Mathematics Department
Permanent URI for this community
Browse
Browsing Mathematics Department by Title
Now showing 1 - 20 of 70
Results Per Page
Sort Options
Item A universal Cannon-Thurston map and the surviving curve complex(American Mathematical Society, 2022) Gültepe, Funda; Leininger, Christopher; Pho-on, WitsarutUsing the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon-Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer [Comment. Math. Helv. 86 (2011), pp. 769–816].Item Abelian quotients of subgroups of the mapping class group and higher Prym representations(London Mathematical Society, 2013-08) Putman, Andrew; Wieland, BenA well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto Z if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counterexample to this conjecture, then there must be a counterexample of a particularly simple form. We prove these results by relating the conjecture to a family of linear representations of the mapping class group that we call the higher Prym representations. They generalize the classical symplectic representation.Item Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line(Springer Nature, 2022) Damanik, David; Lenz, DanielWe 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 continuous spectrum.Item Absolutely continuous spectrum for CMV matrices with small quasi-periodic Verblunsky coefficients(American Mathematical Society, 2022) Li, Long; Damanik, David; Zhou, QiWe 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.Item Anderson localization for quasi-periodic CMV matrices and quantum walks(Elsevier, 2019) Wang, Fengpeng; Damanik, DavidWe 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 walks on the integer lattice with suitable analytic quasi-periodic coins.Item Anderson localization for radial tree graphs with random branching numbers(Elsevier, 2019) Damanik, David; Sukhtaiev, SelimWe 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.Item Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups(Elsevier, 2014) Varilly-Alvarado, Anthony; Viray, BiancaItem Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds(Springer Nature, 2023) Bader, Uri; Fisher, David; Miller, Nicholas; Stover, MatthewFor n≥2, we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.Item A Birman exact sequence for Aut(Fn)(Elsevier, 2012) Day, Matthew; Putman, AndrewThe Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The key to many of our proofs are several new generalizations of the Johnson homomorphisms.Item Borel-Cantelli Sequences(Springer, 2012-06) Boshernitzan, Michael; Chaika, JonItem Campana points of bounded height on vector group compactifications(Wiley, 2021) Pieropan, Marta; Smeets, Arne; Tanimoto, Sho; Várilly-Alvarado, AnthonyWe 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 that delineates well different types of behavior of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our setup, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.Item Cocycle rigidity and splitting for some discrete parabolic actions(American Institute of Mathematical Sciences, 2014) Damjanović, Danijela; Tanis, JamesItem Cubic fourfolds containing a plane and a quantic del Pezzo surface(Foundation Compositio Mathematica, 2014) Auel, Asher; Bernardara, Marcello; Bolognesi, Michele; Várilly-Alvarado, AnthonyWe isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C8 ∩ C14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov’s derived categorical conjecture on the rationality of cubic fourfolds.Item Dichotomy for arithmetic progressions in subsets of reals(American Mathematical Society, 2016) Boshernitzan, Michael; Chaika, JonLet 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 that the first alternative holds if and only if the set A is meager (a countable union of nowhere dense sets).Item Effective Computation of Picard Groups and Brauer-Manin Obstructions of Degree Two K3 Surfaces Over Number Fields(Springer, 2013) Hassett, Brendan; Kresch, Andrew; Tschinkel, YuriUsing the Kuga-Satake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields.Item End-periodic homeomorphisms and volumes of mapping tori(Wiley, 2023) Field, Elizabeth; Kim, Heejoung; Leininger, Christopher; Loving, MarissaGiven an irreducible, end-periodic homeomorphism f:S→S$f: S \rightarrow S$ of a surface with finitely many ends, all accumulated by genus, the mapping torus, Mf$M_f$, is the interior of a compact, irreducible, atoroidal 3-manifold M¯f$øverlineM_f$ with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of M¯f$øverlineM_f$ in terms of the translation length of f$f$ on the pants graph of S$S$. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.Item Ergodic properties of compositions of interval exchange maps and rotations(IOP Publishing, 2012) Athreya, Jayadev S; Boshernitzan, MichaelWe study the ergodic properties of compositions of interval exchange transformations (IETs) and rotations. We show that for any IET T, there is a full measure set of α ∈ [0, 1) so that T Rα is uniquely ergodic, where Rα is rotation by α.Item Ergodic Schrödinger operators in the infinite measure setting(EMS Press, 2021) Boshernitzan, Michael; Damanik, David; Fillman, Jake; Lukic, MilivojeWe 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 well as a version of the Pastur–Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.Item Explicit computation of symmetric differentials and its application to quasihyperbolicity(Mathematical Science Publishers, 2022) Bruin, Nils; Thomas, Jordan; Várilly-Alvarado, AnthonyWe 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 through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. We also improve lower bounds on the dimension of the space of symmetric differentials on surfaces with A1-singularities, and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing 3×3 magic squares of squares are all algebraically quasihyperbolic.Item Failure of the Hasse Principle on General K3 Surfaces(Oxford University Press, 2013) Hassett, Brendan; Varilly-Alvarado, AnthonyWe show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X, ) is constructed from a double cover of P2 × P2 ramified over a hypersurface of bi-degree (2, 2).