# Spectral Geometry: Theory, Numerical Analysis and Applications (18w5090)

Arriving in Banff, Alberta Sunday, July 1 and departing Friday July 6, 2018

## Organizers

Iosif Polterovich (Université de Montréal)

Nilima Nigam (Simon Fraser University)

Justin Solomon (Massachusetts Institute of Technology)

## Objectives

Beyond its vibrancy as a research discipline and mathematical challenges, spectral geometry serves as a model for the advantages of interdisciplinary collaboration. Below we document several examples of joint developments in theory, numerical analysis, and applications. Our workshop will be the first of its kind, intended to bring interactions between researchers with these different focuses to community scale.\\

There is a long history of fruitful interaction between geometric spectral theory and numerical analysis. The interplay between these fields recently has reached new heights due to the availability of powerful computers and to the development of efficient numerical algorithms. From the theoretical standpoint, numerical calculations serve as an indispensable source of intuition and are behind many advances in spectral geometry. In particular, numerical experiments stimulated recent breakthrough research on nodal portraits of random linear combinations of eigenfunctions [Na, So, SW, CS]. Numerical methods are actively used in the rapidly-developing theory of spectral minimal partitions [BH1, BH2]. For low eigenvalues, numerical experiments are a major source of conjectures in shape optimization [JLNNP, AF, AKO, BBG2]. Finally, an exciting new trend is to use numerical analysis not only to formulate conjectures, but actually to prove theorems in spectral geometry using rigorous computer-assisted methods [BF, Poly].\\ New numerical algorithms have been developed for fast, precise computation of higher eigenvalues and eigenfunctions of the Laplacian (see [BH, BHT]). In some special cases – such as for the Laplacian on hyperbolic surfaces – algorithms were found for computation of eigenvalues and other spectral quantities with rigorous error estimates [SU, S].\\ Many modern applications of spectral geometry are inspired by problems in computer graphics, shape analysis and machine learning. These techniques are based on the properties of spectral quantities such as heat kernels and eigenfunctions associated with the Laplacian [BBG, JMS]. For instance, popular techniques for surface-to-surface correspondence in computer graphics and medical imaging make use of spectral techniques to identify features to be matched [SOG, OMMG], to leverage and detect symmetry [OMPG, OSG, WSSZ], to find differences between features of different surfaces [ROABCG, CSBGO] and to provide low-frequency approximations of mappings [OBSBG]. An algorithm for quadrilateral meshing makes use of the shape and topology of Laplacian nodal domains [DBGPH]. In the higher-dimensional machine learning context, spectral geometry appears when propagating labels through a partially labeled dataset [BNS, ZGL], reducing the dimensionality of a dataset [BN, CL], or constructing wavelet bases on unstructured clouds of data points [CM].\\

The Laplacian on a Riemannian manifold carries considerable information about intrinsic geometry. To capture the extrinsic geometry of shapes using spectral quantities, one needs to use other operators. A natural choice is the Dirichlet-to-Neumann operator, which has been the focus of intensive research in modern theoretical spectral geometry [PS, GPPS, FS1, FS2, GP, AKO, BBG2]. The collection of Dirichlet-to-Neumann eigenvalues is called the Steklov spectrum, as it is precisely the spectrum of the Steklov boundary value problem. Recent advances in the study of the Steklov spectrum, particularly, of the corresponding geometric invariants, open up a number of promising new applications to shape analysis.\\ Applications of spectral geometry to shape analysis and machine learning heavily rely on numerical methods. For a successful implementation of a numerical algorithm, it is crucial to use an appropriate discretization of the corresponding differential operator. Understanding the properties of discretizations of differential operators defined on complex geometries such as Riemannian manifolds is highly nontrivial; new advances in this direction for the Laplace Beltrami operator were made in [BIK1, BIK2].\\

