Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology
Luis
A.
Cordero;
Marisa
Fernández;
Alfred
Gray;
Luis
Ugarte
5405-5433
Abstract: We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if $\Gamma \backslash G$ is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}(\Gamma \backslash G)$ is canonically isomorphic to the ${\bar{\partial}}$-cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}({\mathfrak g}^{{\mathbb C}})$of the bigraded complex $(\Lambda ^{\ast ,\ast } ({\mathfrak g}^{{\mathbb C}})^{\ast }, {\bar{\partial}})$of complex valued left invariant differential forms on the nilpotent Lie group $G$.
Weakly o-minimal structures and real closed fields
Dugald
Macpherson;
David
Marker;
Charles
Steinhorn
5435-5483
Abstract: A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.
Curves on normal surfaces
Gunnar
Fløystad
5485-5510
Abstract: We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces.
Germs of holomorphic vector fields in $\mathbb{C}^m$ without a separatrix
I.
Luengo;
J.
Olivares
5511-5524
Abstract: We prove the existence of families of germs of holomorphic vector fields in $\mathbb{C}^m$ without a separatrix, in every complex dimension $m$ bigger than or equal to 4.
Ribbon tile invariants
Igor
Pak
5525-5561
Abstract: Let $\mathbf{T}$ be a finite set of tiles, and $\mathcal{B}$ a set of regions $\Gamma$ tileable by $\mathbf{T}$. We introduce a tile counting group $\mathbb{G} (\mathbf{T}, \mathcal{B})$ as a group of all linear relations for the number of times each tile $\tau \in \mathbf{T}$ can occur in a tiling of a region $\Gamma \in \mathcal{B}$. We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group. The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group. The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.
Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series
Alvaro
Alvarez-Parrilla
5563-5582
Abstract: Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series $\varphi_{ir}$ with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of $\varphi_{ir}$; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter $\lambda (\lambda=\frac{1}{4}+r^2$is the eigenvalue of $\varphi_{ir}$) for a cubic phase. As applications we find sets of asymptotic relations for divisor functions.
Computing the $p$-Selmer group of an elliptic curve
Z.
Djabri;
Edward
F.
Schaefer;
N.
P.
Smart
5583-5597
Abstract: In this paper we explain how to bound the $p$-Selmer group of an elliptic curve over a number field $K$. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most $p^2-1$. Our method is practical for $p=3$, but for larger values of $p$it becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the $p$-Selmer group of the curve, and so one can use the method to find the Mordell-Weil rank of the curve when the usual method of $2$-descent fails.
On the dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space
A.
N.
Dranishnikov
5599-5618
Abstract: For every two compact metric spaces $X$ and $Y$, both with dimension at most $n-3$, there are dense $G_{\delta}$-subsets of mappings $f:X \to \mathbb{R}^n$ and $g:Y\to \mathbb{R}^n$ with $dimf(X)\cap g(Y)\leq dim(X\times Y)-n$.
Power operations in elliptic cohomology and representations of loop groups
Matthew
Ando
5619-5666
Abstract: Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.
On cobordism of manifolds with corners
Gerd
Laures
5667-5688
Abstract: This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.
The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic $K$-theory of Bianchi groups
E.
Berkove;
F.
T.
Farrell;
D.
Juan-Pineda;
K.
Pearson
5689-5702
Abstract: We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic $n$-space $\mathbb{H} ^n$ with finite volume orbit space. We then apply this result to show that, for any Bianchi group $\Gamma$, $Wh(\Gamma)$, $\tilde K_0(\mathbb{Z}\Gamma)$, and $K_i(\mathbb{Z}\Gamma)$ vanish for $i\leq -1$.
Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents
N.
Ghoussoub;
C.
