On model complete differential fields
E.
Hrushovski;
M.
Itai
4267-4296
Abstract: We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE $X$. We show that this sub-family is usually definable (in particular if $X$ lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.
On cubic lacunary Fourier series
Joseph
L.
Gerver
4297-4347
Abstract: For $2<\beta <4$, we analyze the behavior, near the rational points $x=p\pi /q$, of $\sum^\infty_{n=1}n^{-\beta }\exp (ixn^{3})$, considered as a function of $x$. We expand this series into a constant term, a term on the order of $(x-p\pi /q)^{(\beta -1)/3}$, a term linear in $x-p\pi /q$, a ``chirp" term on the order of $(x-p\pi /q)^{(2\beta -1)/4}$, and an error term on the order of $(x-p\pi /q)^{\beta /2}$. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when $\beta \le (\sqrt {97}-1)/4=2.212\dots$, both the real and imaginary parts of the cubic series are differentiable almost nowhere.
Rigidity in holomorphic and quasiregular dynamics
Gaven
J.
Martin;
Volker
Mayer
4349-4363
Abstract: We consider rigidity phenomena for holomorphic functions and then more generally for uniformly quasiregular maps.
Hyperplane arrangement cohomology and monomials in the exterior algebra
David
Eisenbud;
Sorin
Popescu;
Sergey
Yuzvinsky
4365-4383
Abstract: We show that if $X$ is the complement of a complex hyperplane arrangement, then the homology of $X$ has linear free resolution as a module over the exterior algebra on the first cohomology of $X$. We study invariants of $X$ that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.
Group actions on one-manifolds, II: Extensions of Hölder's Theorem
Benson
Farb;
John
Franks
4385-4396
Abstract: This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on $\mathbf R$ with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in $\mathrm{Diff}^2(\mathbf R)$ as those groups whose elements have at most one fixed point.
Sheaf cohomology and free resolutions over exterior algebras
David
Eisenbud;
Gunnar
Fløystad;
Frank-Olaf
Schreyer
4397-4426
Abstract: We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual'' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part'' of a resolution over the exterior algebra. We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree $>1$ in the complex yields a new complex which is eventually exact. As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.
Examples for the mod $p$ motivic cohomology of classifying spaces
Nobuaki
Yagita
4427-4450
Abstract: Let $BG$ be the classifying space of a compact Lie group $G$. Some examples of computations of the motivic cohomology $H^{*,*}(BG;\mathbb{Z}/p)$ are given, by comparing with $H^*(BG;\mathbb{Z}/p)$, $CH^*(BG)$ and $BP^*(BG)$.
Fitting's Lemma for $\mathbb{Z}/2$-graded modules
David
Eisenbud;
Jerzy
Weyman
4451-4473
Abstract: Let $\phi :\; R^{m}\to R^{d}$be a map of free modules over a commutative ring $R$. Fitting's Lemma shows that the ``Fitting ideal,'' the ideal of $d\times d$ minors of $\phi$, annihilates the cokernel of $\phi$ and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a $\mathbb{Z}/2$-graded skew-commutative algebra and prove corresponding theorems about the annihilator; for example, the Fitting ideal and the annihilator of the cokernel are equal in the generic case. Our results generalize the classical Fitting Lemma in the commutative case and extend a key result of Green (1999) in the exterior algebra case. They depend on the Berele-Regev theory of representations of general linear Lie superalgebras. In the purely even and purely odd cases we also offer a standard basis approach to the module $\operatorname{coker}\phi$ when $\phi$ is a generic matrix.
The differential Galois theory of strongly normal extensions
Jerald
J.
Kovacic
4475-4522
Abstract: Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product. We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms. This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields. On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.
Twisted sums with $C(K)$ spaces
F.
Cabello
Sánchez;
J.
M. F.
Castillo;
N.
J.
Kalton;
D.
T.
