Browsing Mathematics Department by Title
Now showing items 120 of 50

Abelian quotients of subgroups of the mapping class group and higher Prym representations
(201308)A wellknown 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 ... 
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. 
A Birman exact sequence for Aut(Fn)
(2012)The 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 ... 
BorelCantelli Sequences
(201206) 
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 ... 
The complex of partial bases for Fn and nite generation of the Torelli subgroup of Aut(Fn)
(2013)We study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these results, we give a new, topological proof of a theorem of ... 
Cubic fourfolds containing a plane and a quantic del Pezzo surface
(2014)We 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 ... 
The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian
(2012)We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V , this measure is exactdimensional and the almost everywhere value dV of the local scaling exponent is a smooth function of V , is strictly smaller than the Hausdor dimension of the spectrum, and converges to one as V ... 
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 ... 
Effective Computation of Picard Groups and BrauerManin Obstructions of Degree Two K3 Surfaces Over Number Fields
(2013)Using the KugaSatake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields. 
Ergodic properties of compositions of interval exchange maps and rotations
(2012)We 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. 
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 ... 
Failure of the Hasse Principle on General K3 Surfaces
(2013)We 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 twotorsion 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 ... 
Filtering smooth concordance classes of topologically slice knots
(2013)We propose and analyze a structure with which to organize the difference between a knot in S3 bounding a topologically embedded 2–disk in B4 and it bounding a smoothly embedded disk. The n–solvable filtration of the topological knot concordance group, due to Cochran–Orr–Teichner, may be complete in the sense that any knot in the intersection of its ... 
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, ... 
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 ... 
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 ...