@article{14278,
  abstract     = {The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral uniqueness of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish local uniqueness for almost all of them.},
  author       = {Koval, Illya},
  issn         = {1432-1297},
  journal      = {Inventiones Mathematicae},
  pages        = {221--298},
  publisher    = {Springer Nature},
  title        = {{Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse}},
  doi          = {10.1007/s00222-025-01397-y},
  volume       = {244},
  year         = {2026},
}

@article{17475,
  abstract     = {As a discrete analogue of Kac’s celebrated question on ‘hearing the shape of a drum’ and towards a practical
graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by
their spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers,
is that ‘almost all graphs are determined by their spectrum’, meaning that the fraction of unlabelled n-vertex
graphs which are determined by their spectrum converges to 1 as n → ∞.
In this paper, we make a step towards this conjecture, showing that there are exponentially many n-vertex
graphs which are determined by their spectrum. This improves on previous bounds (of shape e
c
√
n
). We also
propose a number of further directions of research.
},
  author       = {Koval, Illya and Kwan, Matthew Alan},
  issn         = {1464-3847},
  journal      = {Quarterly Journal of Mathematics},
  number       = {3},
  pages        = {869--899},
  publisher    = {Oxford University Press},
  title        = {{Exponentially many graphs are determined by their spectrum}},
  doi          = {10.1093/qmath/haae030},
  volume       = {75},
  year         = {2024},
}