Electronic Research Announcements

 2012 , Volume 19

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Higher pentagram maps, weighted directed networks, and cluster dynamics
Michael Gekhtman, Michael Shapiro, Serge Tabachnikov and Alek Vainshtein
2012, 19: 1-17 doi: 10.3934/era.2012.19.1 +[Abstract](4000) +[PDF](262.3KB)
The pentagram map was extensively studied in a series of papers by V. Ovsienko, R. Schwartz and S. Tabachnikov. It was recently interpreted by M. Glick as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of geometrically meaningful discrete integrable maps.
Constructing automorphic representations in split classical groups
David Ginzburg
2012, 19: 18-32 doi: 10.3934/era.2012.19.18 +[Abstract](2175) +[PDF](192.9KB)
In this paper we introduce a general construction for a correspondence between certain Automorphic representations in classical groups. This construction is based on the method of small representations, which we use to construct examples of CAP representations.
On GIT quotients of Hilbert and Chow schemes of curves
Gilberto Bini, Margarida Melo and Filippo Viviani
2012, 19: 33-40 doi: 10.3934/era.2012.19.33 +[Abstract](4367) +[PDF](163.9KB)
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree $d$ and genus $g$ in the projective space of dimension $d-g$, whose full details will appear in [6]. In particular, we extend the previous results of L. Caporaso up to $d>4(2g-2)$ and we observe that this is sharp. In the range $2(2g-2) < d < \frac{7}{2} (2g-2)$, we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.
Boundaries, Weyl groups, and Superrigidity
Uri Bader and Alex Furman
2012, 19: 41-48 doi: 10.3934/era.2012.19.41 +[Abstract](3259) +[PDF](161.5KB)
This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $\Gamma$ and a boundary action $\Gamma$ ↷ $B$ we associate a certain generalized Weyl group $W_{{\Gamma}{B}}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \bf{H}$, defines a special homomorphism $W_{{\Gamma}{B}}\to Weyl_{\bf H}$. This general fact allows the deduction of the aforementioned superrigidity results.
Operator representations of logmodular algebras which admit $\gamma-$spectral $\rho-$dilations
Adina Juratoni, Flavius Pater and Olivia Bundău
2012, 19: 49-57 doi: 10.3934/era.2012.19.49 +[Abstract](2404) +[PDF](327.5KB)
This paper deals with some semi-spectral representations of logmodular algebras. More exactly, we characterize such representations by the corresponding scalar semi-spectral measures. In the case of a logmodular algebra we obtain, for $0<\rho \leq 1,$ several results which generalize the corresponding results of Foiaş-Suciu [2] in the case $\rho =1.$
Integration of exact Courant algebroids
David Li-Bland and Pavol Ševera
2012, 19: 58-76 doi: 10.3934/era.2012.19.58 +[Abstract](6386) +[PDF](431.5KB)
In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [32] inverts our integration.
Upper bounds for Steklov eigenvalues on surfaces
Alexandre Girouard and Iosif Polterovich
2012, 19: 77-85 doi: 10.3934/era.2012.19.77 +[Abstract](3724) +[PDF](126.0KB)
We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.
The pentagram map in higher dimensions and KdV flows
Boris Khesin and Fedor Soloviev
2012, 19: 86-96 doi: 10.3934/era.2012.19.86 +[Abstract](2288) +[PDF](398.8KB)
We extend the definition of the pentagram map from 2D to higher dimensions and describe its integrability properties for both closed and twisted polygons by presenting its Lax form. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D.
Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links
Donghi Lee and Makoto Sakuma
2012, 19: 97-111 doi: 10.3934/era.2012.19.97 +[Abstract](2213) +[PDF](1019.1KB)
Following Riley's work, for each $2$-bridge link $K(r)$ of slope $r∈\mathbb{R}$ and an integer or a half-integer $n$ greater than $1$, we introduce the Heckoid orbifold $S(r;n)$and the Heckoid group $G(r;n)=\pi_1(S(r;n))$ of index $n$ for $K(r)$. When $n$ is an integer, $S(r;n)$ is called an even Heckoid orbifold; in this case, the underlying space is the exterior of $K(r)$, and the singular set is the lower tunnel of $K(r)$ with index $n$. The main purpose of this note is to announce answers to the following questions for even Heckoid orbifolds. (1) For an essential simple loop on a $4$-punctured sphere $S$ in $S(r;n)$ determined by the $2$-bridge sphere of $K(r)$, when is it null-homotopic in $S(r;n)$? (2) For two distinct essential simple loops on $S$, when are they homotopic in $S(r;n)$? We also announce applications of these results to character varieties, McShane's identity, and epimorphisms from $2$-bridge link groups onto Heckoid groups.
On Totally integrable magnetic billiards on constant curvature surface
Misha Bialy
2012, 19: 112-119 doi: 10.3934/era.2012.19.112 +[Abstract](3545) +[PDF](286.6KB)
We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrable then the boundary curve is necessarily a circle. This result shows that the so-called Hopf rigidity phenomenon which was recently obtained for classical billiards on constant curvature surfaces holds true also in the presence of constant magnetic field.
Locally decodable codes and the failure of cotype for projective tensor products
Jop Briët, Assaf Naor and Oded Regev
2012, 19: 120-130 doi: 10.3934/era.2012.19.120 +[Abstract](2939) +[PDF](405.2KB)
It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $l_p\hat\otimes X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $l_{p_1}\hat\otimes l_{p_2} \hat\otimes l_{p_3}$ does not have finite cotype. This is proved via a connection to the theory of locally decodable codes.

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