Spectral Geometry in the Clouds
Intergalactic seminar
on spectral theory and geometric analysis
on spectral theory and geometric analysis
| Home | Program | Useful Links |
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August 29 - September 2 |
GEMSTONE mini-course |
| Michael Levitin (University of Reading) | |
| The Steklov problem on non-smooth domains | |
| September 5 | Egor Shelukhin (Université de Montréal) |
| Coarse nodal count and topological persistence | |
View AbstractDirect generalizations of Courant's celebrated nodal domain theorem are often false. A notable example is the case of linear combinations of eigenfunctions in dimension 2 or more, known as the "Courant-Herrmann conjecture".It turns out that a bound which is consistent with Courant's theorem and Weyl's law remains true in this case if we ignore nodal domains where the function is small. It also extends in the following sought-after directions: to elliptic operators acting on sections of vector bundles, to higher Betti numbers of nodal domains, and to products of linear combinations of eigenfunctions. In particular, we obtain a coarse version of Bézout's theorem for eigenfunctions. Our methods combine new results about persistence modules, a notion originating in topological data analysis, and multiscale polynomial approximation in Sobolev spaces. Joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich, and V. Stojisavljevic. |
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| September 12 | Maxime Ingremau (Université de Nice Sophia-Antipolis) |
| How Lagrangian states evolve into random waves | |
View AbstractIn 1977, Berry conjectured that eigenfunctions of the Laplacian on manifolds of negative curvature behave, in the high-energy (or semiclassical) limit, as a random superposition of plane waves. This conjecture, central in quantum chaos, is still completely open.In this talk, we will consider a much simpler situation. On a manifold of negative curvature, we will consider a Lagrangian state associated to a generic phase. We show that, when evolved during a long time by the Schrödinger equation, these functions do behave, in the semiclassical limit, as a random superposition of plane waves. This talk is based on joint work with Alejandro Rivera, and on work in progress with Martin Vogel. |
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| September 26 | Semyon Dyatlov (Massachusetts Institute of Technology) |
| Ruelle zeta at zero for nearly hyperbolic 3-manifolds | |
View AbstractFor a compact negatively curved Riemannian manifold \((\Sigma,g)\), the Ruelle zeta function \(\zeta_{\mathrm R}(\lambda)\) of its geodesic flow is defined for \(\Re\lambda\gg 1\) as a convergent product over the periods \(T_{\gamma}\) of primitive closed geodesics \(\zeta_{\mathrm R}(\lambda)=\prod_\gamma(1-e^{-\lambda T_{\gamma}})\) and extends meromorphically to the entire complex plane. If \(\Sigma\) is hyperbolic (i.e. has sectional curvature \(-1\)), then the order of vanishing \(m_{\mathrm R}(0)\) of \(\zeta_{\mathrm R}\) at \(\lambda=0\) can be expressed in terms of the Betti numbers \(b_j(\Sigma)\). In particular, Fried proved in 1986 that when \(\Sigma\) is a hyperbolic 3-manifold, \(m_{\mathrm R}(0)=4-2b_1(\Sigma)\). I will present a recent result joint with Mihajlo Cekić, Benjamin Küster, and Gabriel Paternain: when \(\dim\Sigma=3\) and \(g\) is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely \(m_{\mathrm R}(0)=4-b_1(\Sigma)\). This is in contrast with dimension 2 where \(m_{\mathrm R}(0)=b_1(\Sigma)-2\) for all negatively curved metrics. The proof uses the microlocal approach of expressing \(m_{\mathrm R}(0)\) as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott—Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations. |
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| October 3 | Bram Petri (Sorbonne Université) |
| How do you efficiently chop a hyperbolic surface in two? | |
View AbstractThe Cheeger constant of a Riemannian manifold measures how hard it is to cut out a large part of the manifold. If the Cheeger constant of a manifold is large, then, through Cheeger's inequality, this implies that Laplacian of the manifold has a large spectral gap. In this talk, I will discuss how large Cheeger constants of hyperbolic surfaces can be. In particular, I will discuss recent joint work with Thomas Budzinski and Nicolas Curien in which we prove that the Cheeger constant of a closed hyperbolic surface of large genus cannot be much larger than \(2/\pi\) (approximately \(0.6366\)). This in particular proves that there is a uniform gap between the maximal possible Cheeger constant of a hyperbolic surface of large enough genus and the Cheeger constant of the hyperbolic plane (which is equal to \(1\)). |
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| October 10 | Mostafa Sabri (New York University Abu Dhabi) |
| Quantum ergodicity for periodic graphs | |
