Volume 18, Issue 2

The restriction of an isotropic α-stable Lévy process X α in ℝ n on the fractal support F of certain finite Borel measures μ is introduced in three different ways: by means of the trace of its Dirichlet form on F and related Besov spaces, by its μ-local time on F and the corresponding time changed process, and by restricting on the fine support of μ via balayage spaces and potentials. Moreover, for compact upper d -regular sets F with d > n − α an approximation by means of restrictions of X α to small ɛ-neighborhoods of F is given.

In this work, the assembly map in L -theory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by Carlsson-Pedersen to split the assembly map in the case of torsion free groups. Here, the continuously controlled techniques and results are extended to handle groups with torsion.

We apply tilting theory to study modules of finite projective dimension. We introduce the notion of finite and cofinite type for tilting and cotilting classes of modules, respectively, showing that, for each dimension, there is a bijection between these classes and resolving classes of modules. We then focus on Iwanaga-Gorenstein rings. Using tilting theory, we prove the first finitistic dimension conjecture for these rings. Moreover, we characterize them among noetherian rings by the property that Gorenstein injective modules form a tilting class. Finally, we give an explicit construction of families of (co)tilting modules of (co)finite type for one-dimensional commutative Gorenstein rings.

Cauchy-Riemann equations for smooth maps ϕ: ℂ → ℂ k are proved to be of variational type if and only if k is even. This fact is seen to be related to a complex differential form of degree (3, 0) on ℂ × ℂ k , which exists only for an even k . The Lie algebra of infinitesimal symmetries of the Hamiltonian structure associated with Cauchy-Riemann equations is also determined.

Let p, q be such that 2≤ p, q ≤ + ∞. We prove in this paper the L p – L q version of Hardy's Theorem for an arbitrary nilpotent Lie group G extending then earlier cases and the classical Hardy theorem proved recently by E. Kaniuth and A. Kumar. The case where 1 ≤ p , q ≤ + ∞ is studied for a restricted class of nilpotent Lie groups.

We prove the Lipschitz regularity of solutions to a class of elliptic problems characterized by weak growth, differentiability and ellipticity assumptions.

We investigate the rigidity and asymptotic properties of quantum SU (2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU (2) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan's property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of SL (2,ℤ) on L 2 (ℝ). As a consequence we obtain a curious analytic fact about the Fourier transform on ℝ which may not have been previously observed.

A free Γ- complex is a connected complex X together with an action of Γ which permutes freely the cells of X . Let Γ be a group in the class and X be an infinite-dimensional free Γ-complex which is homotopy equivalent to some sphere S m , m > 1, and let Ω be the Euler class of X /Γ. Then we prove the following main results: Theorem B. Suppose Γ induces a trivial action on H * ( X ). Then X/ Γ is homotopy equivalent to a finite-dimensional complex if and only if Γ is torsion-free, or else the natural map H m +1 (Γ,ℤ) → Ĥ m +1 (Γ,ℤ) sends Ω to Ωˆ, which is an invertible element of the generalized Farrell-Tate cohomology ring of Γ, and m is odd. Theorem C. Suppose Γ induces a nontrivial action on H* (X). Then X/ Γ is homotopy equivalent to a finite-dimensional complex if and only if either (1) Γ is torsion-free, (2) Γ≅Γ 0 ⋊ H where Γ 0 is torsion-free and H is isomorphic to ℤ/2, res Γ H (Ω) ≠ 0, and m is even, or else (3) all the torsion elements of Γ lie in Γ 0 , and Ω is mapped to Ωˆ 0 for which some power of Ωˆ 0 is an invertible element of the generalized Farrell-Tate cohomology ring of Γ 0 , and m is odd.

Let S and T be locally compact transitive countable state Markov shifts. When is there a factor map from S onto T ? In the compact setting a trivial periodic point condition and the topological entropy settle this question. For the non-compact case the behaviour of the Markov shifts at infinity plays a central role. To capture some of this we introduce a new entropy at infinity and use it to solve the factoring problem in several situations.