Accompanying these developments, the design and analysis of high-accuracy discretization methods for spectral problems and optimal design has been the focus of intense research in numerical analysis. A plethora of discretization techniques have been developed, including methods based on particular solutions [BHT], spectral methods, finite element methods [B, CGPS] and integral equations [ABN]. Each presents distinct advantages and challenges in numerical analysis, which in turn leads to improved understanding of the properties of the eigenvalues and eigenfunctions of the operators under investigation. For example, the use of common Lagrange finite elements for the Maxwell eigenvalue problem was shown to yield the wrong spectrum even on simple geometries [B], motivating the large-scale adoption of the edge finite elements. The design of numerical methods for inverse problems lead to the investigation of nonlinear and non-self-adjoint transmission eigenvalue problems, which in turn has inspired novel methods of discretizations for them [CMS]. The computations were used extensively to enhance our understanding of the situations under which the existence of such eigenvalues could be proven. The design of a boundary integral method for mixed Dirichlet-Neumann eigenvalues of the Laplacian is necessarily intertwined with a careful study of the asymptotic behaviour of the eigenfunctions [ABN]. A related area of investigation seeks to provide computable error bounds and eigenvalue enclosures [L]. These bounds are crucial if numerical methods are to be used for formulating conjectures.\\ Upon discretization, the resulting large discrete systems have been coupled with state-of-the-art eigenvalue solvers and optimization methods. Naive approaches do not suffice, and the design of algorithms in this area has yielded many interesting and challenging research directions, due to the potentially nonlinear/non-convex nature of the objective function to be optimized. If the eigenvalues are clustered, or if the original spectrum is not discrete, these problems are much harder [GGO]. Application-specific solvers for spectral geometry problems also have been developed to support the transition of these techniques from academia to practice [BBK, CNO, KFS].\\

Practitioners in geometry processing, computer graphics, medical imaging, machine learning, and other computational disciplines have employed spectral geometry to great effect, improving the quality and flexibility of algorithms for a broad range of geometric tasks. The most exciting research in this field requires synthesis of theoretical, numerical, and applied ideas to motivate practical research problems, extract detailed understanding of underlying models, and formulate discrete approximations that provide a stable, faithful link between theory and practice.\\ The primary objective of this workshop is to bring together researchers in theoretical, numerical, and applied spectral geometry to facilitate collaboration between these closely-linked communities. The workshop is jointly organized by faculty with expertise in each discipline: Iosif Polterovich (Professor and Canada Research Chair in Geometry and Spectral Theory, Université de Montréal), Nilima Nigam (Professor of Mathematics, Simon Fraser University) and Justin Solomon (Assistant Professor of Electrical Engineering and Computer Science, MIT). Invited attendees and speakers will be evenly divided among these groups and will be asked to communicate their research in language accessible to all three communities.\\ The workshop will be structured around identifying open problems and avenues for research of interest to the full audience. In particular:\\ - Researchers whose work focuses on theoretical aspects of spectral geometry will present some important recent advances in the field that could open new paths of interaction among theory, numerical analysis and applications.\\

- Researchers studying discrete or discretized aspects of spectral geometry will discuss how their work approximates or preserves structure from underlying theoretical models and to show how it might be used to bridge between theory and engineering-oriented practice.\\

- Researchers in applications areas will be asked to identify key theoretical questions and discretization challenges raised in their experiments with spectral geometry.\\

References\\ [ABN] E. Akhmetgaliyev, O. Bruno, and N. Nigam. A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem. Journal of Computational Physics (2015).\\

[AF] P. Antunes and P. Freitas. Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory 154 (2012), No. 1, 235-257.\\

[AKO] E. Akhmetgaliyev, C.-Y. Kao, and B. Oxting. Computational methods for extremal Steklov problems. ArXiv: 1601.00605.\\

[B] D. Boffi. Finite element approximation of eigenvalue problems. Acta Numerica 19 (2010), 1-120.\\

[BBG] P. Berard, G. Besson and S. Gallot. Embedding Riemannian manifolds by their heat kernel. GAFA 4 (1994), 373 398.\\

[BBG2] B. Bogosel, D. Bucur and A. Giacomini. Optimal shapes maximizing the Steklov eigenvalues. Preprint (2016).\\

[BBK] M. Botsch, D. Bommes, and L. Kobbelt. Efficient linear system solvers for mesh processing. International Conference on Mathematics of Surfaces (2005), 62-83.\\

[BF] D. Bucur and I. Fragala. Blaschke-Santalo and Mahler inequalities for the first eigenvalue of the Dirichlet Laplacian. Proc. London Math. Soc. (2016).\\

[BH] A. Barnett and A. Hassell. Fast computation of high-frequency Dirichlet eigenmodes via spectral flow of the interior Neumann-to-Dirichlet map. Comm. Pure Appl. Math. 67.3 (2014), 351-507.\\

