A conjecture, known as the Shokurov–Kollár connectedness principle, predicts the following. Let $(\mathit{X},\mathit{B})$ be a pair, and let $\mathit{f}:\mathit{X}\to \mathit{S}$ be a contraction with $-({\mathit{K}}_{\mathit{X}}+\mathit{B})$ nef over S; then, for any point $\mathit{s}\in \mathit{S}$, the intersection ${\mathit{f}}^{-1}(\mathit{s})\cap Nklt(\mathit{X},\mathit{B})$ has at most two connected components, where $Nklt(\mathit{X},\mathit{B})$ denotes the non-klt locus of $(\mathit{X},\mathit{B})$. This conjecture has been extensively studied in characteristic zero, and it has been recently settled in that context. In this work, we consider this conjecture in the setup of positive characteristic algebraic geometry. We prove that this conjecture holds for threefolds in positive and mixed characteristic, where the residue fields are assumed to have characteristic $\mathit{p}>5$. Under the same assumptions, we characterize the cases in which $Nklt(\mathit{X},\mathit{B})$ fails to be connected.

In [Xu16], Xu defines the dlt motivic zeta function associated with a regular function f on a smooth variety X over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser [DL98]. The dlt motivic zeta function is defined on a dlt modification via a Denef–Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in [Xu16], and that it is therefore not well defined.

Let k be a field of positive characteristic p with degree of imperfection 1 and a nontrivial discrete valuation. We show that the Galois and flat cohomology of unipotent k-groups over k of dimension $<\mathit{p}-1$ are finite (in fact, trivial) if and only if they are k-split. We give some examples to show that the bound $\mathit{p}-1$ is best possible and present several applications to local-global principles.

We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces that recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk–Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the ham sandwich theorem and (iv) the Lyusternik–Shnirel’man–Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.

Given $\mathrm{\Sigma }\subset \mathit{R}:=\mathbb{K}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{k}}]$, where $\mathbb{K}$ is a field, any finite collection of linear forms, some possibly proportional, and any $1\le \mathit{a}\le |\mathrm{\Sigma }|$, we prove that ${\mathit{I}}_{\mathit{a}}(\mathrm{\Sigma })$, the ideal generated by all a-fold products of Σ, has linear graded free resolution, and we exhibit a combinatorial primary decomposition for ${\mathit{I}}_{\mathit{a}}(\mathrm{\Sigma })$. The linear graded free resolution allows us to determine a generating set for the defining ideal of the Orlik–Terao algebra of the second order of a line arrangement in ${\mathbb{P}}_{\mathbb{K}}^2$, and to conclude that for the case $\mathit{k}=3$, and Σ defining such a line arrangement, the ideal ${\mathit{I}}_{|\mathrm{\Sigma }|-2}(\mathrm{\Sigma })$ is of fiber type. Finally, with the help of the primary decomposition, we prove several conjectures of symbolic powers for the defining ideals of star configurations of any codimension c, a special class of ideals generated by a-fold products of linear forms.

A ${\mathbb{Z}}^{\mathit{d}}$-graded differential R-module is a ${\mathbb{Z}}^{\mathit{d}}$-graded R-module D with a morphism $\mathit{\delta }:\mathit{D}\to \mathit{D}$ such that ${\mathit{\delta }}^2=0$. For $\mathit{R}=\mathit{k}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{d}}]$, this paper establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology $ker\mathit{\delta }/im\mathit{\delta }$ of D is nonzero and finite dimensional over k, there is an inequality ${rk}_{\mathit{R}}\mathit{D}\ge 2^{\mathit{d}}$.

We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.

The level N elliptic KZB connection is a flat connection over the universal elliptic curve in level N with its N-torsion sections removed. Its fiber over the point $(\mathit{E},\mathit{x})$ is the unipotent completion of ${\mathit{\pi }}_1(\mathit{E}-\mathit{E}[\mathit{N}],\mathit{x})$. It was constructed by Calaque and Gonzalez. In this paper, we show that the connection underlies an admissible variation of mixed Hodge structure and that it degenerates to the cyclotomic KZ connection over the singular fibers of the compactified universal elliptic curve. These are the first steps in a larger project to compute the action of the Galois group of mixed Tate motives unramified over $\mathbb{Z}[{\mathit{\mu }}_{\mathit{N}},1/\mathit{N}]$ on the unipotent fundamental group of ${\mathbb{P}}^1-\{0,{\mathit{\mu }}_{\mathit{N}},\infty \}$ and to better understand Goncharov’s higher cyclotomy.