Research

Eigenvalues of differential operators such as the Laplacian and bi-Laplacian represent frequencies of vibration, rates of decay to equilibrium, stability indices, and quantum energy levels. Much of my research aims at discovering and proving inequalities for spectral functionals and determining the extremal configurations in which equality reigns.

Click for open problems from workshops at the: American Institute of Mathematics, the Oberwolfach Research Institute (pages 45-71), and links from the Queen Dido Conference.
Here is an overview of 2000 years of work on isoperimetric problems in mathematical physics, and a more technical survey by Grebenkov and Nguyen of properties of eigenvalues and eigenfunctions of the Laplacian.

Many of my papers can be found on the ArXiv here.

Publications

  1. Spherical caps do not always maximize Neumann eigenvalues on the sphere
    (With D. Bucur, E. Martinet and M. Nahon.)
    Submitted (23 pages).
    ArXiv
  2. Fourth order Saint-Venant inequalities: maximizing compliance and mean deflection among clamped plates
    (With M. Ashbaugh, D. Bucur and R. Leylekian.)
    Submitted (35 pages).
    ArXiv
  3. Two disks maximize the third Robin eigenvalue: positive parameters
    (With H. N. Kim.)
    Submitted (16 pages).
    ArXiv
  4. Maximizing Riesz capacity ratios: conjectures and theorems
    (With C. Clark.)
    Submitted (34 pages).
    ArXiv
  5. Hausdorff measure and decay rate of Riesz capacity
    (With Q. Fan.)
    Journal of Mathematical Analysis and Applications, 550 (2025), article 129625, 16 pages.
    open access
  6. Riesz capacity: monotonicity, continuity, diameter and volume
    (With C. Clark.)
    Analysis and Mathematical Physics, 15 (2025), article 4, 21 pages.
    doi or read-only link
  7. Balls minimize moments of logarithmic and Newtonian equilibrium measures
    (With C. Clark.)
    Potential Analysis (2024), 24 pages.
    doi or read-only link
  8. An internship network in the mathematical sciences
    (With F. Santosa.)
    SIAM News, Dec 2023.
    Article     (8 pages)
  9. Maximizing the second Robin eigenvalue of simply connected curved membranes
    (With J. J. Langford.)
    Computational Methods and Function Theory, 25 (2023), 83-117.
    doi or read-only link
  10. Scaling inequalities for spherical and hyperbolic eigenvalues
    (With J. J. Langford.)
    Journal of Spectral Theory, 13 (2023), 263-296.
    doi
  11. Maximizers beyond the hemisphere for the second Neumann eigenvalue
    (With J. J. Langford.)
    Mathematische Annalen, 386 (2023), 2255–2281.
    doi or read-only link
  12. Minimizing capacity among linear images of rotationally invariant conductors
    Analysis and Mathematical Physics, 12 (2022), article 21, 25 pages.
    doi or read-only link
  13. Two balls maximize the third Neumann eigenvalue in hyperbolic space
    (With P. Freitas.)
    Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 23(3) (2022), 1325-1355.
    doi
  14. Well-posedness of Hersch-Szego’s center of mass by hyperbolic energy minimization
    Annales mathématiques du Québec, 45(2) (2021), 363-390.
    doi or read-only link
  15. Well-posedness of Weinberger’s center of mass by euclidean energy minimization
    Journal of Geometric Analysis, 31(9) (2021), 8762-8779.
    doi or read-only link
  16. Robin spectrum: two disks maximize the third eigenvalue
    (With A. Girouard.)
    Indiana University Mathematics Journal, 70 (2021), 2711–2742.
    doi
  17. From Neumann to Steklov and beyond, via Robin: the Weinberger way
    (With P. Freitas.)
    American Journal of Mathematics, 143 (2021), 969-994.
    doi
  18. From Steklov to Neumann and beyond, via Robin: the Szegő way
    (With P. Freitas.)
    Canadian Journal of Mathematics, 72(4) (2020), 1024-1043.
    doi
  19. A new method for calculating the effective reproduction number for COVID-19
    (With P. Grice and S. Grice)
    technical report (11 pages)
    This report develops the mathematical and statistical methods that were applied to various countries in paper 69 below.
  20. Calculating the effective reproduction number for COVID-19 using a new process for various countries
    (With P. Grice and S. Grice)
    technical report (16 pages)
  21. The effect of social distancing, isolation and digital contact tracing on COVID-19
    (With H. Locke, P. Grice and S. Grice)
    technical report (11 pages)
  22. The Robin Laplacian — spectral conjectures, rectangular theorems
    Journal of Mathematical Physics, 60 (2019), article 121507, 32 pages.
    doi
  23. BIG career developments for mathematics graduate students
