Research
When an object such as a musical instrument vibrates, it does so at a particular set of resonance frequencies, each with a corresponding mode or modes of vibration. Mathematically, these resonance frequencies and modes correspond to the eigenvalues and eigenfunctions of a differential operator known as the Laplacian. My area of research, spectral geometry, is the study of these eigenvalues and eigenfunctions, and their relation to the geometry of the object in question. These questions are often generalized to the setting of Riemannian manifolds, as well as to other operators arising from considerations of physics or geometry.
Research collaborators since 2012 (in alphabetical order):
Pierre Albin, University of Illinois.
Bruno Colbois, Université de Neuchatel (Switzerland).
Nikolay Filonov, St. Petersburg State University.
Alexandre Girouard, Université Laval (Québec).
Katie Gittins, University of Durham.
Carolyn Gordon, Dartmouth College.
Colin Guillarmou, Ecole Normale Superieure (Paris).
Asma Hassannezhad, University of Bristol.
Stanislav Krymski, St. Petersburg State University.
Corentin Léna, University of Padova (Italy).
Michael Levitin, University of Reading.
Sam Looi, Caltech.
Medet Nursultanov, University of Helsinki.
Leonid Parnovski, University College London.
Iosif Polterovich, Université de Montréal.
Frédéric Rochon, Université du Québec à Montréal.
Julie Rowlett, Chalmers University (Gothenburg, Sweden).
Luis Silva, UNAM (Mexico).
John Toth, McGill University.
Alejandro Uribe, University of Michigan.
Boris Vertman, University of Oldenburg (Germany).
Carlos Villegas-Blas, UNAM (Mexico).
Monika Winklmeier, Universidad de los Andes (Bogota, Colombia).
Full publication list:
28. David Sher, Luis Silva, Boris Vertman, and Monika Winklmeier. Scattering theory for difference equations with operator coefficients. To appear, Journal of the London Mathematical Society (accepted January 2026). Preprint, arXiV:2501.11194, p. 1-47.
27. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Polya's conjecture for Dirichlet eigenvalues of annuli. To appear, Journal of the London Mathematical Society (accepted December 2025). Preprint, arXiV:2505.21737, p. 1-31.
26. Katie Gittins, Asma Hassannzehad, Corentin Lena, and David Sher. Nodal counts for the Robin problem on Lipschitz domains. To appear, Pure and Applied Functional Analysis (accepted April 2025). Preprint, arXiV:2411.11427, p. 1-36.
25. Asma Hassannezhad and David Sher. Nodal count for Dirichlet-to-Neumann operators with potential. To appear, Proceedings of the American Mathematical Society (accepted July 2022). Preprint, arXiV:2107.03370, p. 1-10.
24. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Uniform enclosures for the phase and zeroes of Bessel functions and their derivatives. SIAM Journal of Mathematical Analysis 56, no. 6, p. 7644-7682 (2024). DOI 10.1137/24M1642032.
23. Medet Nursultanov, Julie Rowlett, and David Sher. The heat kernel for curvilinear polygonal domains in surfaces. Annales Mathematiques du Quebec p. 1-61 (2024). DOI 10.1007/s40316-024-00237-4.
22. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Inequalities a la Polya for the Aharonov-Bohm eigenvalues of the disk. Journal of Spectral Theory 14, no. 2, p. 597-618 (2024). DOI 10.4171/JST/506.
21. Asma Hassannezhad and David Sher. On Pleijel's nodal domain theorem for the Robin problem. Bulletin of the London Mathematical Society 56, p. 1449-1467 (2024). DOI 10.1112/blms.13005.
20. (Survey article) Bruno Colbois, Alexandre Girouard, Carolyn Gordon, and David Sher. Some recent developments on the Steklov eigenvalue problem. Revista Matematica Complutense (2023). DOI 10.1007/s13163-023-00480-3.
19. David A. Sher. Joint asymptotic expansions for Bessel functions. Pure and Applied Analysis 5, no. 2, p. 461-505 (2023). DOI 10.2140/paa.2023.5.461.
18. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Polya's conjecture for Euclidean balls. Inventiones mathematicae 234, p. 169-209 (2023). DOI 10.1007/s00222-023-01198-1.
17. Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. Sloshing, Steklov and corners: asymptotics of Steklov eigenvalues for curvilinear polygons. Proceedings of the London Mathematical Society 125, no. 3, p. 359-487 (June 2022). DOI 10.1112/plms.12461.
16. Pierre Albin, Frederic Rochon, and David A. Sher. A Cheeger-Muller theorem for manifolds with wedge singularities. Analysis & PDE 15, no. 3, p. 567-642 (2022). DOI 10.2140/apde.2022.15.567.
15. Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. Sloshing, Steklov and corners: asymptotics of sloshing eigenvalues. Journale d'Analyse Mathématique 146, no. 1, p. 65-125 (2022). DOI 10.1007/s11854-021-0188-x.
14. David Sher, Alejandro Uribe, and Carlos Villegas-Blas. On the pseudospectra of Schrodinger operators on Zoll manifolds. Memorias de la reunión de Matemáticos Mexicanos en el Mundo 2018, Contemporary Mathematics Series volume 775, American Mathematical Society (2021). DOI 10.1090/conm/775.
13. Pierre Albin, Frederic Rochon, and David A. Sher. Resolvent, heat kernel, and torsion under degeneration to fibered cusps. Memoirs of the American Mathematical Society 269, no. 1314 (2021). DOI 10.1090/memo/1314.
12. Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. Inverse Steklov spectral problem for curvilinear polygons. International Mathematics Research Notices, 2021, no. 1, p. 1--37 (2021). DOI 10.1093/imrn/rnaa200.
11. Medet Nursultanov, Julie Rowlett, and David Sher. How to hear the corners of a drum. In 2017 Matrix Annals, p. 243-278, MATRIX, Melbourne, Australia (2019). Available online here.
10. Iosif Polterovich, David A. Sher, and John A. Toth. Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces. J. Reine Angew. Math. (Crelle’s Journal), 754, p. 14-47 (2019). DOI 10.1515/crelle-2017-0018.
9. Pierre Albin, Frederic Rochon, and David A. Sher. Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Duke Mathematical Journal, 167, no. 10, 1883-1950 (2018). DOI 10.1215/00127094-2018-0009.
8. Alexandre Girouard, Leonid Parnovski, Iosif Polterovich, and David A. Sher. The Steklov spectrum of surfaces: asymptotics and invariants. Mathematical Proceedings of the Cambridge Philosophical Society, 157, no. 3, p. 379-389 (2014). DOI 10.1017/S030500411400036X.
7. Colin Guillarmou and David A. Sher. Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion. International Mathematics Research Notices, p. 1-75 (2015). DOI 10.1093/imrn/rnu119
6. Iosif Polterovich and David A. Sher. Heat invariants of the Steklov problem. Journal of Geometric Analysis 25, no. 2, p. 924-950 (2015). DOI 10.1007/s12220-013-4951-4.
5. David A. Sher. The determinant on flat conic surfaces with excision of disks. Proceedings of the American Mathematical Society 143, no. 3, p. 1333-1346 (2015).
4. David A. Sher. Conic degeneration and the determinant of the Laplacian. Journale d'Analyse Mathématique 126, no. 1, p. 175-226 (2015). DOI 10.1007/s11854-015-0015-3.
3. David A. Sher. The heat kernel on an asymptotically conic manifold. Analysis & PDE 6, no. 7, 1755-1791 (2013).
2. Charles R. Johnson, Christopher Jordan-Squire, and David A. Sher. Eigenvalue assignments and the two largest multiplicities in an Hermitian matrix whose graph is a tree. Discrete Applied Mathematics 158, no. 6, pp. 681-691 (2010).
1. Charles R. Johnson, António Leal Duarte, Carlos M. Saiago, and David Sher. Eigenvalues, multiplicities and graphs. In Algebra and its applications, 167--183, Contemporary Mathematics, 419, American Mathematical Society, Providence, RI, 2006.
27. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Polya's conjecture for Dirichlet eigenvalues of annuli. To appear, Journal of the London Mathematical Society (accepted December 2025). Preprint, arXiV:2505.21737, p. 1-31.
26. Katie Gittins, Asma Hassannzehad, Corentin Lena, and David Sher. Nodal counts for the Robin problem on Lipschitz domains. To appear, Pure and Applied Functional Analysis (accepted April 2025). Preprint, arXiV:2411.11427, p. 1-36.
25. Asma Hassannezhad and David Sher. Nodal count for Dirichlet-to-Neumann operators with potential. To appear, Proceedings of the American Mathematical Society (accepted July 2022). Preprint, arXiV:2107.03370, p. 1-10.
24. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Uniform enclosures for the phase and zeroes of Bessel functions and their derivatives. SIAM Journal of Mathematical Analysis 56, no. 6, p. 7644-7682 (2024). DOI 10.1137/24M1642032.
23. Medet Nursultanov, Julie Rowlett, and David Sher. The heat kernel for curvilinear polygonal domains in surfaces. Annales Mathematiques du Quebec p. 1-61 (2024). DOI 10.1007/s40316-024-00237-4.
22. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Inequalities a la Polya for the Aharonov-Bohm eigenvalues of the disk. Journal of Spectral Theory 14, no. 2, p. 597-618 (2024). DOI 10.4171/JST/506.
