@article{8682,
  abstract     = {It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least 3 over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least 3. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.},
  author       = {Browning, Timothy D and Boudec, Pierre Le and Sawin, Will},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  number       = {3},
  pages        = {1115--1203},
  publisher    = {Princeton University},
  title        = {{The Hasse principle for random Fano hypersurfaces}},
  doi          = {10.4007/annals.2023.197.3.3},
  volume       = {197},
  year         = {2023},
}

@article{8421,
  abstract     = {The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.},
  author       = {Kaloshin, Vadim and Sorrentino, Alfonso},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {1},
  pages        = {315--380},
  publisher    = {Annals of Mathematics, Princeton U},
  title        = {{On the local Birkhoff conjecture for convex billiards}},
  doi          = {10.4007/annals.2018.188.1.6},
  volume       = {188},
  year         = {2018},
}

@article{8427,
  abstract     = {We show that any sufficiently (finitely) smooth ℤ₂-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.},
  author       = {De Simoi, Jacopo and Kaloshin, Vadim and Wei, Qiaoling},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  number       = {1},
  pages        = {277--314},
  publisher    = {Annals of Mathematics},
  title        = {{Dynamical spectral rigidity among Z2-symmetric strictly convex domains close to a circle}},
  doi          = {10.4007/annals.2017.186.1.7},
  volume       = {186},
  year         = {2017},
}

@article{8496,
  author       = {Avila, Artur and De Simoi, Jacopo and Kaloshin, Vadim},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  number       = {2},
  pages        = {527--558},
  publisher    = {Princeton University Press},
  title        = {{An integrable deformation of an ellipse of small eccentricity is an ellipse}},
  doi          = {10.4007/annals.2016.184.2.5},
  volume       = {184},
  year         = {2016},
}

@article{8503,
  abstract     = {We prove there are finitely many isometry classes of planar central configurations (also called relative equilibria) in the Newtonian 5-body problem, except perhaps if the 5-tuple of positive masses belongs to a given codimension 2 subvariety of the mass space.},
  author       = {Albouy, Alain and Kaloshin, Vadim},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  number       = {1},
  pages        = {535--588},
  publisher    = {Princeton University Press},
  title        = {{Finiteness of central configurations of five bodies in the plane}},
  doi          = {10.4007/annals.2012.176.1.10},
  volume       = {176},
  year         = {2012},
}

@article{8512,
  abstract     = {For diffeomorphisms of smooth compact finite-dimensional manifolds, we consider the problem of how fast the number of periodic points with period n grows as a function of n. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for C2 or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call “prevalence”, the growth is not much faster than exponential. Specifically, we show that for each ρ,δ>0, there is a prevalent set of C1+ρ (or smoother) diffeomorphisms for which the number of periodic n points is bounded above by exp(Cn1+δ) for some C independent of n. We also obtain a related bound on the decay of hyperbolicity of the periodic points as a function of n, and obtain the same results for 1-dimensional endomorphisms. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity show this to be a subtle and complex phenomenon, reminiscent of KAM theory. Here in Part I we state our results and describe the methods we use. We complete most of the proof in the 1-dimensional C2-smooth case and outline the remaining steps, deferred to Part II, that are needed to establish the general case.

The novel feature of the approach we develop in this paper is the introduction of Newton Interpolation Polynomials as a tool for perturbing trajectories of iterated maps.},
  author       = {Kaloshin, Vadim and Hunt, Brian},
  issn         = {0003-486X},
  journal      = {Annals of Mathematics},
  number       = {1},
  pages        = {89--170},
  publisher    = {Princeton University Press},
  title        = {{Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms I}},
  doi          = {10.4007/annals.2007.165.89},
  volume       = {165},
  year         = {2007},
}

@article{8526,
  author       = {Kaloshin, Vadim},
  issn         = {0003-486X},
  journal      = {The Annals of Mathematics},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {2},
  pages        = {729--741},
  publisher    = {JSTOR},
  title        = {{An extension of the Artin-Mazur theorem}},
  doi          = {10.2307/121093},
  volume       = {150},
  year         = {1999},
}