Yost
4523-4541
Abstract: If $X$ is a separable Banach space, we consider the existence of non-trivial twisted sums $0\to C(K)\to Y\to X\to 0$, where $K=[0,1]$ or $\omega^{\omega}.$For the case $K=[0,1]$ we show that there exists a twisted sum whose quotient map is strictly singular if and only if $X$ contains no copy of $\ell_1$. If $K=\omega^{\omega}$ we prove an analogue of a theorem of Johnson and Zippin (for $K=[0,1]$) by showing that all such twisted sums are trivial if $X$ is the dual of a space with summable Szlenk index (e.g., $X$ could be Tsirelson's space); a converse is established under the assumption that $X$ has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with $C(\omega^{\omega})$ with strictly singular quotient map.
Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds
Hui
Li
4543-4568
Abstract: Assume $(M, \omega)$ is a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim H^2(M)<3$. We give a complete list of the possible manifolds, and determine their equivariant cohomology rings and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.
A constructive Schwarz reflection principle
Jeremy
Clark
4569-4579
Abstract: We prove a constructive version of the Schwarz reflection principle. Our proof techniques are in line with Bishop's development of constructive analysis. The principle we prove enables us to reflect analytic functions in the real line, given that the imaginary part of the function converges to zero near the real line in a uniform fashion. This form of convergence to zero is classically equivalent to pointwise convergence, but may be a stronger condition from the constructivist point of view.
Maximal complexifications of certain homogeneous Riemannian manifolds
S.
Halverscheid;
A.
Iannuzzi
4581-4594
Abstract: Let $M=G/K$ be a homogeneous Riemannian manifold with $\dim_{\mathbb{C}} G^{\mathbb{C}} = \dim_{\mathbb{R}} G$, where $G^{\mathbb{C}}$ denotes the universal complexification of $G$. Under certain extensibility assumptions on the geodesic flow of $M$, we give a characterization of the maximal domain of definition in $TM$ for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.
Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds
Sylvain
Maillot
4595-4638
Abstract: Our main result is a characterization of open Seifert fibered $3$-manifolds in terms of the fundamental group and large-scale geometric properties of a triangulation. As an application, we extend the Seifert Fiber Space Theorem and the Torus Theorem to a class of $3$-orbifolds.
Baxter algebras and Hopf algebras
George
E.
Andrews;
Li
Guo;
William
Keigher;
Ken
Ono
4639-4656
Abstract: By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.
On the Diophantine equation $G_n(x)=G_m(P(x))$: Higher-order recurrences
Clemens
Fuchs;
Attila
Petho;
Robert
F.
Tichy
4657-4681
Abstract: Let $\mathbf{K}$ be a field of characteristic $0$ and let $(G_{n}(x))_{n=0}^{\infty}$ be a linear recurring sequence of degree $d$ in $\mathbf{K}[x]$ defined by the initial terms $G_0,\ldots,G_{d-1}\in\mathbf{K}[x]$ and by the difference equation \begin{displaymath}G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots+A_0(x)G_{n}(x), \quad \mbox{for} \,\, n\geq 0,\end{displaymath} with $A_0,\ldots,A_{d-1}\in\mathbf{K}[x]$. Finally, let $P(x)$ be an element of $\mathbf{K}[x]$. In this paper we are giving fairly general conditions depending only on $G_0,\ldots,G_{d-1},$ on $P$, and on $A_0,\ldots,A_{d-1}$ under which the Diophantine equation \begin{displaymath}G_{n}(x)=G_{m}(P(x))\end{displaymath} has only finitely many solutions $(n,m)\in \mathbb{Z}^{2},n,m\geq 0$. Moreover, we are giving an upper bound for the number of solutions, which depends only on $d$. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.
Compact composition operators on Besov spaces
Maria
Tjani
4683-4698
Abstract: We give a Carleson measure characterization of the compact composition operators on Besov spaces. We use this characterization to show that every compact composition operator on a Besov space is compact on the Bloch space. Finally we give conditions that guarantee that the converse holds.