    (With R. Levy and F. Santosa.)
    Notices of the American Mathematical Society, 66 (2019) no. 4, 523-524.
    reprint
  24. Symmetrization in Analysis
    (By Albert Baernstein II, with David Drasin and Richard Laugesen.)
    Cambridge University Press, 2019. 474 pages.
    purchase
  25. Pólya’s conjecture fails for the fractional Laplacian
    (With M. Kwaśnicki and B. A. Siudeja.)
    Journal of Spectral Theory, 9 (2019) no. 1, 127–135.
    doi
  26. BIG Jobs Guide: Business, Industry, and Government Careers for Mathematical Scientists, Statisticians, and Operations Researchers
    (By Rachel Levy, Richard Laugesen, and Fadil Santosa.)
    SIAM book series, 2018. 141 pages.
    purchase
  27. Spectral theory of partial differential equations
    In: Spectral Theory and Applications, Contemporary Mathematics, vol. 720, American Mathematical Society, Providence, RI, 2018, pp. 23-55. (Proceedings of the 2016 CRM Summer School on Spectral Theory and Applications.)
  28. Shifted lattices and asymptotically optimal ellipses
    (With S. Liu.)
    The Journal of Analysis, 26 (2018) no. 1, 71-102.
    doi or read-only link
  29. Optimal stretching for lattice points and eigenvalues
    (With S. Liu.)
    Arkiv för Matematik, 56 (2018) no. 1, 111-145.
    doi
  30. Optimal stretching for lattice points under convex curves
    (With S. Ariturk.)
    Portugaliae Mathematica (N.S.), 74 (2017) no. 2, 91-114.
    doi
  31. Triangles and Other Special Domains
    (With B. A. Siudeja.)
    Chapter 6 (pp. 149-200) in the book Shape Optimization and Spectral Theory, edited by Antoine Henrot, De Gruyter Open, 2017.
  32. Math PhD careers: new opportunities emerging amidst crisis
    (With Y. Baryshnikov and R. DeVille.)
    Notices of the American Mathematical Society, 64 (2017) no. 3, 260-264.
    reprint
  33. Preparing graduates for careers in the mathematical sciences
    (With S. Minkoff, W. Menasco, F. Santosa, S. Pankavich.)
    SIAM News, 49 (2016).
    reprint
  34. Torsion and ground state maxima: close but not the same
    (With B. A. Benson, M. Minion and B. A. Siudeja.)
    Bulletin of the Irish Mathematical Society, 78 (2016), 81-88.
    reprint
  35. Steklov eigenvalues and quasiconformal maps of simply connected planar domains
    (With A. Girouard and B. A. Siudeja.)
    Archive for Rational Mechanics and Analysis, 219 (2016), 903-936.
    doi or read-only link
  36. Multivariable feedback particle filter
    (With T. Yang, P. G. Mehta and S. Meyn.)
    Automatica, 71 (2016), 10-23.
    doi
  37. Poisson’s equation in nonlinear filtering
    (With P. G. Mehta, S. Meyn, and M. Raginsky.)
    SIAM Journal on Control and Optimization, 53 (2015), 501-525.
    doi
  38. Magnetic spectral bounds on starlike plane domains
    (With B. A. Siudeja.)
    ESAIM: Control, Optimisation and Calculus of Variations, 21 (2015), 670-689.
    doi
  39. Sharp spectral bounds on starlike domains
    (With B. A. Siudeja.)
    Journal of Spectral Theory, 4 (2014), 309-347.
    doi
  40. Explicit interpolation bounds between Hardy space and L2
    (With H.-Q. Bui.)
    Journal of the Australian Mathematical Society, 95 (2013), 158-168.
    doi
  41. Wavelet frame bijectivity on Lebesgue and Hardy spaces
    (With H.-Q. Bui.)
    Journal of Fourier Analysis and Applications 19 (2013), 376-409.
    doi or read-only link
  42. Multivariable feedback particle filter
    (With T. Yang, P. G. Mehta and S. Meyn.)
    2012 IEEE 51st Annual Conference on Decision and Control (CDC), (Dec. 2012), 4063-4070.
    doi
  43. Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian
    In: Proceedings of the AMSI International Conference on Harmonic Analysis and Applications (Macquarie University, February 2011), pp. 63-82. Edited by X. Duong, J. Hogan, C. Meaney, A. Sikora.
    reprint
  44. Uniqueness for the continuous wavelet transform
    (With H.-Q. Bui.)
    Far East Journal of Applied Mathematics 65 (2012), 1-11.
  45. Sums of magnetic eigenvalues are maximal on rotationally symmetric domains
    (With J. Liang and A. Roy.)
    Annales Henri Poincare 13 (2012), 731-750.
    doi or read-only link
  46. Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals
    (With Z. C. Pan and S. S. Son.)
    Mathematical Inequalities and Applications 15 (2012), 381-394.
    doi
  47. Rebuttal of Donnelly’s paper on the spectral gap
    (With M. S. Ashbaugh and A. Henrot.)
    Mathematische Zeitschrift 269 (2011), 5-7.
    doi or read-only link
  48. Dirichlet eigenvalue sums on triangles are minimal for equilaterals
    (With B. A. Siudeja.)
    Communications in Analysis and Geometry 19 (2011), 855–885.
    doi

