| Critical metrics of eigenvalue functionals via subdifferential | |
View AbstractI will present a joint work with Romain Petrides (Université Paris Cité) where we propose a general approach to study mapping properties of critical points of functionals \(F(g) = F(S_g)\), where \(g\) runs over an open set of Riemannian metrics on a given smooth manifold, \(S_g\) is a set of eigenvalues depending on \(g\) and \(F\) is a locally Lipschitz function. At the core of our approach is Clarke's notion of subdifferential. Our work covers well-known cases, like Laplace and Steklov eigenvalues, and provides promising perspectives on new situations. |
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| May 20 | Ryan Gibara (University of Cincinnati) |
| On the Dirichlet-to-Neumann Map for the \(p\)-Laplacian on a Metric Measure Space | |
View AbstractIn this talk, we will consider joint work with Nageswari Shanmugalingam on the construction of a Dirichlet-to-Neumann map in the setting of a metric measure space. For the Newton-Sobolev space on a bounded, locally compact, uniform domain equipped with a doubling measure supporting a \(p\)-Poincaré inequality, its trace class onto the boundary (which has been equipped with a Radon measure that is codimensional with the measure on the domain) can be identified with the space of Besov functions. As such, in the context of Dirichlet problems we study Besov boundary data and in the context of Neumann problems we study boundary data in the dual of the Besov space. |
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| May 27 | Severin Schraven (Technical University of Munich) |
| Two-sided Lieb-Thirring bounds | |
View AbstractWe discuss upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. For atomic Hamiltonians with Kato potentials one can strengthen the result to obtain two-sided estimates for the sum of the negative eigenvalues. Instead of being in terms of the potential itself, as in the usual Lieb-Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of \((-\Delta + V + M )u_M = 1\) in \(\mathbb{R}^d\); here \(M \in \mathbb{R}\) is chosen so that the operator is positive. This talk is based on the preprint arXiv:2403.19023 which is joint work with S. Bachmann and R. Froese. |
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| June 3 | Gilles Carron (Nantes Université) |
| Hardy inequalities: how and why? | |
View AbstractI will explain different approaches (spectral inequalities, real harmonic analysis, PDE supersolution) to get Hardy inequalities and I will give some applications of such Hardy inequalities (volume growth, weighted \(L^2\)-boundedness of Hodge projectors). |
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| June 10 | Manuel Ruivo de Oliveira (University of British Columbia) |
| New free boundary minimal annuli of revolution in the 3-sphere | |
View AbstractI will describe the main ideas behind my recent preprint establishing the existence of explicit free boundary minimal annuli in the 3-sphere. These arise as compact pieces of complete minimal surfaces of revolution described long ago by Otsuki (1970) and do Carmo and Dajczer (1983) and have interesting connections to a recent extremal eigenvalue problem considered by Lima and Menezes (2023). In contrast to the Euclidean case, in this setting we quickly obtain a variety of examples which may be embedded or self-intersecting, contained in a geodesic ball or not. |