Existence of Bα,βkB^k_{\alpha,\beta}-Structures on CkC^k-Manifolds

In this paper we introduce $B_{\alpha,\beta}^{k}$-manifolds as generalizations of the notion of smooth manifolds with $G$-structure or with $k$-bounded geometry. These are $C^{k}$-manifolds whose transition functions $\varphi_{ji}=\varphi_{j}\circ\varphi_{i}^{-1}$ are such that $\partial^{\mu}\varphi_{ji}\in B_{\alpha(r)}\cap C^{k-\beta(r)}$ for every $\vert\mu\vert=r$, where $B=(B_{r})_{r\in\Gamma}$ is some sequence of presheaves of Fr\'echet spaces endowed with further structures, $\Gamma\subset\mathbb{Z}_{\geq0}$ is some parameter set and $\alpha,\beta$ are functions. We present embedding theorems for the presheaf category of those structural presheaves $B$. The existence problem of $B_{\alpha,\beta}^{k}$-structures on $C^{k}$-manifolds is studied and it is proved that under certain conditions on $B$, $\alpha$ and $\beta$, the forgetful functor from $C^{k}$-manifolds to $B_{\alpha,\beta}^{k}$-manifolds has adjoints.Comment: Geometric motivation added. Some typos have been fixe

Polynomial families of tautological classes on Mg,nrt\mathcal{M}_{g,n}^{rt}

We study classes $P_{g,T}(\alpha;\beta)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. A comparison with classes $Q_{g,T}$ arising from sections of the universal Jacobian shows the classes $P_{g,T}(\alpha;\beta)$ are polynomial in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.Comment: 43 pages, 4 figure

Diffeomorphism groups of critical regularity

Let $M$ be the circle or a compact interval, and let $\alpha=k+\tau\ge1$ be a real number such that $k=\lfloor \alpha\rfloor$. We write $\mathrm{Diff}_+^{\alpha}(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are H\"older continuous with exponent $\tau$. If $\alpha\ge1$, we prove that there exists a finitely generated subgroup $G_\alpha\le\mathrm{Diff}_+^\alpha(M)$ with the property that $G_\alpha$ admits no injective homomorphisms into $\mathrm{Diff}_+^\beta(M)$ for all $\beta>\alpha$. If $\alpha>1$, we also show the dual result: there exists a finitely generated group $H_\alpha\le\bigcap_{\beta<\alpha}\mathrm{Diff}_+^\beta(M)$ with the property that $H_\alpha$ admits no injective homomorphisms into $\mathrm{Diff}_+^\alpha(M)$. We can further require that the same properties are inherited by all finite index subgroups, and also by the commutator subgroups, of $G_\alpha$ and $H_\alpha$. The commutator groups of $G_\alpha$ and of $H_\alpha$ are countable simple groups. As a consequence, whenever $1\le\alpha<\beta$ we have a continuum of isomorphism types of finitely generated subgroups of $\mathrm{Diff}_+^{\alpha}(M)$ whose images under arbitrary homomorphisms to $\mathrm{Diff}_+^{\beta}(M)$ are abelian. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if $k\neq 2$ is an integer and if $k<\beta$ then there is no nontrivial homomorphism $\mathrm{Diff}_+^k(S^1)\to\mathrm{Diff}_+^{\beta}(S^1)$.Comment: 70 pages. To appear in Inventiones mathematica

The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems

We prove that an $\omega$-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations $\alpha$, $\beta$, $s$ satisfying the identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This establishes an algebraic criterion equivalent to the conjectured borderline between P and NP-complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page