    AMS Fellows – a modest proposal
    AMS Notices, Letter to the Editor, December 2011.
    reprint
    Readers might enjoy the original “A Modest Proposal” by Jonathan Swift (1729).

  49. Sums of Laplace eigenvalues – rotations and tight frames in higher dimensions
    (With B. A. Siudeja.)
    Journal of Mathematical Physics 52 (2011), 093703. 13 pages.
    doi
  50. Sums of Laplace eigenvalues – rotationally symmetric maximizers in the plane
    (With B. A. Siudeja.)
    Journal of Functional Analysis 260 (2011), 1795-1823.
    doi
  51. Approximately dual frames in Hilbert spaces and applications to Gabor frames
    (With O. Christensen.)
    Sampling Theory in Signal and Image Processing 9 (2011), 77-90.
    preprint
  52. Wavelets in Littlewood-Paley space, and Mexican hat completeness
    (With H.-Q. Bui.)
    Applied and Computational Harmonic Analysis 30 (2011), 204-213.
    doi (note: on page 210, the reference to [6,formula (16)] should be to [6,Section 8]).
    For related estimates, see our technical report A note on the solution of the Mexican hat problem.
  53. Frequency-scale frames and the solution of the Mexican hat problem
    (With H.-Q. Bui.)
    Constructive Approximation 33 (2011), 163–189.
    doi
  54. Moment inequalities for equilibrium measures in the plane
    (With A. Baernstein II and I. E. Pritsker.)
    Pure and Applied Mathematics Quarterly 7 (2011), 51-86.
    reprint
  55. Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality
    (With B. A. Siudeja.)
    Journal of Differential Equations 249 (2010), 118-135.
    doi
  56. Maximizing Neumann fundamental tones of triangles
    (With B. A. Siudeja.)
    22 pages. Journal of Mathematical Physics, 50:112903, 2009.
    doi
  57. Gabor dual spline windows
    Applied and Computational Harmonic Analysis, 27:180-194, 2009.
    doi, or an earlier preprint with more details
  58. A computable Fourier condition generating alias-free sampling lattices
    (With Y. M. Lu and M. N. Do.)
    IEEE Transactions on Signal Processing, 57:1768-1782, 2009.
    doi
  59. A note on constructing affine systems for L2
    (With H.-Q. Bui and N. Kaiblinger.)
    Applied and Computational Harmonic Analysis, 25:400-406, 2008.
    doi
  60. Affine synthesis onto Lp when 0 < p <= 1
    Journal of Fourier Analysis and Applications, 14:235-266, 2008.
    doi
  61. Affine synthesis onto Lebesgue and Hardy spaces
    (With H.-Q. Bui.)
    Indiana University Mathematics Journal, 57:2203-2233, 2008.
    reprint; an earlier version that treats Sobolev spaces too is here
  62. Sobolev spaces and approximation by affine spanning systems
    (With H.-Q. Bui.)
    Mathematische Annalen, 341:347-389, 2008.
    doi
  63. Approximation and spanning in the Hardy space, by affine systems
    (With H.-Q. Bui.)
    Constructive Approximation, 28:149-172, 2008.
    doi
  64. On convex surfaces with minimal moment of inertia
    (With P. Freitas and G. F. Liddell.)
    Journal of Mathematical Physics, 48, 122902, 2007.
    doi
  65. On affine frames with transcendental dilations
    Proceedings of the American Mathematical Society, 135:211-216, 2007.
    doi
  66. Affine systems that span Lebesgue spaces
    (With H.-Q. Bui.)
    Journal of Fourier Analysis and Applications, 11:533-556, 2005.
    doi

