Algebraic Goodwillie calculus and a cotriple model for the remainder
Andrew
Mauer-Oats
1869-1895
Abstract: Goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the Taylor series of a function. His $n^{\text{th}}$ order approximation $P_n F$ at a space $X$ depends on the values of $F$ on coproducts of large suspensions of the space: $F(\vee \Sigma^M X)$. We define an ``algebraic'' version of the Goodwillie tower, $P_n^{\text{alg}} F(X)$, that depends only on the behavior of $F$ on coproducts of $X$. When $F$ is a functor to connected spaces or grouplike $H$-spaces, the functor $P_n^{\text{alg}} F$ is the base of a fibration $\displaystyle \vert{\bot^{*+1} F}\vert \rightarrow F \rightarrow P_n^{\text{alg}} F,$ whose fiber is the simplicial space associated to a cotriple $\bot$ built from the $(n+1)^{\text{st}}$ cross effect of the functor $F$. In a range in which $F$ commutes with realizations (for instance, when $F$ is the identity functor of spaces), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor $F$ in many interesting cases.
Non-Moishezon twistor spaces of $4{\mathbf{CP}}^2$ with non-trivial automorphism group
Nobuhiro
Honda
1897-1920
Abstract: We show that a twistor space of a self-dual metric on $4{\mathbf{CP}}^2$ with $U(1)$-isometry is not Moishezon iff there is a $\mathbf{C}^*$-orbit biholomorphic to a smooth elliptic curve, where the $\mathbf C^*$-action is the complexification of the $U(1)$-action on the twistor space. It follows that the $U(1)$-isometry has a two-sphere whose isotropy group is $\mathbf Z_2$. We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.
On topological invariants of stratified maps with non-Witt target
Markus
Banagl
1921-1935
Abstract: The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.
Limiting weak--type behavior for singular integral and maximal operators
Prabhu
Janakiraman
1937-1952
Abstract: The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator $T$ in $\mathbb{R}^n$: for $f\in L^1(\mathbb{R}^n)$, $f\geq 0$, $\displaystyle \lim_{\lambda\rightarrow 0} \lambda\hspace{1mm}m\{x\in\mathbb{R}^... ...a\} = \frac{1}{n} \int_{S^{n-1}}\vert\Omega(x)\vert d\sigma(x)\Vert f\Vert _1.$ For the maximal operator $M$, the corresponding result is $\displaystyle \lim_{\lambda\rightarrow 0} \lambda\hspace{1mm}m\{x\in\mathbb{R}^n: \vert Mf(x)\vert>\lambda\} = \Vert f\Vert _1.$
Symplectic forms invariant under free circle actions on 4-manifolds
Boguslaw
Hajduk;
Rafal
Walczak
1953-1970
Abstract: Let $M$ be a smooth closed 4-manifold with a free circle action generated by a vector field $X.$ Then for any invariant symplectic form $\omega$ on $M$ the contracted form $\iota_X\omega$ is non-vanishing. Using the map $\omega \mapsto \iota_X\omega$ and the related map to $H^1(M \slash S^1,\mathbb{R})$ we study the topology of the space $S_{inv}(M)$ of invariant symplectic forms on $M.$ For example, the first map is proved to be a homotopy equivalence. This reduces examination of homotopy properties of $S_{inv}$ to that of the space $\mathcal{N}_L$ of non-vanishing closed 1-forms satisfying certain cohomology conditions. In particular we give a description of $\pi_0S_{inv}(M)$ in terms of the unit ball of Thurston's norm and calculate higher homotopy groups in some cases. Our calculations show that the homotopy type of the space of non-vanishing 1-forms representing a fixed cohomology class can be non-trivial for some torus bundles over the circle. This provides a counterexample to an open problem related to the Blank-Laudenbach theorem (which says that such spaces are connected for any closed 3-manifold). Finally, we prove some theorems on lifting almost complex structures to symplectic forms in the invariant case.
Torus actions on weakly pseudoconvex spaces
Stefano
Trapani
1971-1981
Abstract: We show that the univalent local actions of the complexification of a compact connected Lie group $K$ on a weakly pseudoconvex space where $K$ is acting holomorphically have a universal orbit convex weakly pseudoconvex complexification. We also show that if $K$ is a torus, then every holomorphic action of $K$ on a weakly pseudoconvex space extends to a univalent local action of $K^{\mathbf{C}}.$
Filtrations in semisimple Lie algebras, I
Y.
Barnea;
D.
S.
Passman
1983-2010
Abstract: In this paper, we study the maximal bounded $\mathbb{Z}$-filtrations of a complex semisimple Lie algebra $L$. Specifically, we show that if $L$ is simple of classical type $A_n$, $B_n$, $C_n$ or $D_n$, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups.
New properties of convex functions in the Heisenberg group
Nicola
Garofalo;
Federico
Tournier
2011-2055
Abstract: We prove some new properties of the weakly $H$-convex functions recently introduced by Danielli, Garofalo and Nhieu. As an interesting application of our results we prove a theorem of Busemann-Feller-Alexandrov type in the Heisenberg groups $\mathbb{H}^n$, $n=1,2$.
Almost complex manifolds and Cartan's uniqueness theorem
Kang-Hyurk
Lee
2057-2069
Abstract: We present a generalization of Cartan's uniqueness theorem to the almost complex manifolds.
Resonances for steplike potentials: Forward and inverse results
T.
Christiansen
2071-2089
Abstract: We consider resonances associated to the one dimensional Schrödinger operator $-\frac{d^2}{dx^2}+V(x)$, where $V(x)=V_+$ if $x>x_M$ and $V(x)=V_-$ if $x<-x_M$, with $V_+\not = V_-$. We obtain asymptotics of the resonance-counting function for several regions. Moreover, we show that in several situations, the resonances, $V_+$, and $V_-$ determine $V$ uniquely up to translation.