View AbstractQuantum ergodicity for graphs is a delocalization result which says the following. Suppose that a sequence of finite graphs \(\Gamma_N\) converges to some infinite graph \(\Gamma\). Then most eigenfunctions \(\psi_j^{(N)}\) of the adjacency matrix \(\mathcal{A}_N\) on \(\Gamma_N\) become equidistributed on \(\Gamma_N\) when \(N\) gets large. More precisely, for most \(j\), the probability measure \(\sum_{v\in \Gamma_N} |\psi_j^{(N)}(v)|^2 \delta_v\) approaches the uniform measure \(\frac{1}{|\Gamma_N|}\sum_{v\in \Gamma_N} \delta_v\), in a weak sense. Potentials \(Q\) can sometimes be added, so that one now considers the eigenfunctions of \(H_N = \mathcal{A}_N+Q_N\). Usually, the proof partially relies on certain nice properties of the infinite graph \(\Gamma\). In particular, quantum ergodicity theorems have previously been established when \(\Gamma\) is a tree.In this talk, I will present recent results of quantum ergodicity when \(\Gamma\) is invariant under translations of some basis of \(\mathbb{Z}^d\), and the "fundamental block" is endowed a potential \(Q\) which is copied across the blocks, so that \(H = \mathcal{A}_\Gamma+Q\) is a periodic Schrödinger operator. This framework includes \(\Gamma=\mathbb{Z}^d\), the honeycomb lattice, strips, cylinders, etc. I will first discuss the Bloch theorem and give some examples of its limitations, presenting along the way some very homogeneous graphs which violate quantum ergodicity. I will then discuss our main results, contrasting them with the tree case, give various examples of applications, and sketch the proof. An open problem concerning Schrödinger operators with a periodic potential on \(\mathbb{Z}^d\), \(d>1\), will also be presented. This talk is based on a joint work with Theo McKenzie (Harvard). |
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| October 17 | Alexei Stepanenko (Cardiff University) |
| Computing eigenvalues of the Laplacian on rough domains | |
View AbstractIn this talk, I shall present a recent paper with Frank Rösler in which we consider the computability and non-computability of eigenvalues of the Dirichlet Laplacian on bounded domains with rough, possibly fractal, boundaries. This problem may be formulated rigorously within the recently introduced framework of Solvability Complexity Indices. In our pursuit to address this question, we are led to the development of new spectral convergence results for the Dirichlet Laplacian on rough domains, as well as a novel Poincaré-type inequality, which shall be the main focus of my talk. |
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| October 24 | Polyxeni Spilioti (Aarhus University) |
| Twisted Ruelle zeta function on locally symmetric spaces, the Fried's conjecture and further applications | |
View AbstractIn this talk, we will present the twisted Ruelle zeta function, associated with representations that are not necessarily unitary, and how its special value at zero is related to the complex-valued analytic torsion. The relation between the twisted Ruelle zeta function and spectral (or topological) invariants is the so called Fried's conjecture. In addition, we will present results that are related to the Fried's conjecture for hyperbolic surfaces and orbisurfaces. If time allows, we will present some recent results concerning the investigation of the spectrum of the twisted Laplacians, as the representation varies in a suitable Teichmueller space. |
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| October 31 | Anke Pohl (University of Bremen) |
| Fractal Weyl bounds | |
View AbstractResonances of Riemannian manifolds are often studied with tools of microlocal analysis. I will discuss some recent results on upper fractal Weyl bounds for certain hyperbolic surfaces of infinite area, obtained with transfer operator techniques, which are tools complementary to microlocal analysis. This is joint work with F. Naud and L. Soares. |
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| November 7 | Clara Lucia Aldana Dominguez (Universidad del Norte) |
| On Quasi-isospectral potentials | |
View AbstractOn this talk I will first talk about the isospectral problem in geometry and about isospectrality of Strum-Liouville operators on a finite interval in the simplify form of a Schrödinger operator. I will mention very interesting known results about isospectral potentials. I will introduce quasi-isospectrality as a generalization of isospectrality, and will present what we know so far about quasi-isospectral potentials. The work presented here is still on-going joint work with Camilo Perez. |
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| November 14 | Marco Michetti (Université Paris Saclay) |
| Maximization of Neumann eigenvalues under diameter constraint | |