[BH1] V. Bonnaillie-Noel and B. Helffer. Nodal and spectral minimal partitions – the state of the art in 2015. ArXiv, 1506.07249.\\

[BH2] V. Bonnaillie-Noel and B. Helffer. Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square with Application to Minimal Partitions. Experiment. Math. 20.3 (2011), 304-322.\\

[BHT] A. Barnett, A. Hassell and M. Tacy, Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues, arXiv:1512.04165.\\

[BIK] D. Burago, S. Ivanov and S. Kurylev. A graph discretization of the Laplace-Beltrami operator. J. Spectr. Theory 4 (2014), 675-714.\\

[BIK2] D. Burago, S. Ivanov and S. Kurylev. Spectral stability of metric-measure Laplacians. ArXiv, 150606781.\\

[BN] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15.6 (2006), 1373-1396.\\

[BNS] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research (2006).\\

[CGPS] C. Carstensen, D. Gallistl, and M. Schedensack. Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems. Math. Comp. 84.293 (2015), 1061-1087.\\

[CL] R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis 21.1 (2006), 5-30.\\

[CM] R. Coifman and M. Maggioni. Diffusion wavelets. Applied and Computational Harmonic Analysis 21.1 (2006), 53 94.\\

[CMS], F. Cakoni, P. Monk and J. Sun. Error analysis of the finite element approximation of transmission eigenvalues. Computational Methods in Applied Mathematics 14 (2014), 419-427.\\

[CNO] L. Castelli Aleardi, A. Nolin, and M. Ovsjanikov. Efficient and practical tree preconditioning for solving Laplacian systems. International Symposium on Experimental Algorithms (2015).\\

[CS] Y. Canzani and P. Sarnak. On the topology of zero sets of monochromatic random waves. ArXiv, 1412.4437.\\

[CSBGO] E. Corman, J. Solomon, M. Ben-Chen, L. Guibas, and M. Ovsjanikov. Functional characterization of intrinsic and extrinsic geometry. Transactions on Graphics, to appear.\\

[DBGPH] S. Dong, P. Bremer, M. Garland, V. Pascucci, and J. Hart. Spectral surface quadrangulation. SIGGRAPH (2006), 1057-1066.\\

[DP] R. B. Platte and T. A. Driscoll. Computing eigenmodes of elliptic operators using radial basis functions. Computers Math. Appl. 48 (2004), 561–576.\\

[FS1] A. Fraser and R. Schoen. The first Steklov eigenvalue, conformal geometry and minimal surfaces. Adv. Math. 226.5 (2011), 4011-4030.\\

[FS2] A. Fraser and R. Schoen. Sharp eigenvalue bounds and minimal surfaces in the ball. Invent. Math. 203.3 (2016), 823-890.\\

[GGO] S. Giani, L. Grubišić, and J. Ovall. Error control for hp-adaptive approximations of semi-definite eigenvalue problems. Computing (2012), 1-23.\\

[GKKLU] A. Greenleaf, H. Kettunen, Y. Kurylev, M. Lassas, G. Uhlmann. Superdimensional Metamaterial Resonators. ArXiV, 1409.3608\\

[GP] A. Girouard and I. Polterovich. Spectral geometry of the Steklov problem. J. Spec. Theory, to appear.\\

[GPPS] A. Girouard, L. Parnovski, I. Polterovich and D. Sher. The Steklov spectrum of surfaces: asymptotics and invariants. Math. Proc. Camb. Philos. Soc. 157 (2014), 379-389.\\

[JLNNP] D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigam and I. Polterovich. How large can the first eigenvalue be on a surface of genus two? Int. Math. Res. Not. 63 (2005), 3967-3985.\\

[JMS] P. Jones, M. Maggioni and R. Schul. Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels. PNAS 105.6 (2008), 1803-1808.\\

[KFS] D. Krishnan, R. Fattal, and R. Szeliski. Efficient preconditioning of Laplacian matrices for computer graphics. SIGGRAPH (2013).\\

[L] Xuefeng Liu. A framework of verified eigenvalue bounds for self-adjoint differential operators. Applied Mathematics and Computation 267 (2015), 341-355.\\