Stable mapping class groups of $4$-manifolds with boundary
Osamu
Saeki
2091-2104
Abstract: We give a complete algebraic description of the mapping class groups of compact simply connected 4-manifolds with boundary up to connected sum with copies of $S^2 \times S^2$.
Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity
Yue
Liu;
Xiao-Ping
Wang;
Ke
Wang
2105-2122
Abstract: This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation) \begin{displaymath}i u_t + \Delta u + V(\epsilon x) \vert u\vert^p u = 0, \; x \in {\mathbf R}^N. \end{displaymath} In the critical and supercritical cases $p \ge 4/N,$ with $N \ge 2,$ it is shown here that standing-wave solutions of (INLS-equation) on $H^1({\mathbf R}^N)$ perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small $\epsilon > 0.$
Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane
Robert
C.
Dalang;
Olivier
Lévêque
2123-2159
Abstract: We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.
Dual decompositions of 4-manifolds II: Linear invariants
Frank
Quinn
2161-2181
Abstract: This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq 2$. Part I (Trans. Amer. Math. Soc. 354 (2002), 1373-1392) gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction theory'' using lower commutator quotients. By thinking of a 2-handlebody as essentially determined by the links used as attaching maps for its 2-handles, this theory can be thought of as giving ``ambient'' link invariants. The moves used are related to the grope cobordism of links developed by Conant-Teichner, and the Cochran-Orr-Teichner filtration of the link concordance groups. The invariants give algebraically sophisticated ``finite type'' invariants in the sense of Vassilaev.
Cohomology theories based on Gorenstein injective modules
Javad
Asadollahi;
Shokrollah
Salarian
2183-2203
Abstract: In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.
Relative entropy functions for factor maps between subshifts
Sujin
Shin
2205-2216
Abstract: Let $(X, S)$ and $(Y, T)$ be topological dynamical systems and $\pi : X \rightarrow Y$ a factor map. A function $F \in C (X)$ is a compensation function for $\pi$ if $P (F + \phi \circ \pi ) = P (\phi)$ for all $\phi \in C(Y)$. For a factor code between subshifts of finite type, we analyze the associated relative entropy function and give a necessary condition for the existence of saturated compensation functions. Necessary and sufficient conditions for a map to be a saturated compensation function will be provided.
Equivalence of domains arising from duality of orbits on flag manifolds
Toshihiko
Matsuki
2217-2245
Abstract: S. Gindikin and the author defined a $G_{\mathbb R}$- $K_{\mathbb C}$ invariant subset $C(S)$ of $G_{\mathbb C}$ for each $K_{\mathbb C}$-orbit $S$ on every flag manifold $G_{\mathbb C}/P$ and conjectured that the connected component $C(S)_0$ of the identity would be equal to the Akhiezer-Gindikin domain $D$ if $S$ is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open $K_{\mathbb C}$-orbit $S$ on an ``arbitrary'' flag manifold generalizing the result of Barchini. This conjecture for closed $S$ was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.
Intersecting curves and algebraic subgroups: Conjectures and more results
E.
Bombieri;
D.
Masser;
U.
Zannier
2247-2257
Abstract: This paper solves in the affirmative, up to dimension $n=5$, a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the Appendix.
The fourth power moment of automorphic $L$-functions for $GL(2)$ over a short interval
Yangbo
Ye
2259-2268
Abstract: In this paper we will prove bounds for the fourth power moment in the $t$ aspect over a short interval of automorphic $L$-functions $L(s,g)$ for $GL(2)$ on the central critical line Re$s=1/2$. Here $g$ is a fixed holomorphic or Maass Hecke eigenform for the modular group $SL_{2}(\mathbb{Z})$, or in certain cases, for the Hecke congruence subgroup $\Gamma _{0}({\mathcal{N}})$ with $\mathcal{N}>1$. The short interval is from a large $K$ to $K+K^{103/135+\varepsilon }$. The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg $L$-function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).
Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
Henry
K.
Schenck;
Alexander
I.
Suciu
2269-2289
Abstract: If $\mathcal A$ is a complex hyperplane arrangement, with complement $X$, we show that the Chen ranks of $G=\pi_1(X)$ are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring $A=H^*(X,\Bbbk)$, viewed as a module over the exterior algebra $E$ on $\mathcal A$: $\displaystyle \theta_k(G) = \dim_{\Bbbk}\operatorname{Tor}^E_{k-1}(A,\Bbbk)_k,$ for $k\ge 2$$\displaystyle ,$ where $\Bbbk$ is a field of characteristic 0. The Chen ranks conjecture asserts that, for $k$ sufficiently large, $\theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$, where $h_r$ is the number of $r$-dimensional components of the projective resonance variety $\mathcal R^{1}(\mathcal A)$. Our earlier work on the resolution of $A$ over $E$ and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of $\mathcal R ^{1}(\mathcal A)$ and a localization argument, we establish the inequality $\displaystyle \theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k},$ for $k\gg 0$$\displaystyle ,$ for arbitrary $\mathcal A$. Finally, we show that there is a polynomial $\mathrm{P}(t)$ of degree equal to the dimension of $\mathcal R^1(\mathcal A)$, such that $\theta_k(G) = \mathrm{P}(k)$, for all $k\gg 0$.
Canard solutions at non-generic turning points
Peter
De Maesschalck;
Freddy
Dumortier
2291-2334
Abstract: This paper deals with singular perturbation problems for vector fields on $2$-dimensional manifolds. ``Canard solutions'' are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a ``turning point'' and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the ``control curve''.