We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For $q\in[1,\infty]$, the $l^{q}$-interpolation method allows to interpolate linear operators that have bounded $l^{q}$-valued extensions. For $q=2$ and if the Banach function spaces are $r$-concave for some $r<\infty$, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the $H^{\infty}$-functional calculus. As a special case, we obtain Triebel–Lizorkin spaces $F^{2\theta}_{p,q}(\mathbb{R}^{d})$ by $l^{q}$-interpolation between $L^{p}(\mathbb{R}^{d})$ and $W^{2}_{p}(\mathbb{R}^{d})$ where $p\in(1,\infty)$. A similar result holds for the recently introduced generalized Triebel–Lizorkin spaces associated with $R_{q}$-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel–Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces.

We compare the Grushin geometry to Euclidean geometry, through quasisymmetric parametrization, bilipschitz parametrization and bilipschitz embedding, highlighting the role of the exponents and the fractal nature of the singular hyperplanes in Grushin geometry.

We study a half-space problem related to graphs in $\mathbb{H}^{2}\times\mathbb{R}$, where $\mathbb{H}^{2}$ is the hyperbolic plane, having constant mean curvature $H$ defined over unbounded domains in $\mathbb{H}^{2}$.

We introduce a higher rank analog of the Joyce–Song theory of stable pairs. Given a nonsingular projective Calabi–Yau threefold $X$, we define the higher rank Joyce–Song pairs given by ${O}^{\oplus r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg0$ is a fixed integer. We equip the higher rank pairs with a Joyce–Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of “weakly” semistable objects.

We develop the ring-theoretic notion of Invariant Basis Number in the context of unital $C^{*}$-algebras and their Hilbert $C^{*}$-modules. Characterization of $C^{*}$-algebras with Invariant Basis Number is given in $K$-theoretic terms, closure properties of the class of $C^{*}$-algebras with Invariant Basis Number are given, and examples of $C^{*}$-algebras both with and without the property are explored. For $C^{*}$-algebras without Invariant Basis Number, we determine structure in terms of a “Basis Type” and describe a class of $C^{*}$-algebras which are universal in an appropriate sense. We conclude by investigating properties which are strictly stronger than Invariant Basis Number.

We prove that any quadratic complete intersection with a certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. Furthermore, we discuss under what conditions its ring of invariants by a Young subgroup is a homogeneous complete intersection with a standard grading.

For an algebraic variety $X$ in characteristic $p>0$, the push-forward $F^{e}_{*}\mathcal{O}_{X}$ of the structure sheaf by an iterated Frobenius endomorphism $F^{e}$ is closely related to the geometry of $X$. We study the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ into direct summands when $X$ is obtained by blowing up the projective plane $\mathbb{P}^{2}$ at four points in general position. We explicitly describe the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ and show that there appear only finitely many direct summands up to isomorphism, when $e$ runs over all positive integers. We also prove that these summands generate the derived category $D^{b}(X)$. On the other hand, we show that there appear infinitely many distinct indecomposable summands of iterated Frobenius push-forwards on a ten-point blowup of $\mathbb{P}^{2}$.

In this paper, new correlation inequalities are obtained for massless Euclidean fields on the $d$ dimensional integer lattice. Some of the inequalities have been obtained previously, in the case where the Lagrangian is a very small perturbation of a quadratic, using the renormalization group method. The results of the present paper apply provided the Lagrangian is uniformly convex. They therefore hold for the Coulomb dipole gas in which particle density can be of order $1$. The approach of the present paper is based on the methodology of Naddaf–Spencer, which relates second moment correlation functions for the Euclidean field to expectations of Green’s functions for parabolic PDE with random coefficients.

Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^{n}$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where $z_{0}$ is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of $z_{0}$. In particular, we find how many curves there are that are fixed by $f$ and that land at $z_{0}$.

It is known that, under strong combinatorial axioms, $O_{N}\subset O_{N}^{*}\subset O_{N}^{+}$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N-1}_{\mathbb{R}}\subset S^{N-1}_{\mathbb{R},*}\subset S^{N-1}_{\mathbb{R},+}$, the noncommutative projective spaces $P^{N-1}_{\mathbb{R}}\subset P^{N-1}_{\mathbb{C}}\subset P^{N-1}_{+}$, and the projective orthogonal quantum groups $PO_{N}\subset PO_{N}^{*}\subset PO_{N}^{+}$.

Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathbb{P}}_{\mu}(\Vert X_{t}\Vert \ne0)=\nu^{-1}\langle\phi_{0},\mu\rangle$, where $\phi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator $L$ of the mean semigroup of $X$, and $\nu$ is a positive constant. Then we show that, for a large class of functions $f$, conditioning on $\Vert X_{t}\Vert \ne0$, $t^{-1}\langle f,X_{t}\rangle$ converges in distribution to $\langle f,\psi_{0}\rangle_{m}W$, where $W$ is an exponential random variable, and $\psi_{0}$ is the eigenfunction corresponding to the first eigenvalue of the dual of $L$. Finally, if $\langle f,\psi_{0}\rangle_{m}=0$, we prove that, conditioning on $\Vert X_{t}\Vert \ne0$, $(t^{-1}\langle\phi_{0},X_{t}\rangle,t^{-1/2}\langle f,X_{t}\rangle )$ converges in distribution to $(W,G(f)\sqrt{W})$, where $G(f)\sim\mathcal{N}(0,\sigma_{f}^{2})$ is a normal random variable, and $W$ and $G(f)$ are independent.