[Na] M. Nastasescu, The number of ovals of a real plane curve. Senior Thesis, Princeton University (2011).\\

[OBSBG] M. Ovsjanikov, M. Ben-Chen, J. Solomon, A. Butscher, and L. Guibas. Functional maps: A flexible representation of maps between shapes. SIGGRAPH (2012).\\

[OMMG] M. Ovsjanikov, Q. Merigot, F. Memoli, and L. Guibas. One point isometric matching with the heat kernel. Symposium on Geometry Processing (2010).\\

[OMPG] M. Ovsjanikov, Q. Merigot, V. Patraucean, and L. Guibas. Shape matching via quotient spaces. Symposium on Geometry Processing (2013).\\

[OSG] M. Ovsjanikov, J. Sun, and L. Guibas. Global intrinsic symmetries of shapes. Symposium on Geometry Processing (2008).\\

[PS] I. Polterovich and D. Sher. Heat invariants of the Steklov problem. J. Geom. Analysis 25.2 (2015), 924-950.\\

[Poly] Polymath7 project: The hot spots conjecture. http://michaelnielsen.org/polymath1/index. php? title=The\_hot\_spots\_conjecture\\

[ROABCG] R. Rustamov, M. Ovsjanikov, O. Azencot, M. Ben-Chen, F. Chazal, and L. Guibas. Map-based exploration of intrinsic shape differences and variability. SIGGRAPH (2013).\\

[S] A. Strohmaier. Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces. ArXiv:1604.02722.\\

[So] M. Sodin. Lectures on random nodal portraits. Lecture Notes for a Mini-course Given at the St. Petersburg Summer School in Probability and Statistical Physics (June, 2012).\\

[SOG] J. Sun, M. Ovsjanikov, and L. Guibas. A concise and provably informative multi-scale signature based on heat diffusion. Symposium on Geometry Processing (2009), 1383-1392.\\

[SW] P. Sarnak and I. Wigman. Topologies of nodal sets of random band limited functions. In: Contemporary Mathematics 664 (2016).\\

[SU] A. Strohmaier and V. Uski. An algorithm for the computation of eigenvalues, spectral zeta functions and zeta determinants on hyperbolic surfaces. Comm. Math. Phys. 317.3 (2013), 827-869.\\

[WSSZ] H. Wang, P. Simari, Z. Su, and H. Zhang. Spectral global intrinsic symmetry invariant functions. Graphics Interface (2014), 209-215.\\

[ZGL] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields and harmonic functions. International Conference on Machine Learning (2003).

There is a long history of fruitful interaction between geometric spectral theory and numerical analysis. The interplay between these fields recently has reached new heights due to the availability of powerful computers and to the development of efficient numerical algorithms. From the theoretical standpoint, numerical calculations serve as an indispensable source of intuition and are behind many advances in spectral geometry. In particular, numerical experiments stimulated recent breakthrough research on nodal portraits of random linear combinations of eigenfunctions [Na, So, SW, CS]. Numerical methods are actively used in the rapidly-developing theory of spectral minimal partitions [BH1, BH2]. For low eigenvalues, numerical experiments are a major source of conjectures in shape optimization [JLNNP, AF, AKO, BBG2]. Finally, an exciting new trend is to use numerical analysis not only to formulate conjectures, but actually to prove theorems in spectral geometry using rigorous computer-assisted methods [BF, Poly].\\ New numerical algorithms have been developed for fast, precise computation of higher eigenvalues and eigenfunctions of the Laplacian (see [BH, BHT]). In some special cases – such as for the Laplacian on hyperbolic surfaces – algorithms were found for computation of eigenvalues and other spectral quantities with rigorous error estimates [SU, S].\\ Many modern applications of spectral geometry are inspired by problems in computer graphics, shape analysis and machine learning. These techniques are based on the properties of spectral quantities such as heat kernels and eigenfunctions associated with the Laplacian [BBG, JMS]. For instance, popular techniques for surface-to-surface correspondence in computer graphics and medical imaging make use of spectral techniques to identify features to be matched [SOG, OMMG], to leverage and detect symmetry [OMPG, OSG, WSSZ], to find differences between features of different surfaces [ROABCG, CSBGO] and to provide low-frequency approximations of mappings [OBSBG]. An algorithm for quadrilateral meshing makes use of the shape and topology of Laplacian nodal domains [DBGPH]. In the higher-dimensional machine learning context, spectral geometry appears when propagating labels through a partially labeled dataset [BNS, ZGL], reducing the dimensionality of a dataset [BN, CL], or constructing wavelet bases on unstructured clouds of data points [CM].\\

