We consider versions of the local duality theorem in ${\mathbb{C}}^n$. We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff–Herrera and Andersson–Wulcan, and we give several different proofs of non-degeneracy of the pairings. One of the proofs of non-degeneracy uses the theory of linkage, and conversely, we can use the non-degeneracy to obtain results about linkage for modules. We also discuss a variant of such pairings based on residues considered by Passare, Lejeune-Jalabert and Lundqvist.

We consider the problem of characterizing the extreme points of the set of analytic functions $f$ on the bidisk with positive real part and $f\left(0\right)=1$. If one restricts to those $f$ whose Cayley transform is a rational inner function, one gets a more tractable problem. We construct families of such $f$ that are extreme points and conjecture that these are all such extreme points. These extreme points are constructed from polynomials dubbed ${\mathbb{T}}^2$-saturated, which roughly speaking means they have no zeros in the bidisk and as many zeros as possible on the boundary without having infinitely many zeros.

By modifying a coupling method developed by the third author with much more delicate analysis, we prove that a family of stochastic partial differential equations (SPDEs) driven by highly degenerate pure jump Lévy noises are exponential mixing. These pure jump Lévy noises include a finite dimensional $\alpha$-stable process with $\alpha \in (0,2)$.

The module cancellation problem asks whether, given modules $X$, $X^{\prime }$ and $Y$ over a ring $\Lambda$, the existence of an isomorphism $X\oplus Y\cong X^{\prime }\oplus Y$ implies that $X\cong X^{\prime }$. When $\Lambda$ is the integral group ring of a metacyclic group $G(p,q)$, results of Klingler show that the answer to this question is generally negative. By contrast, in this case we show that cancellation holds when $Y=\Lambda$ and $X$ is a generalized Swan module.

Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent perversities with some growth conditions, verifying $\overline{p}\left(1\right)=\overline{p}\left(2\right)=0$. King reproves this invariance by associating an intrinsic pseudomanifold $X^{\ast }$ to any pseudomanifold $X$. His proof consists of an isomorphism between the associated intersection homologies $H_{\ast }^{\overline{p}}\left(X\right)\cong H_{\ast }^{\overline{p}}\left(X^{\ast }\right)$ for any perversity $\overline{p}$ with the same growth conditions verifying $\overline{p}\left(1\right)\ge 0$.

In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, $\overline{p}$, which corresponds to the classical topological invariance if $\overline{p}$ is a GM-perversity. We also extend it to the tame intersection homology, a variation of the intersection homology, particularly suited for “large” perversities, if there is no singular strata on $X$ becoming regular in $X^{\ast }$. In particular, under the above conditions, the intersection homology and the tame intersection homology are invariant under a refinement of the stratification.

We identify all semidualizing modules over certain classes of ladder determinantal rings over a field $\mathsf{k}$. Specifically, given a ladder of variables $Y$, we show that the ring $\mathsf{k}\left[Y\right]/I_t\left(Y\right)$ has only trivial semidualizing modules up to isomorphism in the following cases: (1) $Y$ is a one-sided ladder, and (2) $Y$ is a two-sided ladder with $t=2$ and no coincidental inside corners.