We establish the logarithmic foundation for compactifying the moduli stacks of the gauged linear sigma model using stable log maps. We then illustrate our method via the key example of Witten’s $r$–spin class to construct a proper moduli stack with a reduced perfect obstruction theory whose virtual cycle recovers the $r$–spin virtual cycle of Chang, Li and Li. Indeed, our construction of the reduced virtual cycle is built upon their work by appropriately extending and modifying the Kiem–Li cosection along certain logarithmic boundary. In a follow-up article, we push the technique to a general situation.

One motivation of our construction is to fit the gauged linear sigma model in the broader setting of Gromov–Witten theory so that powerful tools such as virtual localization can be applied. A project along this line is currently in progress, leading to applications including computing loci of holomorphic differentials, and calculating higher-genus Gromov–Witten invariants of quintic threefolds.

Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the trivial band.

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel–Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperkähler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel–Mukai variety is equivalent to the derived category of a K3 surface.

Gromov’s systolic inequality asserts that the length, ${\text{sys}}_1(M^n)$, of the shortest noncontractible curve in a closed essential Riemannian manifold $M^n$ does not exceed $c(n){ \text{vol}}^{1∕n}(M^n)$ for some constant $c(n)$. (Essential manifolds is a class of non–simply connected manifolds that includes all non–simply connected closed surfaces, tori and projective spaces.)

Here we prove that all closed essential Riemannian manifolds satisfy ${\text{sys}}_1(M^n)\le n{\text{vol} }^{1∕n}(M^n)$. (The best previously known upper bound for $c(n)$ was exponential in $n$.)

We similarly improve a number of related inequalities. We also give a qualitative strengthening of Guth’s theorem (2011, 2017) asserting that if volumes of all metric balls of radius $r$ in a closed Riemannian manifold $M^n$ do not exceed ${(r∕c(n))}^n$, then the $(n-1)$–dimensional Urysohn width of the manifold does not exceed $r$. In our version the assumption of Guth’s theorem is relaxed to the assumption that for each $x\in M^n$ there exists $\varrho (x)\in (0,r]$ such that the volume of the metric ball $B(x,\varrho (x))$ does not exceed ${(\varrho (x)∕c(n))}^n$, where one can take $c(n)=\frac{1}{2}n$.

A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two.

Given $d\in \mathbb{N}$, $g\in \mathbb{N}\cup \{0\}$ and an integral vector $\kappa =(k_1,\dots ,k_n)$ such that and $k_1+\cdots +k_n=d(2g-2)$, let ${\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )$ denote the moduli space of meromorphic $d$–differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\kappa$. We show that ${\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )$ carries a canonical volume form that is parallel with respect to its affine complex manifold (orbifold) structure, and that the total volume of $\mathbb{P}{\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )={\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )∕{ℂ}^{\ast }$ with respect to the measure induced by this volume form is finite.

Let $\mathcal{𝒳}∕S$ be a flat algebraic stack of finite presentation. We define a new étale fundamental pro-groupoid ${\mathrm{\Pi }}_1(\mathcal{𝒳}∕S)$, generalizing Grothendieck’s enlarged étale fundamental group from SGA 3 to the relative situation. When $S$ is of equal positive characteristic $p$, we prove that ${\mathrm{\Pi }}_1(\mathcal{𝒳}∕S)$ naturally arises as colimit of the system of relative Frobenius morphisms $\mathcal{𝒳}\to {\mathcal{𝒳}}^{p∕S}\to {\mathcal{𝒳}}^{p^2∕S}\to \cdots$ in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between ${\mathrm{\Pi }}_1$ and the stack $\text{Fdiv}$ of $F$–divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.