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 ${rk}_{\mathit{R}}\mathit{D}\ge 2^{\mathit{d}}$.

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.

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.

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 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.

The aim of this paper is a construction of new explicit annihilators of the minus part of the ideal class group of an imaginary Abelian number field M, that is, annihilators that are outside of the Stickelberger ideal, their usual source. This construction works for quite a large class of Abelian fields M; a sufficient condition to get a new annihilator is that there is an odd prime $\ell \mid [\mathit{M}:\mathbb{Q}]$, unramified in $\mathit{M}/\mathbb{Q}$, and two primes q and ${\mathit{q}}^{\prime }$ ramifying in $\mathit{M}/\mathbb{Q}$, having their decomposition groups ${\mathit{D}}_{\mathit{q}}\subseteq {\mathit{D}}_{{\mathit{q}}^{\prime }}$ cyclic of ℓ-power order.

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

We improve on the result of Hacon and Witaszek by showing that the MMP for semistable fourfolds in mixed characteristic terminates in several new situations. In particular, we show the validity of the MMP for strictly semistable fourfolds over excellent Dedekind schemes globally when the residue fields are perfect and have characteristics $\mathit{p}>5$.

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.

The ${\tilde{\mathit{A}}}_{\mathit{n}}$ Coxeter groups are known to not be systolic or cocompactly cubulated for $\mathit{n}\ge 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic complexes. To prove weak modularity, we describe the canonical embeddings of the 1-skeleta of ${\tilde{\mathit{A}}}_{\mathit{n}}$ Coxeter complexes into the Euclidean spaces ${\mathbb{R}}^{\mathit{n}+1}$. We also prove that the 1-skeleta of buildings of type ${\tilde{\mathit{A}}}_3$ are weakly modular.

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 abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple $(\mathit{G},\mathit{X},\mathit{H})$ consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H that is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of $\mathrm{Out}({\mathit{F}}_{\mathit{n}})$, and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich–Rafi and Dowdall–Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.

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.

In response to W. Kuperberg’s question, for every dimension at least two, there are uncountably many compact metric spaces, each of which (1) has the shape of the sphere of that dimension and (2) contains an infinite sequence of pairwise disjoint closed subsets irreducibly having nonzero top-dimensional integer Čech–Alexander–Spanier cohomology.

We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on p-forms by computing the heat invariants associated with the p-spectrum. We show that the heat invariants of the 0-spectrum together with those of the 1-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $\le 3$. This is enough to distinguish orbifolds from manifolds for dimension $\le 3$.

For a finitely generated group G, let $\mathit{H}(\mathit{G})$ denote Bowditch’s taut loop length spectrum. We prove that if $\mathit{G}=(\mathit{A}\ast \mathit{B})/⟨⟨\mathcal{R}⟩⟩$ is a ${\mathit{C}}^{\prime }(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $\mathit{H}(\mathit{G})$ is equivalent to $\mathit{H}(\mathit{A})\cup \mathit{H}(\mathit{B})$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch’s construction of continuously many non-quasi-isometric ${\mathit{C}}^{\prime }(1/6)$ small cancellation 2-generated groups to obtain our main result: Let $\mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups:

(1) $\{\mathit{G}\in \mathcal{G}:\underset{\_}{\mathrm{cd}}(\mathit{G})=2\text{and}\underset{\_}{\mathrm{gd}}(\mathit{G})=3\}$;

(2) $\{\mathit{G}\in \mathcal{G}:\underset{\_}{\underset{\_}{\mathrm{cd}}}(\mathit{G})=2\text{and}\underset{\_}{\underset{\_}{\mathrm{gd}}}(\mathit{G})=3\}$;

(3) $\{\mathit{G}\in \mathcal{G}:{\mathrm{cd}}_{\mathbb{Q}}(\mathit{G})=2\text{and}{\mathrm{cd}}_{\mathbb{Z}}(\mathit{G})=3\}$.

On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented ${\mathit{C}}^{\prime }(1/12)$ small cancellation quotients of free products; in particular, this produces new examples of groups exhibiting an Eilenberg–Ganea phenomenon for families.

We also show that if there is a finitely presented counterexample to the Eilenberg–Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counterexamples.

We establish a congruence satisfied by the integer group determinants for the non-Abelian Heisenberg group of order ${\mathit{p}}^3$. We characterize all determinant values coprime to p, give sharp divisibility conditions for multiples of p, and determine all values when $\mathit{p}=3$. We also provide new sharp conditions on the power of p dividing the group determinants for ${\mathbb{Z}}_{\mathit{p}}^2$.

For a finite group, the integer group determinants can be understood as corresponding to Lind’s generalization of the Mahler measure. We speculate on the Lind–Mahler measure for the discrete Heisenberg group and for two other infinite non-Abelian groups arising from symmetries of the plane and 3-space.

