Let $X$ be a mixture of independent Brownian motion and symmetric stable process. In this paper we establish sharp bounds for transition density of $X$, and prove a parabolic Harnack inequality for nonnegative parabolic functions of $X$.

We first present an alternative derivation of a local Weierstrass representation for flat surfaces in the real hyperbolic three-space, $\mathbb{H}^3$, using as a starting point an old result due to Luigi Bianchi. We then prove the following: let $M\subset \mathbb{H}^3$ be a flat compact connected smooth surface with $\partial M\neq \emptyset$, transversal to a foliation of $\mathbb{H}^3$ by horospheres. If, along $\partial M$, $M$ makes a constant angle with the leaves of the foliation, then $M$ is part of an equidistant surface to a geodesic orthogonal to the foliation. We also consider the caustic surface associated with a family of parallel flat surfaces and prove that the caustic of such a familyis also a flat surface (possibly with singularities). Finally, a rigidity result for flat surfaces with singularities and a geometrical application of Schwarz's reflection principle are shown.

We establish finite generation of certain invariant graded algebras defined on toric log Fano fibrations. These are the toric version of FGA algebras, introduced by Shokurov in connection with the existence of flips.

In 1981, Hamachi introduced an interesting class of type III shifts. Since these are not measure-preserving, one cannot use metric entropy to study them. As a possible substitute, we give estimates for the upper and lower critical dimensions of Hamachi shifts. These invariants have previously been used by the authors to study odometer actions.

Let $K$ be a local field and $k$ an algebraically closed field. We prove the finiteness of isomorphism classes of semisimple Galois representations of $K$ into $\GL_d(k)$ with bounded Artin conductor and residue degree. We calculate explicitly the number of totally ramified finite abelian extensions of $K$ with bounded conductor. Using this result, we give an upper bound for the number of certain Galois extensions of $K$.

We prove that a discrete group $G$ is amenable if and only if it is strongly unitarizable in the following sense: every unitarizable representation $\pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $\pi$. Analogously, a $C^*$-algebra $A$ is nuclear if and only if any bounded homomorphism $u: A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $\xi$ in the von Neumann algebra generated by the range of $u$ such that $a\to \xi u(a) \xi^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $L(A\otimes_{\max} B)$ of the maximal tensor product $A\otimes_{\max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $L(A\otimes_{\max} B)<\infty$ either for $B=B(\ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non-Abelian free group, then $A$ must be nuclear. We also show that $L(A\otimes_{\max} B)\le d$ if and only if the canonical quotient map from the unital free product $A\,{\ast}\, B$ onto $A\otimes_{\max} B$ remains a complete quotient map when restricted to the closed span of the words of length at most $d$.

We show that a homogeneous Siegel domain is symmetric if and only if its Cayley transform image is convex. Moreover, this convexity forces the parameter of the Cayley transform to be specific, so that the Cayley transform coincides with the inverse of the Cayley transform introduced by Korányi and Wolf.

We show that the classification of $A$-hypergeometric systems and that of multi-graded simple modules (up to shift) over the ring of differential operators on an affine toric variety are the same. We then show that the set of multi-homogeneous primitive ideals of the ring of differential operators is finite. Furthermore, we give conditions for the algebra being simple.