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 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.

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 ${\mathrm{HP}}_{\mathrm{\#}}$.

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].

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].

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.

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.

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 ${\mathit{HFK}}_{\mathit{n}}$.