Given a unipotent bundle of smooth manifolds we construct its secondary transfer map and show that this map determines the higher smooth torsion of the bundle. This approach to higher torsion provides a new perspective on some of its properties. In particular it yields in a natural way a formula for torsion of a composition of two bundles.

Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed $\pm 1$ according to the parity of the covering surface. These numbers were first defined by Eskin, Okounkov and Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov–Witten invariants of complex $2$–folds by work of Lee and Parker and work of Maulik and Pandharipande. In this paper, we construct a (spin) TQFT which computes these numbers, and deduce a formula for any genus in terms of the combinatorics of the Sergeev algebra, generalizing the formula of Eskin, Okounkov and Pandharipande. During the construction, we describe a procedure for averaging any TQFT over finite covering spaces based on the finite path integrals of Freed, Hopkins, Lurie and Teleman.

Burago and Kleiner and, independently, McMullen, gave examples of Delone sets (that is, subsets of Euclidean space that are discrete and separated in a uniform way) that are non-bi-Lipschitz equivalent to the standard lattice. We refine their methods of construction via a discretization technique, thus giving the first examples of Delone sets as above that are also repetitive, in the sense that a translated copy of each patch appears in every large enough ball.

We define a class of noncompact Fano toric manifolds which we call admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets.

We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, $S{H}^{\ast }\left(M\right)$ is obtained from $Q{H}^{\ast }\left(M\right)$ by localizing at the toric divisors. We give explicit presentations of $S{H}^{\ast }\left(M\right)$ and $Q{H}^{\ast }\left(M\right)$, using ideas of Batyrev, McDuff and Tolman.

Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, ie is split-generated by the natural Lagrangian torus fibers of the moment map taken with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite.

We prove a generic generation theorem: a generic deformation of the monotone toric symplectic form defines a local system for which the twisted wrapped Fukaya category satisfies the toric generation criterion. This theorem, together with a limiting argument about continuity of eigenspaces, are used to prove the untwisted generation results.

We prove that for any closed Fano toric manifold, and a generic local system, the twisted Fukaya category satisfies the toric generation criterion. If the superpotential is Morse (or assuming semisimplicity), also the untwisted Fukaya category satisfies the criterion.

The key ingredients are nonvanishing results for the open-closed string map, using tools from the paper by Ritter and Smith; we also prove a conjecture from that paper that any monotone toric negative line bundle contains a nondisplaceable monotone Lagrangian torus. The above presentation results require foundational work: we extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in $S{H}^{\ast }\left(M\right)$. Computing $S{H}^{\ast }\left(M\right)$ is notoriously hard and there are very few known examples beyond the cases of cotangent bundles and subcritical Stein manifolds. So this computation is significant in itself, as well as being the key ingredient in proving the above results in homological mirror symmetry.

We show that the cusp volume of a hyperbolic alternating knot can be bounded above and below in terms of the twist number of an alternating diagram of the knot. This leads to diagrammatic estimates on lengths of slopes, and has some applications to Dehn surgery. Another consequence is that there is a universal lower bound on the cusp density of hyperbolic alternating knots.

We give the first examples of finite groups $G$ such that the Chow ring of the classifying space $BG$ depends on the base field, even for fields containing the algebraic closure of $\mathbb{Q}$. As a tool, we give several characterizations of the varieties that satisfy Künneth properties for Chow groups or motivic homology.

We define the (compactly supported) motive of a quotient stack in Voevodsky’s derived category of motives. This makes it possible to ask when the motive of $BG$ is mixed Tate, which is equivalent to the motivic Künneth property. We prove that $BG$ is mixed Tate for various “well-behaved” finite groups $G$, such as the finite general linear groups in cross-characteristic and the symmetric groups.

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQEs) that governs the Gromov–Witten invariants of the Fano orbifold projective curve ${\mathbb{P}}_{a_1,a_2,a_3}^1$. The vertex operators in our construction are given in terms of the $K$–theory of ${\mathbb{P}}_{a_1,a_2,a_3}^1$ via Iritani’s $\Gamma$–class modification of the Chern character map. We also identify our HQEs with an appropriate Kac–Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of ${\mathbb{P}}^1$ to all Fano orbifold curves.

Using Taubes’ periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We use Heegaard Floer homology to give an obstruction to a homology sphere being surgery on a knot, and then use this obstruction to construct infinitely many small Seifert fibered examples.

Inspired by Le Calvez’s theory of transverse foliations for dynamical systems on surfaces, we introduce a dynamical invariant, denoted by $\mathcal{N}$, for Hamiltonians on any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces.

Along the way, we obtain several results of independent interest: we show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with $\mathcal{N}$ on autonomous Hamiltonians, thus establishing a certain uniqueness result for spectral invariants; we obtain a “max formula” for spectral invariants on aspherical manifolds; we give a very simple description of the Entov–Polterovich partial quasi-state on aspherical surfaces, and we characterize the heavy and super-heavy subsets of such surfaces.

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic $4$–manifolds as multisections, such as Seiberg–Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the $4$–ball. Various problems regarding the topology of symplectic $4$–manifolds, such as the smooth classification of symplectic Calabi–Yau $4$–manifolds, can be translated to combinatorial problems in this manner. After producing special monodromy factorizations of Lefschetz pencils on symplectic Calabi–Yau homotopy $K3$ and Enriques surfaces, and introducing monodromy substitutions tailored for generating multisections, we obtain several novel applications, allowing us to construct: new counterexamples to Stipsicz’s conjecture on fiber sum indecomposable Lefschetz fibrations, nonisomorphic Lefschetz pencils of the same genera on the same new symplectic $4$–manifolds, the very first examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.

In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.

This note corrects some omissions in Section 2 of the paper “Lipschitz connectivity and filling invariants in solvable groups and buildings” [Geom. Topol. 18 (2014) 2375–2417].