Browsing Mathematics Department by Title
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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 ... 
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 this ... 
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 ... 
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. 
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 ... 
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. 
New Anomalous LiebRobinson Bounds in Quasiperiodic XY Chains
(2014)We announce and sketch the rigorous proof of a new kind of anomalous (or subballistic) LiebRobinson (LR) bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light ... 
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 ... 
On the unirationality of del Pezzo surfaces of degree two
(2014)Among geometrically rational surfaces, del Pezzo surfaces of degree 2 over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for krational curves on ... 
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. ... 
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 spectrum ... 
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 ... 
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 ... 
The WeilPeterson Hessian of Length on Teichmuller Space
(2012)We present a brief but nearly selfcontained proof of a formula for the WeilPetersson 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 ...