The Laplacian on a Riemannian manifold carries considerable information about intrinsic geometry. To capture the extrinsic geometry of shapes using spectral quantities, one needs to use other operators. A natural choice is the Dirichlet-to-Neumann operator, which has been the focus of intensive research in modern theoretical spectral geometry [PS, GPPS, FS1, FS2, GP, AKO, BBG2]. The collection of Dirichlet-to-Neumann eigenvalues is called the Steklov spectrum, as it is precisely the spectrum of the Steklov boundary value problem. Recent advances in the study of the Steklov spectrum, particularly, of the corresponding geometric invariants, open up a number of promising new applications to shape analysis.\\ Applications of spectral geometry to shape analysis and machine learning heavily rely on numerical methods. For a successful implementation of a numerical algorithm, it is crucial to use an appropriate discretization of the corresponding differential operator. Understanding the properties of discretizations of differential operators defined on complex geometries such as Riemannian manifolds is highly nontrivial; new advances in this direction for the Laplace Beltrami operator were made in [BIK1, BIK2].\\

Accompanying these developments, the design and analysis of high-accuracy discretization methods for spectral problems and optimal design has been the focus of intense research in numerical analysis. A plethora of discretization techniques have been developed, including methods based on particular solutions [BHT], spectral methods, finite element methods [B, CGPS] and integral equations [ABN]. Each presents distinct advantages and challenges in numerical analysis, which in turn leads to improved understanding of the properties of the eigenvalues and eigenfunctions of the operators under investigation. For example, the use of common Lagrange finite elements for the Maxwell eigenvalue problem was shown to yield the wrong spectrum even on simple geometries [B], motivating the large-scale adoption of the edge finite elements. The design of numerical methods for inverse problems lead to the investigation of nonlinear and non-self-adjoint transmission eigenvalue problems, which in turn has inspired novel methods of discretizations for them [CMS]. The computations were used extensively to enhance our understanding of the situations under which the existence of such eigenvalues could be proven. The design of a boundary integral method for mixed Dirichlet-Neumann eigenvalues of the Laplacian is necessarily intertwined with a careful study of the asymptotic behaviour of the eigenfunctions [ABN]. A related area of investigation seeks to provide computable error bounds and eigenvalue enclosures [L]. These bounds are crucial if numerical methods are to be used for formulating conjectures.\\ Upon discretization, the resulting large discrete systems have been coupled with state-of-the-art eigenvalue solvers and optimization methods. Naive approaches do not suffice, and the design of algorithms in this area has yielded many interesting and challenging research directions, due to the potentially nonlinear/non-convex nature of the objective function to be optimized. If the eigenvalues are clustered, or if the original spectrum is not discrete, these problems are much harder [GGO]. Application-specific solvers for spectral geometry problems also have been developed to support the transition of these techniques from academia to practice [BBK, CNO, KFS].\\

Practitioners in geometry processing, computer graphics, medical imaging, machine learning, and other computational disciplines have employed spectral geometry to great effect, improving the quality and flexibility of algorithms for a broad range of geometric tasks. The most exciting research in this field requires synthesis of theoretical, numerical, and applied ideas to motivate practical research problems, extract detailed understanding of underlying models, and formulate discrete approximations that provide a stable, faithful link between theory and practice.\\ The primary objective of this workshop is to bring together researchers in theoretical, numerical, and applied spectral geometry to facilitate collaboration between these closely-linked communities. The workshop is jointly organized by faculty with expertise in each discipline: Iosif Polterovich (Professor and Canada Research Chair in Geometry and Spectral Theory, Université de Montréal), Nilima Nigam (Professor of Mathematics, Simon Fraser University) and Justin Solomon (Assistant Professor of Electrical Engineering and Computer Science, MIT). Invited attendees and speakers will be evenly divided among these groups and will be asked to communicate their research in language accessible to all three communities.\\ The workshop will be structured around identifying open problems and avenues for research of interest to the full audience. In particular:\\ - Researchers whose work focuses on theoretical aspects of spectral geometry will present some important recent advances in the field that could open new paths of interaction among theory, numerical analysis and applications.\\