    Papers 23 and 27 are based on the following research report:
    Spanning and sampling in Lebesgue and Sobolev spaces
    (With H.-Q. Bui.)
    University of Canterbury Research Report UCDMS2004/8, 64 pages.
    preprint

  67. New dissipated energies for the thin fluid film equation
    Communications on Pure and Applied Analysis, 4:613-634, 2005.
    doi
  68. Another way to say subsolution: the maximum principle and sums of Green functions
    (With N. A. Watson.)
    Mathematica Scandinavica, 97:127-153, 2005.
    doi
  69. Potential theory of the farthest-point distance function
    (With Igor E. Pritsker.)
    Canadian Mathematical Bulletin, 46:373-387, 2003.
    doi
  70. Heteroclinic orbits, mobility parameters and stability for thin film type equations
    (With Mary C. Pugh.)
    Electronic Journal of Differential Equations, 2002: No. 95, 1-29.
    reprint
  71. Energy levels of steady states for thin film type equations
    (With Mary C. Pugh.)
    Journal of Differential Equations, 182:377-415, 2002.
    doi
  72. A characterization of the higher dimensional groups associated with continuous wavelets
    (With Nik Weaver, Guido Weiss and Edward Wilson.)
    Journal of Geometric Analysis 12:89-102, 2002.
    doi
  73. Translational averaging for completeness, characterization and oversampling of wavelets
    Collectanea Mathematica, 53:211-249, 2002.
    doi
  74. Completeness of orthonormal wavelet systems, for arbitrary real dilations
    Applied and Computational Harmonic Analysis, 11:455-473, 2001.
    doi
  75. Linear stability of steady states for thin film and Cahn-Hilliard type equations
    (With Mary C. Pugh.)
    Archive for Rational Mechanics and Analysis, 154:3-51, 2000.
    doi
  76. Properties of steady states for thin film equations
    (With Mary C. Pugh.)
    European Journal of Applied Mathematics, 11(3):293-351, 2000.
    reprint
  77. Binary forms, equiangular polygons and harmonic measure
    (With Michael A. Bean.)
    Rocky Mountain Journal of Mathematics, 30:15-62, 2000.
    reprint
  78. Eigenvalues of strings and cylinders with variable mass density
    Communications in Analysis and Geometry, 8:393-443, 2000.
    reprint
  79. Eigenvalues of the Laplacian on inhomogeneous membranes
    American Journal of Mathematics, 120:305-344, 1998.
    reprint
  80. Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
    Illinois Journal of Mathematics, 42:19-39, 1998.
    reprint
  81. Extremals for eigenvalues of Laplacians under conformal mapping
    (With Carlo Morpurgo.)
    Journal of Functional Analysis, 155:64-108, 1998.
    reprint
  82. Planar harmonic maps with inner and Blaschke dilatations
    Journal of the London Mathematical Society, 56:37-48, 1997.
    reprint
  83. Inequalities for the first eigenvalues of the clamped plate and buckling problems
    (With Mark S. Ashbaugh and Rafael D. Benguria.)
    International Series of Numerical Mathematics, Volume 123 (Proceedings of the Oberwolfach Conference “General Inequalities 7”), pp. 95-110, 1997.
    reprint
  84. Fundamental tones and buckling loads of clamped plates
    (With Mark S. Ashbaugh.)
    Annali della Scuola Normale Superiore di Pisa, 23:383-402, 1996.
    reprint
  85. The argument principle for harmonic functions
    (With Peter L. Duren and Walter Hengartner.)
    American Mathematical Monthly, 103:411-415, 1996
    reprint
  86. Injectivity can fail for higher-dimensional harmonic extensions
    Complex Variables, 28:357-369, 1996.
    reprint
  87. Conformal mapping of long quadrilaterals and thick doubly connected domains
    Constructive Approximation, 10:523-554, 1994.
    reprint
  88. Extremal problems involving logarithmic and Green capacity
    Duke Mathematical Journal, 70:445-480, 1993.
    reprint

Support from the NSF, Simons Foundation, the University of Illinois Research Board, the New Zealand Institute for Mathematics and its Applications, and the Visiting Erskine Fellowship at the University of Canterbury, is gratefully acknowledged.