Model projective twists and generalised lantern relations

We use Picard-Lefschetz theory to introduce a new local model for the planar projective twists $\tau_{\mathbb{A}\mathbb{P}^2} \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2), \ \mathbb{A} \in \{ \mathbb{R}, \mathbb{C} \}$. In each case, we construct an exact Lefschetz fibration $\pi\colon T^*\mathbb{A}\mathbb{P}^2\to \mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $\varphi \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $\Delta_{\alpha},\Delta_{\beta} \subset T^*\mathbb{A}\mathbb{P}^2$, we compute the Floer cohomology groups $\mathrm{HF}(\varphi^k(\Delta_{\alpha}), \Delta_{\beta}; \mathbb{Z}/2\mathbb{Z})$ and verify (partially for $\mathbb{C}\mathbb{P}^2$) that $\varphi$ is indeed isotopic to (a power of) the projective twist in its local model. The constructions we present are governed by generalised lantern relations, which provide an isotopy between the total monodromy of a Lefschetz fibration and a fibred twist along an $S^1$-fibred coisotropic submanifold of the smooth fibre. We also use these relations to generate non-exact fillings for the contact manifolds $(ST^*\mathbb{C}\mathbb{P}^2, \xi_{std}), (ST^*\mathbb{R}\mathbb{P}^3,\xi_{std})$, and to study two classes of monotone Lagrangian submanifolds of $(T^*\mathbb{C}\mathbb{P}^2, d\lambda_{\mathbb{C}\mathbb{P}^2})$.Comment: 46 pages, 2 appendices, 9 figure

Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.Comment: 50 pages, revised version, accepted for publication on Mathematical Structures in Computer Scienc

Quadruple covers and Gorenstein stable surfaces with K^2=1 and χ=2

In this thesis we study Gorenstein stable surfaces with K 2X = 1 and \chi(\ko_X) = 2. These arise as quadruple covers of the projective plane and we give the precise relation between the structure of the cover and the canonical ring. We then use these results to study some strata of the moduli space \overline{\mathfrak{M}_1,2

ELKI: A large open-source library for data analysis - ELKI Release 0.7.5 "Heidelberg"

This paper documents the release of the ELKI data mining framework, version 0.7.5. ELKI is an open source (AGPLv3) data mining software written in Java. The focus of ELKI is research in algorithms, with an emphasis on unsupervised methods in cluster analysis and outlier detection. In order to achieve high performance and scalability, ELKI offers data index structures such as the R*-tree that can provide major performance gains. ELKI is designed to be easy to extend for researchers and students in this domain, and welcomes contributions of additional methods. ELKI aims at providing a large collection of highly parameterizable algorithms, in order to allow easy and fair evaluation and benchmarking of algorithms. We will first outline the motivation for this release, the plans for the future, and then give a brief overview over the new functionality in this version. We also include an appendix presenting an overview on the overall implemented functionality

Laguerre and Meixner orthogonal bases in the algebra of symmetric functions

Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of symmetric functions are studied. This is a detailed exposition of part of the results announced in arXiv:1009.2037. The work is motivated by a connection with a model of infinite-dimensional Markov dynamics.Comment: Latex, 52p

Heights on elliptic curves over number fields, period lattices, and complex elliptic logarithms

This thesis presents some major improvements in the following computations: a lower bound for the canonical height, period lattices, and elliptic logarithms. On computing a lower bound for the canonical height, we have successfully generalised the existing algorithm of Cremona and Siksek [CS06] to elliptic curves over totally real number fields, and then to elliptic curves over number fields in general. Both results, which are also published in [Tho08] and [Tho10] respectively, will be fully explained in Chapter 2 and 3. In Chapter 4, we give a complete method on computing period lattices of elliptic curves over C, whereas this was only possible for elliptic curves over R in the past. Our method is based on the concept of arithmetic-geometric mean (AGM). In addition, we extend our method further to find elliptic logarithms of complex points. This work is done in collaboration with Professor John E. Cremona; another version of this chapter has been submitted for publication [CT]. In Chapter 5, we finally illustrate the applications of our main results towards certain computations which did not work well in the past due to lack of some information on elliptic curves. This includes determining a Mordell{Weil basis, finding integral points on elliptic curves over number fields [SS97], and finding elliptic curves with everywhere good reduction [CL07]. A number of computer programs have been implemented for the purpose of illustration and verification. Their source code (written in MAGMA) can be found in Appendix A.EThOS - Electronic Theses Online ServiceInstitute for the Promotion of Teaching Science and Technology (Thailand)GBUnited Kingdo