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  • 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.
  • 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.
  • 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.
  • 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 ...
  • Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals 

    Adiceam, Faustin; Damanik, David; Gähler, Franz; Grimm, Uwe; Haynes, Alan; (2016)
    This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding ...
  • Wigner-von Neumann type perturbations of periodic Schrödinger operators 

    Lukic, Milivoje; Ong, Darren C. (2015)
    Schrödinger operators on the half line. More precisely, the perturbations we consider satisfy a generalized bounded variation condition at infinity and an LP decay condition. We show that the absolutely continuous spectrum is preserved, and give bounds on the Hausdorff dimension of the singular part of the resulting perturbed measure. Under additional ...
  • Homology cobordism and Seifert fibered 3-manifolds 

    Cochran, Tim D.; Tanner, Daniel (2014)
    It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first ...
  • Dichotomy for arithmetic progressions in subsets of reals 

    Boshernitzan,ᅠMichael; Chaika, Jon (2016)
    Let H stand for the set of homeomorphisms φ:[0, 1] → [0, 1]. We prove the following dichotomy for Borel subsets A ⊂ [0, 1]: • either there exists a homeomorphism φ ∈ Hsuch that the image φ(A) contains no 3-term arithmetic progressions; • or, for every φ ∈ H, the image φ(A) contains arithmetic progressions of arbitrary finite length. In fact, we show ...
  • Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=−1 

    Brendle, Tara; Margalit, Dan; Putman, Andrew (2015)
    We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=−1 and also the fundamental group of the branch locus of ...
  • Generating the Johnson filtration 

    Church, Thomas; Putman, Andrew (2015)
    For k≥1, let J1g(k) be the k th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k≥1, there exists some Gk≥0 such that J1g(k) is generated by elements which are supported on subsurfaces whose genus is at most Gk. We also prove similar theorems for the Johnson filtration ...
  • Fundamental domains and generators for lattice Veech groups 

    Mukamel, Ronen E. (2017)
    The moduli space QMg of non-zero genus g quadratic differentials has a natural action of G=GL+2(R) / ⟨±(1001) ⟩. The Veech group PSL(X,q) is the stabilizer of (X,q)∈QMg in G. We describe a new algorithm for finding elements of PSL(X,q) which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, ...
  • Higher-dimensional analogs of Chatelet surfaces 

    Varilly-Alvarado, A.; Viray, B. (2012)
    We discuss the geometry and arithmetic of higher-dimensional analogs of Chatelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In addition, we use these varieties to give straightforward generalizations of two recent results of Poonen. Specifically, we prove that, ...

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