View AbstractIn this talk we study the maximization problem of the Neumann eigenvalues under diameter constraint. We start by presenting a sequence of domains \(\Omega_\epsilon\) for which \(D(\Omega_\epsilon)^2 \mu_1(\Omega_\epsilon)\) goes to infinity.We then define the profile function \(f\) associated to a domain \(\Omega \subset \mathbb{R}^d\), assuming that this function is \(\beta\)-concave, with \(0 < \beta \leq 1\), we will give sharp upper bounds of the quantity \(D(\Omega)^2 \mu_k (\Omega)\) in terms of \(β\). The bounds will go to infinity when \(\beta\) goes to zero. This will also give a new proof of a result by Kröger, namely sharp upper bounds for \(D(\Omega)^2 \mu_k (\Omega)\) when \(\Omega\) is convex (that correspond to \(\beta=(d - 1)^{-1}\)). The proof of this results are based on a maximization problem for relaxed Sturm-Liouville eigenvalues. This talk is based on a joint work with Antoine Henrot. |
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| November 21 | Frédéric Naud (Sorbonne Université) |
| Spectrum of random covers of hyperbolic surfaces and GOE/GUE statistics | |
View AbstractIn this talk we will explain how the "number variance" for the twisted laplace spectrum of compact hyperbolic surfaces can be investigated using the model of random covers. In the large degree regime we recover, in a probabilistic sense, the variance of GOE/GUE models depending on the twist behaviour and the time symmetry invariance. Generalizations to Laplace operators acting on flat vector bundles will be also mentioned. |
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| November 28 | María del Mar González Nogueras (Universidad Autónoma de Madrid) |
| Third order boundary eigenvalue problems | |
View AbstractWe consider two problems on third-order boundary operators: the first one arising when studying dynamic boundary conditions for a Cahn-Hilliard type equation; the second one associated to the fourth-order Paneitz operator in conformal geometry. We study the corresponding eigenvalue problems, showing, in particular, a bifurcation phenomenon in annular domains. |
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| December 5 | Mickaël Nahon (MPI for Mathematics in the Sciences) |
| Sharp stability of higher order Dirichlet eigenvalues | |
View AbstractLet \(\Omega\subset\mathbb{R}^n\) be an open set with same area as the unit ball \(B\) and call \(\lambda_k(\Omega)\) the \(k\)-th eigenvalue of the Laplacian with Dirichlet condition on \(\Omega\). Suppose \(\lambda_1(\Omega)-\lambda_1(B)\) is small, how large can \(|\lambda_k(\Omega)-\lambda_k(B)|\) be? We establish bounds with sharp exponents depending on the multiplicity of \(\lambda_k(B)\) through the study of a perturbed shape optimization problem. This is a joint work with Dorin Bucur, Jimmy Lamboley and Raphaël Prunier. |
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| December 12 | Mikhail Dubashinskiy (Saint-Petersburg State University) |
| Growth and divisor of complexified horocycle eigenfunctions | |
View AbstractFurstenberg Theorem on unique ergodicity of horocycle flow over compact hyperbolic surfaces can be passed through a semiclassical quantization. We then arrive to a plenty of horocycle eigenfunctions \(u\) defined at the hyperbolic plane \(\mathbb{C}^+\). They enjoy \( \left(-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+ 2i\tau y\frac{\partial}{\partial x}\right)u(x+iy)=s^2 u(x+iy)\), \(x+iy\in\mathbb{C}^+\), with \(\tau\to\infty\), \(s=o(\tau)\), \(s,\tau\in\mathbb{R}\), and possess Quantum Unique Ergodicity (\(\hbar=1/\tau\)). At the left-hand side, we recognize magnetic Hamiltonian at hyperbolic plane.Such functions can be analytically continued to a neighborhood of \(\mathbb{C}^+\) in its complexification. The latter is just \(\{(X,Y)\colon X, Y\in\mathbb{C}\}\). We establish asymptotic estimates for the growth of these continuations as \(\tau\to\infty\), and for de Rham currents given by their divisors. The main feature making this setting different from that of free quantum particle (and geodesic flow) is presence of gauge factors in calculations and in answers. |
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| January 16 | Nikolay Filonov (Saint-Petersburg State University) |
| Polya's conjecture for Euclidean balls | |
View AbstractIn 1954, G. Pólya conjectured that the estimate \(N_{\mathcal{D}}(\Omega, \Lambda) \leq C_W \Lambda^{d/2}\) holds true for all \(\Lambda\) > 0. Here \(\Omega \subset \mathbb{R}^d\) is a bounded domain, \(\Lambda\) is the spectral parameter, \(N_{\mathcal{D}}(\Omega, \Lambda)\) is the counting function of the Laplace operator of the Dirichlet problem in \(\Omega\), and \(C_W\) is the constant in the Weyl law. We prove this conjecture for the balls of arbitrary dimension. This is a joint work with M. Levitin, I. Polterovich and D. A. Sher. |
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| January 23 | Badreddine Benhellal (Universität Oldenburg) |
| Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions | |
View AbstractThis talk is devoted to the characterization of the essential spectrum of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported on a smooth compact surface.After giving the rigorous definition and basic spectral properties of the perturbed operator, we show that its essential spectrum within the gap of the free Dirac operator is nonempty and depends on the surface geometry. More precisely, we show that the criticality of the interaction leads to a new interval of the essential spectrum whose position and length are explicitly controlled by the coupling constants and the principal curvatures of the surface, reducing to a single point only in the case of a sphere. This work was carried out jointly with Konstantin Pankrashkin (Carl von Ossietzky Universität Oldenburg). |
