In this paper, we determine when a natural torsor arising in the work of Kato and Usui on partial compactification of period domains of pure Hodge structure is trivial, and we give an application to cycle spaces.

Following the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

We introduce an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation tight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

We define a $p$-adic character to be a continuous homomorphism from $1+t{\mathbb{F}}_q\left[\right[t\left]\right]$ to ${\mathbb{Z}}_p^{\ast }$. For $p>2$, we use the ring of big Witt vectors over ${\mathbb{F}}_q$ to exhibit a bijection between $p$-adic characters and sequences $(c_i{)}_{(i,p)=1}$ of elements in ${\mathbb{Z}}_q$, indexed by natural numbers relatively prime to $p$, and for which ${lim\hspace{0.17em}}_{i\to \infty }{c}_i=0$. To such a $p$-adic character we associate an $L$-function, and we prove that this $L$-function is $p$-adic meromorphic if the corresponding sequence $\left(c_i\right)$ is overconvergent. If more generally the sequence is $Clog$-convergent, we show that the associated $L$-function is meromorphic in the open disk of radius $q^C$. Finally, we exhibit examples of $Clog$-convergent sequences with associated $L$-functions which are not meromorphic in the disk of radius $q^{C+ϵ}$ for any $ϵ>0$.

We introduce a class of ideals generated by a set of $2$-minors of an $(m\times n)$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

Let $S=K[x_1,x_2,\dots ,x_n]$ be a polynomial ring over a field $K$. Let $\Delta$ be a simplicial complex whose vertex set is contained in $\{1,2,\dots ,n\}$. For an integer $k\ge 0$, we investigate the $k$-Buchsbaum property of residue class rings $S/I^{\left(t\right)}$ and $S/I^t$ for the Stanley–Reisner ideal $I=I_{\Delta }$. We characterize the $k$-Buchsbaumness of such rings in terms of the simplicial complex $\Delta$ and the power $t$. We also give a characterization in the case where $I$ is the edge ideal of a simple graph.

Let $\mathcal{X}$ be a smooth proper Deligne–Mumford stack over $\mathbb{C}$. One can define twisted orbifold Gromov–Witten invariants of $\mathcal{X}$ by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable maps ${\mathcal{X}}_{g,n,d}$, cupping them with evaluation and cotangent line classes, and then integrating against the virtual fundamental class. These are more general than the twisted invariants introduced by Tseng. We express the generating series of the twisted invariants in terms of the generating series of the untwisted ones. We derive the corollaries which are used in a paper with Givental about the quantum $K$-theory of a complex compact manifold $X$.