Extender-based Radin forcing
Carmi
Merimovich
1729-1772
Abstract: We define extender sequences, generalizing measure sequences of Radin forcing. Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.
Castelnuovo-Mumford regularity and extended degree
Maria Evelina
Rossi;
Ngô
Viêt
Trung;
Giuseppe
Valla
1773-1786
Abstract: Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring $A$ is effectively bounded by the dimension and any extended degree of $A$. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.
On a problem of W. J. LeVeque concerning metric diophantine approximation
Michael
Fuchs
1787-1801
Abstract: We consider the diophantine approximation problem \begin{displaymath}\left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2} \end{displaymath} where $f$ is a fixed function satisfying suitable assumptions. Suppose that $x$ is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdos). Here, we are going to extend and solve LeVeque's problem.
Cyclotomic units and Stickelberger ideals of global function fields
Jaehyun
Ahn;
Sunghan
Bae;
Hwanyup
Jung
1803-1818
Abstract: In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.
Humbert surfaces and the Kummer plane
Christina
Birkenhake;
Hannes
Wilhelm
1819-1841
Abstract: A Humbert surface is a hypersurface of the moduli space $\mathcal A_2$ of principally polarized abelian surfaces defined by an equation of the form $az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0$ with integers $a,\ldots,e$. We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.
The stringy E-function of the moduli space of rank 2 bundles over a Riemann surface of genus 3
Young-Hoon
Kiem
1843-1856
Abstract: We compute the stringy E-function (or the motivic integral) of the moduli space of rank 2 bundles over a Riemann surface of genus 3. In doing so, we answer a question of Batyrev about the stringy E-functions of the GIT quotients of linear representations.
Hyperbolic $2$-spheres with conical singularities, accessory parameters and Kähler metrics on ${\mathcal{M}}_{0,n}$
Leon
Takhtajan;
Peter
Zograf
1857-1867
Abstract: We show that the real-valued function $S_\alpha$ on the moduli space ${\mathcal{M}}_{0,n}$ of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic $2$-sphere with $n\geq 3$ conical singularities of arbitrary orders $\alpha=\{\alpha_1,\dots, \alpha_n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on ${\mathcal{M}}_{0,n}$ parameterized by the set of orders $\alpha$, explicitly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kähler potentials.
Steenrod operations in Chow theory
Patrick
Brosnan
1869-1903
Abstract: An action of the Steenrod algebra is constructed on the mod $p$ Chow theory of varieties over a field of characteristic different from $p$, answering a question posed in Fulton's Intersection Theory. The action agrees with the action of the Steenrod algebra used by Voevodsky in his proof of the Milnor conjecture. However, the construction uses only basic functorial properties of equivariant intersection theory.
Mappings of finite distortion: The sharp modulus of continuity
Pekka
Koskela;
Jani
Onninen
1905-1920
Abstract: We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.
Holomorphic extensions from open families of circles
Josip
Globevnik
1921-1931
Abstract: For a circle $\Gamma =\{ z\in \mathbb{C}\colon \vert z-c\vert=\rho \}$ write $\Lambda (\Gamma )=\{ (z,w)\colon (z-a)(w-\overline{a}) =\rho ^{2}, 0<\vert z-a\vert<\rho \}$. A continuous function $f$ on $\Gamma$ extends holomorphically from $\Gamma$(into the disc bounded by $\Gamma$) if and only if the function $F(z,\overline{z})=f(z)$ defined on $\{(z,\overline{z})\colon z\in \Gamma \}$ has a bounded holomorphic extension into $\Lambda (\Gamma )$. In the paper we consider open connected families of circles $\mathcal{C}$, write $U=\bigcup \{ \Gamma \colon \Gamma \in \mathcal{C}\}$, and assume that a continuous function on $U$ extends holomorphically from each $\Gamma \in \mathcal{C}$. We show that this happens if and only if the function $F(z, \overline{z})=f(z)$ defined on $\{ (z,\overline{z})\colon z\in U\}$ has a bounded holomorphic extension into the domain $\bigcup \{ \Lambda (\Gamma )\colon \Gamma \in \mathcal{Q}\}$ for each open family $\mathcal{Q}$ compactly contained in $\mathcal{C}$. This allows us to use known facts from several complex variables. In particular, we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions.
Ricci flatness of asymptotically locally Euclidean metrics
Lei
Ni;
Yuguang
Shi;
Luen-Fai
Tam
1933-1959
Abstract: In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.
Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors
Marius
Mitrea;
Michael
Taylor
1961-1985
Abstract: We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor $g_{jk} dx_j\otimes dx_k$ has low regularity. Under the assumption that \begin{displaymath}\vert g_{jk}(x)-g_{jk}(y)\vert\leq C\,\omega(\vert x-y\vert),\end{displaymath} where the modulus of continuity $\omega$ satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with $L^p$ boundary data, for sharp ranges of $p$'s and with optimal nontangential maximal function estimates.