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| February 6 | Romain Petrides (Université Paris Diderot) |
| A variational method for functionals depending on eigenvalues | |
View AbstractProblems of optimisation of eigenvalues (or combination of eigenvalues) associated to an operator (Laplacian, Dirichlet-to-Neumann, etc) depending on a Riemannian metric on a differentiable manifold are often addressed in spectral geometry. For instance, they made it possible to build new minimal surfaces in the past decade. It is about finding the optimal bounds which depend on geometric and topologic data of the manifold and - when they are realized - extremal metrics.We will explain a new variational method on functionals depending on eigenvalues. The classical methods do not work because these functionals are not \(C^1\) at metrics which have eigenvalues with multiplicity bigger than 2 involved in the functional. We will then use in this context a subdifferential which provides the generalization of classical notions of gradient, critical points and Palais-Smale sequences. As an application, we will explain how this new approach simplifies and unifies previous optimization results in dimension 2 and generalizes them in dimension higher than 3. |
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| February 13 | Peter McGrath (North Carolina State University) |
| Free Boundary Minimal Annuli Immersed in the 3-Ball | |
View AbstractI will discuss a recent construction (with N. Kapouleas) for immersed free boundary minimal annuli in the unit 3-ball. These surfaces are constructed using singular perturbation PDE methods by "doubling" an equatorial half-disk, and have arbitrarily small first Steklov eigenvalue. |
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| February 20 | Maxime Fortier Bourque (Université de Montréal) |
| Linear programming bounds for hyperbolic surfaces | |
View AbstractI will discuss joint work with Bram Petri in which we obtain new upper bounds on several geometric and spectral invariants associated with closed hyperbolic surfaces, both in low and high genus. The proofs are based on finding good test functions to use in the Selberg trace formula (a strategy heavily inspired by work of Cohn and Elkies on the density of Euclidean sphere packings). |
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| February 27 | Laura Monk (University of Bristol) |
| Towards an optimal spectral gap result for random compact hyperbolic surfaces | |
View AbstractThe first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus \(g\), equivalently), we know that the spectral gap is bounded above by \(1/4 + o_g(1)\).The aim of this talk is to present recent progress in my project with Nalini Anantharaman, where we aim to prove that most hyperbolic surfaces have a near-optimal spectral gap, i.e. for all \(\epsilon >0\), the limit of \(\mathbb{P}_g \left(\lambda_1 \geq \frac 14 - \epsilon \right)\) as \(g\) goes to infinity is equal to \(1\). Here, \(\mathbb{P}_g\) denotes the Weil--Petersson probability measure on the moduli space of compact hyperbolic surfaces of genus \(g\). This statement is analogous to Alon's 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which is based on the trace method and shares many similarities with Friedman's work. I will explain some of the challenges that are encountered, and introduce new tools that we have developed in order to tackle them. So far, our methods allow us to obtain a spectral gap \(2/9 - \epsilon\), hence improving on the bound \(3/16 - \epsilon\) obtained by Wu—Xue and Lipnowski—Wright in 2021. To conclude the talk, we will discuss the final steps that need to be taken to obtain the optimal result. |
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| March 6 | Yaiza Canzani (University of North Carolina at Chapel Hill) |
| Counting closed geodesics and improving Weyl's law for predominant sets of metrics | |
View AbstractWe discuss the typical behavior of two important quantities on compact manifolds with a Riemannian metric \(g\): the number, \(c(T, g)\), of primitive closed geodesics of length smaller than \(T\), and the error, \(E(L, g)\), in the Weyl law for counting the number of Laplace eigenvalues that are smaller than \(L\). For Baire generic metrics, the qualitative behavior of both of these quantities has been understood since the 1970's and 1980's. In terms of quantitative behavior, the only available result is due to Contreras and it says that an exponential lower bound on \(c(T, g)\) holds for \(g\) in a Baire-generic set. Until now, no upper bounds on \(c(T, g)\) or quantitative improvements on \(E(L, g)\) were known to hold for most metrics, not even for a dense set of metrics. In this talk, we will introduce the concept of predominance in the space of Riemannian metrics. This is a notion that is analogous to having full Lebesgue measure in finite dimensions, and which, in particular, implies density. We will then give stretched exponential upper bounds for \(c(T, g)\) and logarithmic improvements for \(E(L, g)\) that hold for a predominant set of metrics. This is based on joint work with J. Galkowski. |