- Researchers studying discrete or discretized aspects of spectral geometry will discuss how their work approximates or preserves structure from underlying theoretical models and to show how it might be used to bridge between theory and engineering-oriented practice.\\

- Researchers in applications areas will be asked to identify key theoretical questions and discretization challenges raised in their experiments with spectral geometry.\\

References\\ [ABN] E. Akhmetgaliyev, O. Bruno, and N. Nigam. A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem. Journal of Computational Physics (2015).\\

[AF] P. Antunes and P. Freitas. Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory 154 (2012), No. 1, 235-257.\\

[AKO] E. Akhmetgaliyev, C.-Y. Kao, and B. Oxting. Computational methods for extremal Steklov problems. ArXiv: 1601.00605.\\

[B] D. Boffi. Finite element approximation of eigenvalue problems. Acta Numerica 19 (2010), 1-120.\\

[BBG] P. Berard, G. Besson and S. Gallot. Embedding Riemannian manifolds by their heat kernel. GAFA 4 (1994), 373 398.\\

[BBG2] B. Bogosel, D. Bucur and A. Giacomini. Optimal shapes maximizing the Steklov eigenvalues. Preprint (2016).\\

[BBK] M. Botsch, D. Bommes, and L. Kobbelt. Efficient linear system solvers for mesh processing. International Conference on Mathematics of Surfaces (2005), 62-83.\\

[BF] D. Bucur and I. Fragala. Blaschke-Santalo and Mahler inequalities for the first eigenvalue of the Dirichlet Laplacian. Proc. London Math. Soc. (2016).\\

[BH] A. Barnett and A. Hassell. Fast computation of high-frequency Dirichlet eigenmodes via spectral flow of the interior Neumann-to-Dirichlet map. Comm. Pure Appl. Math. 67.3 (2014), 351-507.\\

[BH1] V. Bonnaillie-Noel and B. Helffer. Nodal and spectral minimal partitions – the state of the art in 2015. ArXiv, 1506.07249.\\

[BH2] V. Bonnaillie-Noel and B. Helffer. Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square with Application to Minimal Partitions. Experiment. Math. 20.3 (2011), 304-322.\\

[BHT] A. Barnett, A. Hassell and M. Tacy, Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues, arXiv:1512.04165.\\

[BIK] D. Burago, S. Ivanov and S. Kurylev. A graph discretization of the Laplace-Beltrami operator. J. Spectr. Theory 4 (2014), 675-714.\\

[BIK2] D. Burago, S. Ivanov and S. Kurylev. Spectral stability of metric-measure Laplacians. ArXiv, 150606781.\\

[BN] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15.6 (2006), 1373-1396.\\

[BNS] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research (2006).\\

[CGPS] C. Carstensen, D. Gallistl, and M. Schedensack. Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems. Math. Comp. 84.293 (2015), 1061-1087.\\

[CL] R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis 21.1 (2006), 5-30.\\

[CM] R. Coifman and M. Maggioni. Diffusion wavelets. Applied and Computational Harmonic Analysis 21.1 (2006), 53 94.\\

[CMS], F. Cakoni, P. Monk and J. Sun. Error analysis of the finite element approximation of transmission eigenvalues. Computational Methods in Applied Mathematics 14 (2014), 419-427.\\

[CNO] L. Castelli Aleardi, A. Nolin, and M. Ovsjanikov. Efficient and practical tree preconditioning for solving Laplacian systems. International Symposium on Experimental Algorithms (2015).\\

[CS] Y. Canzani and P. Sarnak. On the topology of zero sets of monochromatic random waves. ArXiv, 1412.4437.\\

[CSBGO] E. Corman, J. Solomon, M. Ben-Chen, L. Guibas, and M. Ovsjanikov. Functional characterization of intrinsic and extrinsic geometry. Transactions on Graphics, to appear.\\

[DBGPH] S. Dong, P. Bremer, M. Garland, V. Pascucci, and J. Hart. Spectral surface quadrangulation. SIGGRAPH (2006), 1057-1066.\\

[DP] R. B. Platte and T. A. Driscoll. Computing eigenmodes of elliptic operators using radial basis functions. Computers Math. Appl. 48 (2004), 561–576.\\

