Mathematics Faculty Publications
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Item Lecture notes in mathematics: an elementary approach to bounded symmetric domains(Rice University, 1969) Koecher, Max; Jones, B. Frank; Houston, Tex. : Rice UniversityItem Lecture notes in mathematics: rudiments of Riemann surfaces(Rice University, 1971) Jones, B. Frank; Houston, Tex. : Rice UniversityItem The Teichmüller Theory of Harmonic Maps(Journal of Differential Geometry, 1989) Wolf, MichaelItem Infinite Energy Harmonic Maps and Degeneration of Hyperbolic Surfaces in Moduli Space(Journal of Differential Geometry, 1991) Wolf, MichaelItem The Rational Cohomology of the Mapping Class Group Vanishes in its Virtual Cohomological Dimension(Oxford University Press, 2012) Church, Thomas; Farb, Benson; Putman, AndrewItem 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 The Picard group of the moduli space of curves with level structures(2012) Putman, Andrew; National Science Foundation; Duke University PressFor 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 standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level L subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level L subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod L symplectic group with coefficients in the adjoint representation.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 Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders(Oxford University Press, 2012) Cochran, Tim D.; Harvey, Shelly; Horn, Peter D.Item The second rational homology group of the moduli space of curves with level structures(2012) Putman, Andrew; ElsevierLet Γ be a finite-index 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 for g≥5. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to Q. We also prove analogous results for surface with punctures and boundary components.Item The Weil-Peterson Hessian of Length on Teichmuller Space(International Press, 2012) Wolf, MichaelWe present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argumentWolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist.Item Handle Addition for Doubly-Periodic Scherk Surfaces(De Gruyter, 2012) Weber, Matthias; Wolf, MichaelWe prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus g with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proof of embeddedness is by the conjugate Plateau method.Item Opening gaps in the spectrum of strictly ergodic Schrodinger operators(European Mathematical Society, 2012) Avila, Artur; Bochi, Jairo; Damanik, DavidWe 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 spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap, the sampling function may be continuously deformed so that the gap immediately opens. As a corollary, we conclude that for generic sampling functions, all gaps are open. The proof is based on the analysis of continuous SL.2;R/ cocycles, for which we obtain dynamical results of independent interest.Item Higher-dimensional analogs of Chatelet surfaces(London Mathematical Society, 2012) Varilly-Alvarado, A.; Viray, B.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, assuming Schinzel's hypothesis, the non-mth powers of a number field are diophantine. Also, given a global field k such that Char(k)=p or k contains the pth roots of unity, we construct a (p+1)-fold that has no k-points and no etale-Brauer obstruction to the Hasse principle.Item Small generating sets for the Torelli group(Mathematical Sciences Publisher, 2012) Putman, AndrewProving 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 the handle graph on which Ig acts cocompactly.Item The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian(Springer, 2012) Damanik, David; Gorodetski, AntonWe consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V , this measure is exact-dimensional 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 tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V -dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.Item Borel-Cantelli Sequences(Springer, 2012-06) Boshernitzan, Michael; Chaika, JonItem Filtering smooth concordance classes of topologically slice knots(msp, 2013) Cochran, Tim D.; Harvey, Shelly; Horn, PeterWe 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 terms may well be topologically slice. However, the natural extension of this filtration to what is called the n–solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, fBng, that is simultaneously a refinement of the n–solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn=BnC1 has infinite rank. But our primary interest is in the induced filtration, fTng, on the subgroup, T , of knots that are topologically slice. We prove that T =T0 is large, detected by gauge-theoretic invariants and the , s , –invariants, while the nontriviality of T0=T1 can be detected by certain d –invariants. All of these concordance obstructions vanish for knots in T1 . Nonetheless, going beyond this, our main result is that T1=T2 has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each Tn=TnC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.Item Log minimal model program for the moduli space of stable curves: the first flip(Department of Mathematics, Princeton University, 2013) Hassett, Brendan; Hyeon, DonghoonWe 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 of bicanonical curves; M¯g(7/10−ϵ) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction Ψ:M¯g(7/10+ϵ)→M¯g(7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction Ψ+:M¯g(7/10−ϵ)→M¯g(7/10) that is the Mori flip of Ψ.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.