Electronic Research Announcements

 2009 , Volume 16

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Quasiperiodic motion for the pentagram map
Valentin Ovsienko, Richard Schwartz and Serge Tabachnikov
2009, 16: 1-8 doi: 10.3934/era.2009.16.1 +[Abstract](3200) +[PDF](233.7KB)
The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call twisted polygons. In this note we describe our recent work on the pentagram map, in which we find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.
A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems
Ernest Fontich, Rafael de la Llave and Yannick Sire
2009, 16: 9-22 doi: 10.3934/era.2009.16.9 +[Abstract](3617) +[PDF](222.2KB)
We describe a method to study the existence of whiskered quasi-periodic solutions in Hamiltonian systems. The method applies to finite dimensional systems and also to lattice systems and to PDE's including some ill posed ones. In coupled map lattices, we can also construct solutions of infinitely many frequencies which do not vanish asymptotically.
The spectrum of the weakly coupled Fibonacci Hamiltonian
David Damanik and Anton Gorodetski
2009, 16: 23-29 doi: 10.3934/era.2009.16.23 +[Abstract](4148) +[PDF](143.8KB)
We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.
Descent construction for GSpin groups: Main results and applications
Joseph Hundley and Eitan Sayag
2009, 16: 30-36 doi: 10.3934/era.2009.16.30 +[Abstract](2823) +[PDF](151.5KB)
The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis, and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi [2] on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram [1] on cuspidality of the exterior square lift for representations of $GL_4$. Complete proofs of the results announced in the present note will appear in our forthcoming article(s).
On the analyticity of the bivariant JLO cocycle
Moulay-Tahar Benameur and Alan L. Carey
2009, 16: 37-43 doi: 10.3934/era.2009.16.37 +[Abstract](2705) +[PDF](169.4KB)
The goal of this note is to outline a proof that, for any l $\geq 0$, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth fibration $M\to B$ over the pair of algebras $(C^\infty (M), C^\infty(B))$, is entire when we endow $C^\infty(M)$ with the $C^{l+1}$ topology and $C^\infty(B)$ with the $C^{l}$ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Fréchet smooth topologies on $C^\infty(M)$ and $C^\infty(B)$.
Research announcement: The structure of groups with a quasiconvex hierarchy
Daniel T. Wise
2009, 16: 44-55 doi: 10.3934/era.2009.16.44 +[Abstract](8171) +[PDF](2176.9KB)
Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. We show that $G$ has a finite index subgroup $G'$ that embeds as a quasiconvex subgroup of a right-angled Artin group. It follows that every quasiconvex subgroup of $G$ is a virtual retract, and is hence separable. The results are applied to certain 3-manifold and one-relator groups.
A note on L-series and Hodge spectrum of polynomials
Ricardo García López
2009, 16: 56-62 doi: 10.3934/era.2009.16.56 +[Abstract](2806) +[PDF](148.2KB)
We compare on the one hand the combinatorial procedure described in [1] which gives a lower bound for the Newton polygon of the $L$-function attached to a commode, non-degenerate polynomial with coefficients in a finite field and on the other hand the procedure which gives the Hodge theoretical spectrum at infinity of a polynomial with complex coefficients and with the same Newton polyhedron. The outcome is that they are essentially the same, thus providing a Hodge theoretical interpretation of the Adolphson-Sperber lower bound which was conjectured in [1].
Singular spaces and generalized Poincaré complexes
Markus Banagl
2009, 16: 63-73 doi: 10.3934/era.2009.16.63 +[Abstract](3026) +[PDF](178.8KB)
We introduce a method that associates to a singular space a CW complex whose ordinary rational homology satisfies Poincaré duality across complementary perversities as in intersection homology. The method is based on a homotopy theoretic process of spatial homology truncation, whose functoriality properties are investigated in detail. The resulting homology theory is not isomorphic to intersection homology and addresses certain questions in type II string theory related to massless D-branes. The two theories satisfy an interchange of third and second plus fourth Betti number for mirror symmetric conifold transitions. Further applications of the new theory to K-theory and symmetric L-theory are indicated.

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