Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology for symplectomorphisms of surfaces and a calculation of Seidel’s symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and an asymptotic symplectic invariant. A generalisation of the Poincaré-Birkhoff fixed point theorem and Arnold conjecture is proposed. Dedicated to Vladimir Igorevich Arnold     

A sharp form of Poincaré type inequalities on balls and spheres

Summary Letu be a real valued function on ann-dimensional Riemannian manifoldM ⁿ. We consider an inequality between theL ^q-norm ofu minus its mean value overM ⁿ and theL ^p-norm of the gradient ofu.The best constant in such inequality is exhibited in the following cases: i)M ⁿ is an open ball inIR ⁿ andp=1, 0<qn/(n–1); ii)M ⁿ is a sphere inIR ⁿ +1 and eitherp=1, 0<qn/(n–1) orp>n,q=.

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Sunto Siau una funzione a valori reali dafinita su una varietà riemannianan-dimensionaleM ⁿ. Si considera una disuguaglianza tra la normaL ^q diu meno il suo valor medio suM ⁿ e la normaL ^p del gradiente diu.Si determina la costante ottimale in tale disuguaglianza nei seguenti casi: i)M ⁿ è un disco aperto inIR ⁿ ep=1, 0<qn/(n–1); ii)M ⁿ è una sfera inIR ⁿ +1 ep=1, 0<qn/(n–1) oppurep>n,q=.

     

Rabinowitz–Floer homology for superquadratic Dirac equations on compact spin manifolds

In this paper, we investigate the properties of a semilinear problem on a spin manifold involving the Dirac operator through the construction of Rabinowitz–Floer homology groups. We give several existence results for subcritical and critical nonlinearities as application of the computation of the different homologies.     

Denominators for the Poincaré series of invariants of small matrices

We determine explicit denominators for the Poincaré series of (a) the invariants ofm genericN ×N matrices, and (b) the ring generated bym genericN ×N matrices and their traces, forN≤4. ForN≤3 we prove (and forN=4 we conjecture) that the denominators we obtain are of minimum degree. We also give explicit rational fractions for both series for small values ofm andN. Research supported by NSF grants DMS-9622062 and DMS-9700787.     

The first moment of Salié sums

The main objective of this article is to study the asymptotic behavior of Salié sums over arithmetic progressions. We deduce from our asymptotic formula that Salié sums possess a bias towards being positive. The method we use is based on the Kuznetsov formula for modular forms of half-integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half-integral weight.     

Some inequalities for the Poincaré metric of plane domains

In this paper, the Poincaré (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain In particular, another proof of a recent result of Gardiner and Lakic [7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood’s theorem are derived.This research was carried out during the first-named author’s visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS.     

Fractional Poincaré inequalities for general measures

We prove a fractional version of Poincaré inequalities in the context of ${\mathbb{R}}^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein–Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.     

Poincaré Duality for Logarithmic Crystalline Cohomology

We prove Poincaré duality for logarithmic crystalline cohomology of log smooth schemes whose underlying schemes are reduced. This is a generalization of the result of P. Berthelot for usual smooth schemes and that of O. Hyodo for the special fibers of semi-stable families and trivial coefficients.     

Nonvanishing of Siegel Poincaré series

We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

Poincaré duality for p-adic Lie groups

We establish Poincaré duality for continuous group cohomology of p-adic Lie groups with rational coefficients and compare integral structures under this duality.     

Denominators for the Poincaré series of invariants of matrices with involution

The goal of this paper is to study some Poincaré series associated to the invariants of the symplectic and odd orthogonal groups. These series turn out to be rational functions and our main results will describe the denominators. This work will generalize some known results on the invariants of the general linear groups. In addition to whatever intrinsic interest we hope our results may have, the subject involves an interesting interplay of invariant theory and complex variables. The first author gratefully acknowledges Support from DePaul University Research Council. The second author was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by an Internal Research Grant from Bar-Ilan University.     

A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

We construct an invariant J _M of integral homology spheres M with values in a completion of the polynomial ring ℤ[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten–Reshetikhin–Turaev invariant τ_ζ(M) of M at ζ. Thus J _M unifies all the SU(2) Witten–Reshetikhin–Turaev invariants of M. It also follows that τ_ζ(M) as a function on ζ behaves like an “analytic function” defined on the set of roots of unity.     

Pro-p completions of Poincaré duality groups

We consider some sufficient conditions for the pro-p completion of an orientable Poincaré duality group of dimension n ≥ 3 to be a virtually pro-p Poincaré duality group of dimension at most n ? 2.