[FS1] A. Fraser and R. Schoen. The first Steklov eigenvalue, conformal geometry and minimal surfaces. Adv. Math. 226.5 (2011), 4011-4030.\\

[FS2] A. Fraser and R. Schoen. Sharp eigenvalue bounds and minimal surfaces in the ball. Invent. Math. 203.3 (2016), 823-890.\\

[GGO] S. Giani, L. Grubišić, and J. Ovall. Error control for hp-adaptive approximations of semi-definite eigenvalue problems. Computing (2012), 1-23.\\

[GKKLU] A. Greenleaf, H. Kettunen, Y. Kurylev, M. Lassas, G. Uhlmann. Superdimensional Metamaterial Resonators. ArXiV, 1409.3608\\

[GP] A. Girouard and I. Polterovich. Spectral geometry of the Steklov problem. J. Spec. Theory, to appear.\\

[GPPS] A. Girouard, L. Parnovski, I. Polterovich and D. Sher. The Steklov spectrum of surfaces: asymptotics and invariants. Math. Proc. Camb. Philos. Soc. 157 (2014), 379-389.\\

[JLNNP] D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigam and I. Polterovich. How large can the first eigenvalue be on a surface of genus two? Int. Math. Res. Not. 63 (2005), 3967-3985.\\

[JMS] P. Jones, M. Maggioni and R. Schul. Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels. PNAS 105.6 (2008), 1803-1808.\\

[KFS] D. Krishnan, R. Fattal, and R. Szeliski. Efficient preconditioning of Laplacian matrices for computer graphics. SIGGRAPH (2013).\\

[L] Xuefeng Liu. A framework of verified eigenvalue bounds for self-adjoint differential operators. Applied Mathematics and Computation 267 (2015), 341-355.\\

[Na] M. Nastasescu, The number of ovals of a real plane curve. Senior Thesis, Princeton University (2011).\\

[OBSBG] M. Ovsjanikov, M. Ben-Chen, J. Solomon, A. Butscher, and L. Guibas. Functional maps: A flexible representation of maps between shapes. SIGGRAPH (2012).\\

[OMMG] M. Ovsjanikov, Q. Merigot, F. Memoli, and L. Guibas. One point isometric matching with the heat kernel. Symposium on Geometry Processing (2010).\\

[OMPG] M. Ovsjanikov, Q. Merigot, V. Patraucean, and L. Guibas. Shape matching via quotient spaces. Symposium on Geometry Processing (2013).\\

[OSG] M. Ovsjanikov, J. Sun, and L. Guibas. Global intrinsic symmetries of shapes. Symposium on Geometry Processing (2008).\\

[PS] I. Polterovich and D. Sher. Heat invariants of the Steklov problem. J. Geom. Analysis 25.2 (2015), 924-950.\\

[Poly] Polymath7 project: The hot spots conjecture. http://michaelnielsen.org/polymath1/index. php? title=The\_hot\_spots\_conjecture\\

[ROABCG] R. Rustamov, M. Ovsjanikov, O. Azencot, M. Ben-Chen, F. Chazal, and L. Guibas. Map-based exploration of intrinsic shape differences and variability. SIGGRAPH (2013).\\

[S] A. Strohmaier. Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces. ArXiv:1604.02722.\\

[So] M. Sodin. Lectures on random nodal portraits. Lecture Notes for a Mini-course Given at the St. Petersburg Summer School in Probability and Statistical Physics (June, 2012).\\

[SOG] J. Sun, M. Ovsjanikov, and L. Guibas. A concise and provably informative multi-scale signature based on heat diffusion. Symposium on Geometry Processing (2009), 1383-1392.\\

[SW] P. Sarnak and I. Wigman. Topologies of nodal sets of random band limited functions. In: Contemporary Mathematics 664 (2016).\\

[SU] A. Strohmaier and V. Uski. An algorithm for the computation of eigenvalues, spectral zeta functions and zeta determinants on hyperbolic surfaces. Comm. Math. Phys. 317.3 (2013), 827-869.\\

[WSSZ] H. Wang, P. Simari, Z. Su, and H. Zhang. Spectral global intrinsic symmetry invariant functions. Graphics Interface (2014), 209-215.\\

[ZGL] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields and harmonic functions. International Conference on Machine Learning (2003).