We study trisections of smooth compact nonorientable 4-manifolds and introduce trisections of nonorientable 4-manifolds with boundary. In particular, we prove a nonorientable analogue of a classical theorem of Laudenbach–Poénaru and analogues for some well-known theorems from 3-manifold topology. As a consequence, trisection diagrams and Kirby diagrams of closed nonorientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of nonorientable 4-manifolds with many examples.

Inspired by a conjecture of Vladimir Maz’ya on Φ-inequalities in the spirit of Bourgain and Brezis, we establish some Φ-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of Φ-inequalities for differential operators. The proofs rest on new simple Bellman functions.

Felix Klein in the course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of 60 points in ${\mathbb{P}}^3$. This configuration has appeared in various guises, perhaps most notably as the configuration of points dual to the 60 reflection planes in the group ${\mathit{G}}_{31}$ in the Shephard–Todd list.

In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated with the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities $4,2$ and 2. Secondly, Chiantini and Migliore observed in 2020 that there are nontrivial sets of points in ${\mathbb{P}}^3$ with the surprising property that their general projection to ${\mathbb{P}}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in ${\mathbb{P}}^3$, which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.

The Friedman–Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, as defined by Mann and Rafi, and with 0 or infinite genus, we show that when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.

We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.

In this paper, we provide several results regarding the structure of derived categories of (nested) Hilbert schemes of points. We show that the criteria of Krug–Sosna and Addington for the universal ideal sheaf functor to be fully faithful resp. a $\mathbb{P}$-functor are sharp. Then we show how to embed multiple copies of the derived category of the surface using these fully faithful functors. We also give a semiorthogonal decomposition for the nested Hilbert scheme of points on a surface, and finally we give an alternative proof of a semiorthogonal decomposition due to Toda for the symmetric product of a curve.

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective.

We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding.

Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs, we use ideas that go back to a paper by Banks, Conflitti, and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem established by Bourgain, Demeter, and Guth (2016) and Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a ${\mathit{C}}^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with ${\mathit{C}}^2$ boundary $\mathit{D}\subset {\mathbb{C}}^{\mathit{n}}$, we show the existence of global holomorphic coordinate systems defined on $\bar{\mathit{D}}$ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small ${\mathit{C}}^2$ perturbation of $\partial \mathit{D}$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a ${\mathit{C}}^2$ real hypersurface $\mathit{M}\subset {\mathbb{C}}^{\mathit{n}}$, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.

We show that for any manifold M, compact or homeomorphic to the interior of a compact manifold, the homeomorphism group of M has automatic continuity. The same is true for the relative homeomorphism group $Homeo(\mathit{M},\mathit{X})$ where X is homeomorphic to the union of a Cantor set and a (possibly, empty) finite set, and the big mapping class group $Homeo(\mathit{M},\mathit{X})/{Homeo}_0(\mathit{M},\mathit{X})$. In other words, the algebraic structure of these groups is extremely rigid and determines their topology in a very strong way.

We consider the distribution in residue classes modulo primes p of Euler’s totient function $\mathit{\varphi }(\mathit{n})$ and the sum-of-proper-divisors function $\mathit{s}(\mathit{n}):=\mathit{\sigma }(\mathit{n})-\mathit{n}$. We prove that the values of $\mathit{\varphi }(\mathit{n})$ for $\mathit{n}\le \mathit{x}$ that are coprime to p are asymptotically uniformly distributed among the $\mathit{p}-1$ coprime residue classes modulo p, uniformly for $5\le \mathit{p}\le {(log\mathit{x})}^{\mathit{A}}$ (with A fixed but arbitrary). We also show that the values of $\mathit{s}(\mathit{n})$ for n composite are uniformly distributed among all p residue classes modulo every $\mathit{p}\le {(log\mathit{x})}^{\mathit{A}}$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.

Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin’s “$\mathfrak{sl}(\mathit{n})$-like” Heegaard Floer knot invariants ${\mathit{HFK}}_{\mathit{n}}$ recover both Alexander polynomial evaluations and $\mathfrak{sl}(\mathit{n})$ polynomial evaluations at certain roots of unity for links in ${\mathit{S}}^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $\mathfrak{sl}(\mathit{n})$ homology and ${\mathit{HFK}}_{\mathit{n}}$.

In this paper, we compare the notions of double sliceness for links. The main result is showing that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end, we prove a result classifying Seifert fibered spaces that admit a smooth embedding into integer homology ${\mathit{S}}^1\times {\mathit{S}}^3$s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links are also given.

We explore the structure of invariant measures on compact Kähler manifolds with Hamiltonian torus actions. We derive a formula for conditional measures on the orbits of the complex torus and use it to prove a conditional statement about uniqueness of solutions to the g-Monge–Ampère equation.