Metric character of Hamilton--Jacobi equations
Antonio
Siconolfi
1987-2009
Abstract: We deal with the metrics related to Hamilton-Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an $\inf$-$\sup$ formula involving certain level sets of the Hamiltonian. In the case where these level sets are star-shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation.
Functorial Hodge identities and quantization
M.
J.
Slupinski
2011-2046
Abstract: By a uniform abstract procedure, we obtain integrated forms of the classical Hodge identities for Riemannian, Kähler and hyper-Kähler manifolds, as well as of the analogous identities for metrics of arbitrary signature. These identities depend only on the type of geometry and, for each of the three types of geometry, define a multiplicative functor from the corresponding category of real, graded, flat vector bundles to the category of infinite-dimensional $\mathbf{Z}_{2}$-projective representations of an algebraic structure. We define new multiplicative numerical invariants of closed Kähler and hyper-Kähler manifolds which are invariant under deformations of the metric.
Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group
Detlef
Müller;
Marco
M.
Peloso
2047-2064
Abstract: We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group $\mathbb{H} _n$, of the form \begin{displaymath}\mathcal{P}_\Lambda= \sum_{i,j=1}^{n} \lambda_{ij}X_i Y_j={\,}^t X\Lambda Y, \end{displaymath} where $\Lambda=(\lambda_{ij})$ is a complex $n\times n$matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that $\mathcal{P}_\Lambda$ cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that $\operatorname{Re}\Lambda,$ $\operatorname{Im}\Lambda$ and their commutator are linearly independent, we show that $\mathcal{P}_\Lambda$ is not locally solvable, even in the presence of lower-order terms, provided that $n\ge7$. In the case $n=3$ we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group $\mathbb{H} _3$ a phenomenon first observed by Karadzhov and Müller in the case of $\mathbb{H} _2.$ It is interesting to notice that the analysis of the exceptional operators for the case $n=3$turns out to be more elementary than in the case $n=2.$When $3\le n\le 6$ the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.
Heat kernels on metric measure spaces and an application to semilinear elliptic equations
Alexander
Grigor'yan;
Jiaxin
Hu;
Ka-Sing
Lau
2065-2095
Abstract: We consider a metric measure space $(M,d,\mu )$ and a heat kernel $p_{t}(x,y)$ on $M$ satisfying certain upper and lower estimates, which depend on two parameters $\alpha$ and $\beta$. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space $(M,d,\mu )$. Namely, $\alpha$ is the Hausdorff dimension of this space, whereas $\beta$, called the walk dimension, is determined via the properties of the family of Besov spaces $W^{\sigma ,2}$ on $M$. Moreover, the parameters $\alpha$ and $\beta$ are related by the inequalities $2\leq \beta \leq \alpha +1$. We prove also the embedding theorems for the space $W^{\beta /2,2}$, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on $M$ of the form \begin{displaymath}-\mathcal{L}u+f(x,u)=g(x), \end{displaymath} where $\mathcal{L}$ is the generator of the semigroup associated with $p_{t}$. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in ${\mathbb{R}^{n}}$.
A black-box group algorithm for recognizing finite symmetric and alternating groups, I
Robert
Beals;
Charles
R.
Leedham-Green;
Alice
C.
Niemeyer;
Cheryl
E.
Praeger;
Ákos
Seress
2097-2113
Abstract: We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree $n$ of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of $n$ is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of $S_n$: the conditional probability that a random element $\sigma \in S_n$is an $n$-cycle, given that $\sigma^n=1$, is at least $1/10$.
Oscillation and variation for singular integrals in higher dimensions
James
T.
Campbell;
Roger
L.
Jones;
Karin
Reinhold;
Máté
Wierdl
2115-2137
Abstract: In this paper we continue our investigations of square function inequalities in harmonic analysis. Here we investigate oscillation and variation inequalities for singular integral operators in dimensions $d \geq 1$. Our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator, including upcrossing and $\lambda$ jump inequalities.
Local power series quotients of commutative Banach and Fréchet algebras
Marc
P.
Thomas
2139-2160
Abstract: We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fréchet algebra. In §2 we derive necessary conditions for a commutative Fréchet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has finite closed descent, the algebra cannot be simply a radical algebra with identity adjoined--it must have nontrivial representation theory; if the generating element does not have finite closed descent, then the algebra cannot be a Banach algebra, and the generating element must be locally nilpotent (but non-nilpotent) in an associated quotient algebra. In §3 we consider a fundamental situation which leads to local power series quotients. Let $D$ be a derivation on a commutative radical Fréchet algebra ${\mathcal{R}}^{\sharp }$ with identity adjoined. We show in Theorem 3.10 that if the discontinuity of $D$ is not concentrated in the (Jacobson) radical, then ${\mathcal{R}}^{\sharp }$ has a local power series quotient. The question of whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the affirmative by C. J. Read.