## 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,

Pierre Albin, University of Illinois.

Alexandre Girouard, Université Laval (Québec).

Colin Guillarmou, Ecole Normale Superieure (Paris).

Leonid Parnovski, University College London.

Iosif Polterovich, Université de Montréal.

Frédéric Rochon, Université du Québec à Montréal.

John Toth, McGill University.

Alejandro Uribe, University of Michigan.

Carlos Villegas-Blas, UNAM (Mexico).

*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.__Recent/ongoing collaborators (in alphabetical order)__:Pierre Albin, University of Illinois.

Alexandre Girouard, Université Laval (Québec).

Colin Guillarmou, Ecole Normale Superieure (Paris).

Leonid Parnovski, University College London.

Iosif Polterovich, Université de Montréal.

Frédéric Rochon, Université du Québec à Montréal.

John Toth, McGill University.

Alejandro Uribe, University of Michigan.

Carlos Villegas-Blas, UNAM (Mexico).

__Full publication list__:1. Johnson, Charles R.; Duarte, António Leal; Saiago, Carlos M.; Sher, David. Eigenvalues, multiplicities and graphs. Algebra and its applications, 167--183, Contemp. Math., 419, Amer. Math. Soc., Providence, RI, 2006.

2. Johnson, Charles R.; Jordan-Squire, Christopher; Sher, David A. Eigenvalue Assignments and The Two Largest Multiplicities in an Hermitian Matrix Whose Graph is a Tree. Discrete Applied Mathematics 158 (2010), pp. 681-691.

3. Sher, David A. The heat kernel on an asymptotically conic manifold. Analysis & PDE 6, no. 7, 1755-1991 (2013).

4. Sher, David A. Conic degeneration and the determinant of the Laplacian. Journale d'Analyse Mathematique 126, no. 1, p. 175-226 (2015). DOI 10.1007/s11854-015-0015-3

5. Sher, David A. The determinant on flat conic surfaces with excision of disks. Proc. Amer. Math. Soc. 143, no. 3, p. 1333-1346 (2015).

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.

7. Colin Guillarmou and David A. Sher. Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion. Int. Math. Res. Not., p. 1-75 (2014). DOI 10.1093/imrn/rnu119.

8. Alexandre Girouard, Leonid Parnovski, Iosif Polterovich, and David A. Sher. The Steklov spectrum of surfaces: asymptotics and invariants. Math. Proc. Camb. Phil. Soc., 157, no. 3, p. 379-389 (2014). DOI 10.1017/S030500411400036X.

9. Pierre Albin, Frederic Rochon, and David A. Sher. Resolvent, heat kernel, and torsion under degeneration to fibered cusps. Preprint, arXiv:1410.8406, p. 1-101.

10. Pierre Albin, Frederic Rochon, and David A. Sher. Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Preprint, arXiv:1411.1105, p. 1-45.

11. Iosif Polterovich, David A. Sher, and John A. Toth. Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces. Preprint, arXiv:1506.07600, p. 1-26.

2. Johnson, Charles R.; Jordan-Squire, Christopher; Sher, David A. Eigenvalue Assignments and The Two Largest Multiplicities in an Hermitian Matrix Whose Graph is a Tree. Discrete Applied Mathematics 158 (2010), pp. 681-691.

3. Sher, David A. The heat kernel on an asymptotically conic manifold. Analysis & PDE 6, no. 7, 1755-1991 (2013).

4. Sher, David A. Conic degeneration and the determinant of the Laplacian. Journale d'Analyse Mathematique 126, no. 1, p. 175-226 (2015). DOI 10.1007/s11854-015-0015-3

**.**5. Sher, David A. The determinant on flat conic surfaces with excision of disks. Proc. Amer. Math. Soc. 143, no. 3, p. 1333-1346 (2015).

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.

7. Colin Guillarmou and David A. Sher. Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion. Int. Math. Res. Not., p. 1-75 (2014). DOI 10.1093/imrn/rnu119.

8. Alexandre Girouard, Leonid Parnovski, Iosif Polterovich, and David A. Sher. The Steklov spectrum of surfaces: asymptotics and invariants. Math. Proc. Camb. Phil. Soc., 157, no. 3, p. 379-389 (2014). DOI 10.1017/S030500411400036X.

9. Pierre Albin, Frederic Rochon, and David A. Sher. Resolvent, heat kernel, and torsion under degeneration to fibered cusps. Preprint, arXiv:1410.8406, p. 1-101.

10. Pierre Albin, Frederic Rochon, and David A. Sher. Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Preprint, arXiv:1411.1105, p. 1-45.

11. Iosif Polterovich, David A. Sher, and John A. Toth. Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces. Preprint, arXiv:1506.07600, p. 1-26.