21. Asma Hassannezhad and David Sher. On Pleijel's nodal domain theorem for the Robin problem. Bulletin of the London Mathematical Society 56, p. 1449-1467 (2024). DOI 10.1112/blms.13005.
20. (Survey article) Bruno Colbois, Alexandre Girouard, Carolyn Gordon, and David Sher. Some recent developments on the Steklov eigenvalue problem. Revista Matematica Complutense (2023). DOI 10.1007/s13163-023-00480-3.
19. David A. Sher. Joint asymptotic expansions for Bessel functions. Pure and Applied Analysis 5, no. 2, p. 461-505 (2023). DOI 10.2140/paa.2023.5.461.
18. Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher. Polya's conjecture for Euclidean balls. Inventiones mathematicae 234, p. 169-209 (2023). DOI 10.1007/s00222-023-01198-1.
17. Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. Sloshing, Steklov and corners: asymptotics of Steklov eigenvalues for curvilinear polygons. Proceedings of the London Mathematical Society 125, no. 3, p. 359-487 (June 2022). DOI 10.1112/plms.12461.
16. Pierre Albin, Frederic Rochon, and David A. Sher. A Cheeger-Muller theorem for manifolds with wedge singularities. Analysis & PDE 15, no. 3, p. 567-642 (2022). DOI 10.2140/apde.2022.15.567.
15. Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. Sloshing, Steklov and corners: asymptotics of sloshing eigenvalues. Journale d'Analyse Mathématique 146, no. 1, p. 65-125 (2022). DOI 10.1007/s11854-021-0188-x.
14. David Sher, Alejandro Uribe, and Carlos Villegas-Blas. On the pseudospectra of Schrodinger operators on Zoll manifolds. Memorias de la reunión de Matemáticos Mexicanos en el Mundo 2018, Contemporary Mathematics Series volume 775, American Mathematical Society (2021). DOI 10.1090/conm/775.
13. Pierre Albin, Frederic Rochon, and David A. Sher. Resolvent, heat kernel, and torsion under degeneration to fibered cusps. Memoirs of the American Mathematical Society 269, no. 1314 (2021). DOI 10.1090/memo/1314.
12. Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. Inverse Steklov spectral problem for curvilinear polygons. International Mathematics Research Notices, 2021, no. 1, p. 1--37 (2021). DOI 10.1093/imrn/rnaa200.
11. Medet Nursultanov, Julie Rowlett, and David Sher. How to hear the corners of a drum. In 2017 Matrix Annals, p. 243-278, MATRIX, Melbourne, Australia (2019). Available online here.
10. Iosif Polterovich, David A. Sher, and John A. Toth. Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces. J. Reine Angew. Math. (Crelle’s Journal), 754, p. 14-47 (2019). DOI 10.1515/crelle-2017-0018.
9. Pierre Albin, Frederic Rochon, and David A. Sher. Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Duke Mathematical Journal, 167, no. 10, 1883-1950 (2018). DOI 10.1215/00127094-2018-0009.
8. Alexandre Girouard, Leonid Parnovski, Iosif Polterovich, and David A. Sher. The Steklov spectrum of surfaces: asymptotics and invariants. Mathematical Proceedings of the Cambridge Philosophical Society, 157, no. 3, p. 379-389 (2014). DOI 10.1017/S030500411400036X.
7. Colin Guillarmou and David A. Sher. Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion. International Mathematics Research Notices, p. 1-75 (2015). DOI 10.1093/imrn/rnu119
6. Iosif Polterovich and David A. Sher. Heat invariants of the Steklov problem. Journal of Geometric Analysis 25, no. 2, p. 924-950 (2015). DOI 10.1007/s12220-013-4951-4.
5. David A. Sher. The determinant on flat conic surfaces with excision of disks. Proceedings of the American Mathematical Society 143, no. 3, p. 1333-1346 (2015).
4. David A. Sher. Conic degeneration and the determinant of the Laplacian. Journale d'Analyse Mathématique 126, no. 1, p. 175-226 (2015). DOI 10.1007/s11854-015-0015-3.
3. David A. Sher. The heat kernel on an asymptotically conic manifold. Analysis & PDE 6, no. 7, 1755-1791 (2013).
2. Charles R. Johnson, Christopher Jordan-Squire, and David A. Sher. Eigenvalue assignments and the two largest multiplicities in an Hermitian matrix whose graph is a tree. Discrete Applied Mathematics 158, no. 6, pp. 681-691 (2010).
1. Charles R. Johnson, António Leal Duarte, Carlos M. Saiago, and David Sher. Eigenvalues, multiplicities and graphs. In Algebra and its applications, 167--183, Contemporary Mathematics, 419, American Mathematical Society, Providence, RI, 2006.
