Browsing Mathematics Department by Issue Date
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Multidimensional Schrödinger operators whose spectrum features a halfline and a Cantor set
(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 halfline and a ... 
Random Hamiltonians with arbitrary point interactions in one dimension
(2021)We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random selfadjoint 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 ... 
Campana points of bounded height on vector group compactifications
(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 ... 
Ergodic Schrödinger operators in the infinite measure setting
(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 ... 
Generic spectral results for CMV matrices with dynamically defined Verblunsky coefficients
(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 ... 
Random SturmLiouville operators with generalized point interactions
(2020)In this work we study the point spectra of selfadjoint SturmLiouville operators with generalized point interactions, where the two onesided 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 ... 
Anderson localization for quasiperiodic CMV matrices and quantum walks
(2019)We consider CMV matrices, both standard and extended, with analytic quasiperiodic 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 onedimensional Schrödinger operators. We also prove a similar result for quantum ... 
Anderson localization for radial tree graphs with random branching numbers
(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 halflineﾠJacobi matricesﾠwhose entries are nondegenerate, independent, identically distributed random variables with singular distributions. 
Multidimensional AlmostPeriodic Schrödinger Operators with Cantor Spectrum
(2019)We construct multidimensional almostperiodic Schrödinger operators whose spectrum has zero lower boxcounting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure. 
Limitperiodic Schrödinger operators with Lipschitz continuous IDS
(2019)We show that there exist limitperiodic 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. 
Positive Lyapunov exponents and a Large Deviation Theorem for continuum Anderson models, briefly
(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 ... 
Level structures on Abelian varieties, Kodaira dimensions, and Lang's conjecture
(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 levelpr 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 ... 
Fundamental domains and generators for lattice Veech groups
(2017)The moduli space QMg of nonzero 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, ... 
Dichotomy for arithmetic progressions in subsets of reals
(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 3term arithmetic progressions; • or, for every φ ∈ H, the image φ(A) contains arithmetic progressions of arbitrary finite length. In fact, we show ... 
Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals
(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 ... 
Wignervon Neumann type perturbations of periodic Schrödinger operators
(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 ... 
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=−1
(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
(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 ... 
On the existence and uniqueness of global solutions for the KdV equation with quasiperiodic initial data
(2015)We consider the KdV equation ∂tu+∂3xu+u∂xu=0 with quasiperiodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with exponentially decaying Fourier coefficients, of a solution on a small interval of time, the length of which depends on the given data ... 
Homology cobordism and Seifert fibered 3manifolds
(2014)It is known that every closed oriented 3manifold is homology cobordant to a hyperbolic 3manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3manifolds with no 2torsion in their first ...