Yuan
5703-5743
Abstract: We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: $\left\{ \begin{matrix} {-\triangle_{p} u = \lambda \vert u\vert^{r-2}u + \mu ... ... }, {}} {\hphantom{-} u\vert _{\partial \Omega} = 0, } \end{matrix}\right.$ where $\lambda$ and $\mu$ are two positive parameters and $\Omega$ is a smooth bounded domain in $\mathbf{R}^n$ containing $0$ in its interior. The variational approach requires that $1 < p < n$, $p\leq q\leq p^{*}(s)\equiv \frac{n-s}{n-p}p$ and $p\leq r\leq p^*\equiv p^*(0)=\frac{np}{n-p}$, which we assume throughout. However, the situations differ widely with $q$ and $r$, and the interesting cases occur either at the critical Sobolev exponent ($r=p^*$) or in the Hardy-critical setting ($s=p=q$) or in the more general Hardy-Sobolev setting when $q=\frac{n-s}{n-p}p$. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case $p=2$, especially those corresponding to singularities (i.e., when $0<s\leq p)$.
The quartile operator and pointwise convergence of Walsh series
Christoph
Thiele
5745-5766
Abstract: The bilinear Hilbert transform is given by \begin{displaymath}H(f,g)(x):= p.v. \int f(x-t)g(x+t)\frac{dt}{t}. \end{displaymath} It satisfies estimates of the type \begin{displaymath}\Vert H(f,g)\Vert _r\le C(s,t)\Vert f\Vert _s \Vert g\Vert _t.\end{displaymath} In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in $L^p[0,1]$, with $1<p$ converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.
The range of traces on quantum Heisenberg manifolds
Beatriz
Abadie
5767-5780
Abstract: We embed the quantum Heisenberg manifold $D_{\mu\nu}^{c}$ in a crossed product ${C}^*$-algebra. This enables us to show that all tracial states on $D_{\mu\nu}^{c}$ induce the same homomorphism on $K_0(D_{\mu\nu}^{c})$, whose range is the group $\mathbf{Z} +2\mu\mathbf{Z} + 2\nu\mathbf{Z}$.
Contact topology and hydrodynamics III: knotted orbits
John
Etnyre;
Robert
Ghrist
5781-5794
Abstract: We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian $S^3$ whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on $S^3$. Sufficient review of concepts is included to make this paper independent of the previous works in this series.
The natural representation of the stabilizer of four subspaces
Jozsef
Horvath;
Roger
Howe
5795-5815
Abstract: Consider the natural action of the general linear group $GL(V)$ on the product of four Grassmann varieties of the vector space $V$. In General linear group action on four Grassmannians we gave an algorithm to construct the generic stabilizer $H$ of this action. In this paper we investigate the structure of $V$ as an $H$-module, and we show the effectiveness of the methods developed there, by applying them to the explicit description of $H$.
Intersection theory on non-commutative surfaces
Peter
Jørgensen
5817-5854
Abstract: Consider a non-commutative algebraic surface, $X$, and an effective divisor $Y$ on $X$, as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting. We also apply our theory to some special cases, including the blow up of $X$in a point, and show that the self intersection of the exceptional divisor is $-1$. This is used to give an example of a non-commutative surface with a commutative ${\Bbb P}^1$ which cannot be blown down, because its self intersection is $+1$ rather than $-1$. We also get some results on Hilbert polynomials of modules on $X$.
Quantum $n$-space as a quotient of classical $n$-space
K.
R.
Goodearl;
E.
S.
Letzter
5855-5876
Abstract: The prime and primitive spectra of $\mathcal{O}_{\mathbf q}(k^{n})$, the multiparameter quantized coordinate ring of affine $n$-space over an algebraically closed field $k$, are shown to be topological quotients of the corresponding classical spectra, $\operatorname{spec} \mathcal{O}(k^{n})$ and $\max \mathcal{O}(k^{n})\approx k^{n}$, provided the multiplicative group generated by the entries of $\mathbf{q}$ avoids $-1$.
Correction to ``Geometric groups. I''
Valera
Berestovskii;
Conrad
Plaut;
Cornelius
Stallman
5877