We show that the set of augmentations of the Chekanov–Eliashberg algebra of a Legendrian link underlies the structure of a unital $A_{\infty }$–category. This differs from the nonunital category constructed by Bourgeois and Chantraine (J. Symplectic Geom. 12 (2014) 553–583), but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by Shende, Treumann and Zaslow (Invent. Math. 207 (2017) 1031–1133), who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the $x$–line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.

We construct the first examples of asymmetric L–space knots in $S^3$. More specifically, we exhibit a construction of hyperbolic knots in $S^3$ with both (i) a surgery that may be realized as a surgery on a strongly invertible link such that the result of the surgery is the double branched cover of an alternating link and (ii) trivial isometry group. In particular, this produces L–space knots in $S^3$ which are not strongly invertible. The construction also immediately extends to produce asymmetric L–space knots in any lens space, including $S^1\times S^2$.

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X∕G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge–de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant $K$–theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge–de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau–Ginzburg models.

Given a knot $K\subset S^3$, let $u^{-}\left(K\right)$ (respectively, $u^{+}\left(K\right)$) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for $K$. We use knot Floer homology to construct the invariants ${\mathfrak{𝔩}}^{-}\left(K\right)$, ${\mathfrak{𝔩}}^{+}\left(K\right)$ and $\mathfrak{𝔩}\left(K\right)$, which give lower bounds on $u^{-}\left(K\right)$, $u^{+}\left(K\right)$ and the unknotting number $u\left(K\right)$, respectively. The invariant $\mathfrak{𝔩}\left(K\right)$ only vanishes for the unknot, and satisfies $\mathfrak{𝔩}\left(K\right)\ge {\nu }^{+}\left(K\right)$, while the difference $\mathfrak{𝔩}\left(K\right)-{\nu }^{+}\left(K\right)$ can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.

We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in $3$–dimensional contact topology. This notion allows us to define what an Engel overtwisted disc is, and to prove a complete $h$–principle for overtwisted Engel structures with fixed overtwisted disc.

We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger–Yau–Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma conjecture for Batyrev pairs of mirror Calabi–Yau hypersurfaces.

We prove that the minimum number of critical points of a Weinstein Morse function on a Weinstein domain of dimension at least six is at most two more than the minimum number of critical points of a smooth Morse function on that domain; if the domain has nonzero middle-dimensional homology, these two numbers agree. There is also an upper bound on the number of gradient trajectories between critical points in smoothly trivial Weinstein cobordisms. As an application, we show that the number of generators for the Grothendieck group of the wrapped Fukaya category is at most the number of generators for singular cohomology and hence vanishes for any Weinstein ball. We also give a topological obstruction to the existence of finite-dimensional representations of the Chekanov–Eliashberg DGA for Legendrians.

We exhibit the first examples of compact, orientable, hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n\ge 4$.

The core of the argument is the construction of a compact, oriented, hyperbolic $4$–manifold $M$ that contains a surface $S$ of genus $3$ with self-intersection $1$. The $4$–manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right-angled $120$–cells along a pattern inspired by the minimum trisection of ${ℂ\mathbb{P}}^2$.

The manifold $M$ is also the first example of a compact, orientable, hyperbolic $4$–manifold satisfying either of these conditions:

• $H_2\left(M,\mathbb{Z}\right)$ is not generated by geodesically immersed surfaces.

• There is a covering $\tilde{M}$ that is a nontrivial bundle over a compact surface.