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| March 13 | Francesca Bianchi (Università di Parma) |
| The first eigenvalue of fractional Dirichlet-Laplacian for planar sets with topological constraints | |
View AbstractWe consider the first eigenvalue of the fractional Dirichlet-Laplacian \((-\Delta)^s\) and we show a lower bound in terms of the inradius of the set for certain values of \(s\). In particular, we prove such an estimate for the class of planar sets with a fixed number \(k\) of “holes”. We also discuss the optimality (in some sense) of its dependence with respect to the parameters \(s\) and \(k\). Some of the results presented are obtained in collaboration with Lorenzo Brasco (University of Ferrara). |
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| March 20 | Blake Keeler (Dalhousie University) |
| Off-diagonal spectral cluster asymptotics On Zoll manifolds | |
View AbstractIn this talk, we consider the asymptotic properties of certain spectral cluster kernels on a smooth, compact Riemannian manifold \((M, g)\) without boundary. Let \(\Delta_g\) denote the positive Laplace-Beltrami operator on \(M\), which has discrete spectrum \(\{\lambda_j^2\}_{j=0}^\infty \subset \mathbb{R}\). We study the Schwartz kernel of the orthogonal projection operator \(\Pi_{I_\lambda} : L^2(M) \rightarrow \bigoplus_{\lambda_j \in I_\lambda} \mathrm{ker}(\Delta_g + \lambda_j^2)\) where \(I_\lambda\) is an interval centered around \(\lambda \in \mathbb{R}\) of a small, fixed length. It is conjectured that on any manifold, \(\Pi_{I_\lambda}(x,y)\) has universal asymptotic behavior in a shrinking neighborhood of the diagonal as \(\lambda \rightarrow \infty\), provided that the interval \(I_\lambda\) is chosen to contain sufficiently many eigenvalues. Such asymptotics then imply that certain statistical properties of monochromatic random waves have universal behavior. The conjecture is known to hold for the round sphere and the flat torus, and also under general geometric assumptions which yield a remainder improvement in the off-diagonal Weyl law. In particular, Canzani and Hanin showed that the conjecture holds near non-self focal points. In this talk, we show that in the opposite case of Zoll manifolds, where all geodesics are periodic with the same period, one can still demonstrate universal asymptotic behavior for an appropriate choice of cluster intervals \(I_\lambda\).This talk is based on joint work with Yaiza Canzani and Jeffrey Galkowski. |
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| March 27- 31 | GEMSTONE mini-course |
| Laura Monk (University of Bristol) | |
| Geometry and spectrum of random hyperbolic surfaces | |
| April 3 | Ilaria Fragalà (Politecnico Milano) |
| Riesz inequality for polygons: symmetry and symmetry breaking | |
View AbstractI will discuss counterparts of the classical Hardy-Littlewood and Riesz inequalities when the class of admissible domains is the family of polygons with a fixed number of sides. The latter corresponds to study the polygonal isoperimetric problem in nonlocal version. Based on a joint work with Beniamin Bogosel and Dorin Bucur. |
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| April 10 | Luc Hillairet (Université d’Orléans) |
| Spectral tools for geometric degeneration | |
View AbstractIt is a classical question to study the way the (Laplace) spectrum of a geometric object behaves as the latter degenerates. Depending on the nature of the problem one is led to use either singular or analytic perturbation theory. I will present the basic difference between these two approaches and several well-known or more recent results. I will in particular emphasize the relation between these questions and (semiclassical) concentration properties and present applications to spectral simplicity of simple shapes. Most of this talk will be based on joint work with C. Judge. |
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| April 17 | Jake Fillman (Affiliation) |
| Spectral properties of operators on aperiodic graphs | |
View AbstractWe will discuss Schrödinger operators on graphs generated by aperiodic tilings. For several elements in the MLD class of the Penrose tiling, one finds eigenfunctions whose support is contained in a finite set of tiles. The presence of such eigenfunctions was known for the rhombic Penrose tiling since the work of Kohmoto—Sutherland in the 1980s. These eigenfunctions produce a discontinuity in the integrated density of states at the corresponding energy by linear repetitivity, and therefore the size of the discontinuity can then be estimated from below by the patch frequency corresponding to the support of the eigenfunction in question. [joint work with D. Damanik, M. Embree, and M. Mei.] |