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  • Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence 

    Frei, Sarah; Hassett, Brendan; Várilly-Alvarado, Anthony (2022)
    Given a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial ...
  • Zero measure spectrum for multi-frequency Schrödinger operators 

    Chaika, Jon; Damanik, David; Fillman, Jake; Gohlke, Philipp (2022)
    Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated ...
  • Hierarchical hyperbolicity of graph products 

    Berlyne, Daniel; Russell, Jacob (2022)
    We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups ...
  • Explicit computation of symmetric differentials and its application to quasihyperbolicity 

    Bruin, Nils; Thomas, Jordan; Várilly-Alvarado, Anthony (2022)
    We develop explicit techniques to investigate algebraic quasihyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth’s sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass ...
  • Absolutely continuous spectrum for CMV matrices with small quasi-periodic Verblunsky coefficients 

    Li, Long; Damanik, David; Zhou, Qi (2022)
    We consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from 2005.
  • Hyperbolic cone metrics and billiards 

    Erlandsson, Viveka; Leininger, Christopher J.; Sadanand, Chandrika (2022)
    A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an ...
  • Must the Spectrum of a Random Schrödinger Operator Contain an Interval? 

    Damanik, David; Gorodetski, Anton (2022)
    We consider Schrödinger operators in ℓ2(Z) whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic ...
  • Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line 

    Damanik, David; Lenz, Daniel (2022)
    We show that a generic quasi-periodic Schrödinger operator in L2(ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely ...
  • Ergodic Schrödinger operators in the infinite measure setting 

    Boshernitzan, Michael; Damanik, David; Fillman, Jake; Lukic, Milivoje (2021)
    We develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as ...
  • Campana points of bounded height on vector group compactifications 

    Pieropan, Marta; Smeets, Arne; Tanimoto, Sho; Várilly-Alvarado, Anthony (2021)
    We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version ...
  • Random Hamiltonians with arbitrary point interactions in one dimension 

    Damanik, David; Fillman, Jake; Helman, Mark; Kesten, Jacob; Sukhtaiev, Selim (2021)
    We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous ...
  • Random Sturm-Liouville operators with generalized point interactions 

    Damanik, David; del Rio, Rafael; Franco, Asaf L. (2020)
    In this work we study the point spectra of selfadjoint Sturm-Liouville operators with generalized point interactions, where the two one-sided limits of the solution data are relatedvia a general SL(2,R)matrix. We are particularly interested in the stability of eigenvalues withrespect to the variation of the parameters of the interaction matrix. As a ...
  • Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set 

    Damanik, David; Fillman, Jake; Gorodetski, Anton (2021)
    We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a ...
  • Generic spectral results for CMV matrices with dynamically defined Verblunsky coefficients 

    Fang, Licheng; Damanik, David; Guo, Shuzheng (2020)
    We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral ...
  • Positive Lyapunov exponents and a Large Deviation Theorem for continuum Anderson models, briefly 

    Bucaj, Valmir; Damanik, David; Fillman, Jake; Gerbuz, Vitaly; VandenBoom, Tom; (2019)
    In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik–Sims–Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds ...
  • Anderson localization for quasi-periodic CMV matrices and quantum walks 

    Wang, Fengpeng; Damanik, David (2019)
    We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum ...
  • Anderson localization for radial tree graphs with random branching numbers 

    Damanik, David; Sukhtaiev, Selim (2019)
    We prove Anderson localization for the discrete Laplace operator on radial tree graphs with random branching numbers. Our method relies on the representation of the Laplace operator as the direct sum of half-lineᅠJacobi matricesᅠwhose entries are non-degenerate, independent, identically distributed random variables with singular distributions.
  • Limit-periodic Schrödinger operators with Lipschitz continuous IDS 

    Damanik, David; Fillman, Jake (2019)
    We show that there exist limit-periodic Schrödinger operators such that the associated integrated density of states is Lipschitz continuous. These operators arise in the inverse spectral theoretic KAM approach of Pöschel.
  • Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum 

    Damanik, David; Fillman, Jake; Gorodetski, Anton (2019)
    We construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box-counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.
  • Level structures on Abelian varieties, Kodaira dimensions, and Lang's conjecture 

    Abramovich, Dan; Várilly-Alvarado, Anthony (2018)
    Assuming Lang's conjecture, we prove that for a prime p, number field K, and positive integer g, there is an integer r such that no principally polarized abelian variety A/K has full level-pr structure. To this end, we use a result of Zuo to prove that for each closed subvariety X in the moduli space Ag of principally polarized abelian varieties of ...

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