Mathematics Faculty Publications
Recent Submissions

On the existence and uniqueness of global solutions for the KdV equation with quasiperiodic initial data
(20150629)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 ... 
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 ... 
Log minimal model program for the moduli space of stable curves: the first flip
(2013)We give a geometric invariant theory (GIT) construction of the log canonical model M¯g(α) of the pairs (M¯g,αδ) for α∈(7/10–ϵ,7/10] for small ϵ∈Q+. We show that M¯g(7/10) is isomorphic to the GIT quotient of the Chow variety ... 
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
(2013)We study the ergodic properties of compositions of interval exchange transforma tions and rotations. We show that for any interval exchange transformation T, there is a full measure set of 2 [0; 1) so that T R is ... 
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 ... 
Knot Concordance and Homology Cobordism
(201306)We consider the question: “If the zeroframed surgeries on two oriented knots in S3 are Zhomology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?” We show that ... 
Opening gaps in the spectrum of strictly ergodic Schrodinger operators
(2012)We consider Schrodinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the ... 
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 ... 
Small generating sets for the Torelli group
(2012)Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup Ig of the genus g mapping class group has a finite generating set whose size grows cubically with respect to g. Our main tool is a new space called ... 
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 ... 
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 ... 
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 ... 
The Picard group of the moduli space of curves with level structures
(2012)For 4  L and g large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level L structures. In particular, we determine the divisibility properties of ... 
The second rational homology group of the moduli space of curves with level structures
(2012)Let Γ be a finiteindex subgroup of the mapping class group of a closed genus g surface that contains the Torelli group. For instance, Γ can be the level L subgroup or the spin mapping class group. We show that H2(Γ;Q) ∼= Q ... 
Higherdimensional analogs of Chatelet surfaces
(2012)We discuss the geometry and arithmetic of higherdimensional analogs of Chˆatelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In ...