Basic properties of finite group actions with the Rohlin property on unital C^*-algebras are investigated. A characterization of finite group actions with the Rohlin property on the Cuntz algebra ${\mathcal{0}}_2$ is given in terms of central sequences, which may be considered as an equivariant version of E. Kirchberg and N. C. Phillips's characterization of ${\mathcal{0}}_2$. A large class of symmetries on ${\mathcal{0}}_2$ are classified in terms of the fixed-point algebras for conjugacy and the crossed products for cocycle conjugacy. Model actions of symmetries of ${\mathcal{0}}_2$ are constructed for given K-theoretical invariants.

In this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant, the logarithm of the density of caloric measure with respect to a certain projective Lebesgue measure is of vanishing mean oscillation. We also obtain a partial converse to this result. Our work generalizes to the heat equation recent work of Kenig and Toro for the Laplacian.

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball B⁴(c)⊂R⁴ into 4-dimensional rational symplectic manifolds, where c=πr² is the capacity of the standard ball of radius r. We compute the rational homotopy groups of that space when the 4-manifold has the form M_μ=S²×S², μω₀⊕ω₀, where ω₀ is the area form on the sphere with total area 1 and μ belongs to the interval [1,2]. We show that, when μ is 1, this space retracts to the space of symplectic frames for any value of c. However, for any given 1<μ≤2, the rational homotopy type of that space changes as c crosses the critical parameter λ=μ−1, which is the difference of areas between the two S²-factors. We prove, moreover, that the full homotopy type of that space changes only at that value, that is, that the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value. The same methods apply to all other values of μ and other rational 4-manifolds as well. The methods rely on two different tools: the study of the action of symplectic groups on the stratified space of almost complex structures developed by Gromov, Abreu, and McDuff and the analysis of the relations between the group corresponding to a manifold M, the group corresponding to its blow-up $\tilde{M}$, and the space of symplectic embedded balls in M.

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every integer g≥0, there is a genus-g curve over ${\mathbb{F}}_q$ with at least c_qg rational points over ${\mathbb{F}}_q$. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q=d log q. We show also that there is a constant c>0 such that for every q and every n>0, and for every sufficiently large g there is a genus-g curve over ${\mathbb{F}}_q$ that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to ${(\mathbb{Z}/n\mathbb{Z})}^r$ for some r>cg/n.