Let K be a compact convex domain in the Euclidean plane. The mixed area $\mathit{A}(\mathit{K},-\mathit{K})$ of K and $-\mathit{K}$ can be bounded from above by $1/(6\sqrt{3})\mathit{L}{(\mathit{K})}^2$, where $\mathit{L}(\mathit{K})$ is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil [5]. They also showed that if K is a polygon, then equality holds if and only if K is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality $6\sqrt{3}\mathit{A}(\mathit{K},-\mathit{K})\le \mathit{L}{(\mathit{K})}^2$.

We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden, Livingston, and Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.

Croke–Kleiner admissible groups first introduced by Croke and Kleiner [CK02] belong to a particular class of graphs of groups, which generalizes fundamental groups of three-dimensional graph manifolds. In this paper, we show that if G is a Croke–Kleiner admissible group, then a finitely generated subgroup of G has finite height if and only if it is strongly quasiconvex. We also show that if $\mathit{G}\curvearrowright \mathit{X}$ is a flip CKA action, then G is quasiisometrically embedded into a finite product of quasitrees. With further assumption on the vertex groups of the flip CKA action $\mathit{G}\curvearrowright \mathit{X}$, we show that G satisfies property (QT) introduced by Bestvina, Bromberg, and Fujiwara [BBF21].

A knot (or link) in bridge position is said to be perturbed if there exists a cancelling pair of bridge disks, which gives rise to a lower index bridge position. For some classes of knots, every nonminimal bridge position is perturbed. We study whether such a property is preserved by cabling operation. In this paper, we show that the property is preserved for 2-cable links, that is, if every nonminimal bridge position of a knot K is perturbed, then every nonminimal bridge position of a 2-cable link L of K is also perturbed.

Following Haken [Ha] and Casson and Gordon [CG], it was shown in [Sc] that given a reducing sphere or ∂-reducing disk E in a Heegaard split manifold M, the Heegaard surface T can be isotoped so that it intersects E in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of

∙ isotopies of T rel E,

∙ pushing a stabilizing pair of T through E, and

∙ eyegelass twists of T.

This last move is inspired by one of Powell’s proposed generators for the Goeritz group [Po].

It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieve this bound are called Castelnuovo. On the other hand, a lattice polytope is called Castelnuovo if the associated polarized toric variety is Castelnuovo. Kawaguchi characterized Castelnuovo polytopes having interior lattice points in terms of their ${\mathit{h}}^{\ast }$-vectors. In this paper, as a generalization of this result, we present a characterization of all Castelnuovo polytopes. Finally, as an application of our characterization, we give a sufficient criterion for a lattice polytope to be IDP.

We analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree $(\mathit{n},1)$ case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.

We study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semiexplicit description of the locus of degree $\mathit{n}+2$ rational curves in ${\mathbb{P}}^{\mathit{n}}$ of arithmetic genus two that admit a Khovanskii-finite valuation. Furthermore, we describe an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, we show that rational curves with a single unibranch singularity are always Khovanskii-finite if their arithmetic genus is sufficiently small.

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri [LS84]. This result was extended to the other classical types by Lakshmibai [Lak95, Lak00b], and [Lak00a]. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule $\mathit{G}/\mathit{P}$. Our results extend beyond cominuscule $\mathit{G}/\mathit{P}$; they describe the tangent space to any Schubert variety in $\mathit{G}/\mathit{B}$ at a point $\mathit{x}\mathit{B}$, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of Kazhdan–Lusztig varieties, and then to recover results for Schubert varieties. Our proof uses a relationship between weights of the tangent space of a variety with torus action and factors of the class of the variety in torus equivariant K-theory. The proof relies on a formula for Schubert classes in equivariant K-theory due to Graham [Gra02] and Willems [Wil06] and on a theorem on subword complexes due to Knutson and Miller [KM04, KM05].

We establish a relationship between $\mathrm{HP}(\mathit{Y})$, the Abouzaid–Manolescu sheaf-theoretic $\mathrm{SL}(2,\mathbb{C})$ Floer cohomology for a surgery Y on a small knot in ${\mathit{S}}^3$ and Curtis’ $\mathrm{SL}(2,\mathbb{C})$ Casson invariant. We use this to compute $\mathrm{HP}$ for most surgeries on two-bridge knots. We also compute $\mathrm{HP}$ for surgeries on two nonsmall knots, the granny and square knots. We provide a partial calculation of the framed sheaf-theoretic $\mathrm{SL}(2,\mathbb{C})$ Floer cohomology ${\mathrm{HP}}_{\mathrm{\#}}(\mathit{Y})$ for surgeries on two-bridge knots and apply the data we obtain to show the nonexistence of a surgery exact triangle for ${\mathrm{HP}}_{\mathrm{\#}}$.

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan, and Kazachkov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.