We investigate the relations between algebraic structures, spectral invariants and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov-type low-energy intersection phenomena, and nondegeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.

The Pontryagin classes of an $m$–dimensional real vector bundle satisfy well-known vanishing relations due to their close relationship with the Chern classes of the complexified vector bundle. It is also known that the rational Pontryagin classes qualify as characteristic classes for Euclidean fiber bundles, ie bundles whose fibers are homeomorphic to $m$–dimensional Euclidean space. In this more general setting of fiber bundles, the vanishing relations which one has for vector bundles fail to hold.

We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension-$0$ quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension $0$ and $1$ quotients on surfaces (using also universality, torus localization and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa–Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension-$0$ quotients on surfaces. Dimension-$1$ quotients are studied on K3 surfaces and surfaces of general type, with connections to the Kawai–Yoshioka formula and the Seiberg–Witten invariants, respectively. The dimension-$1$ theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss–Catalan numbers, which is of independent interest.

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$–space ${ℂ\mathbb{P}}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into ${ℂ\mathbb{P}}^3$ is path-connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi–Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into ${ℂ\mathbb{P}}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $\pi :{ℂ\mathbb{P}}^3\to {\mathbb{𝕊}}^4$ onto the $4$–sphere, immersed holomorphic Legendrian curves $M\to {ℂ\mathbb{P}}^3$ are in bijective correspondence with superminimal immersions $M\to {\mathbb{𝕊}}^4$ of positive spin, according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in ${\mathbb{𝕊}}^4$. In particular, superminimal immersions into ${\mathbb{𝕊}}^4$ satisfy the Runge approximation theorem and the Calabi–Yau property.

We prove a formula for the motive of the stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s triangulated category of mixed motives with rational coefficients.

Let $K$ be a knot in an integral homology $3$–sphere $Y$ and $\mathrm{\Sigma }$ the corresponding $n$–fold cyclic branched cover. Assuming that $\mathrm{\Sigma }$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $\mathrm{\Sigma }$. The proof relies on a careful analysis of the Seiberg–Witten equations on $3$–orbifolds and of various $\eta$–invariants. We give several applications of our formula: (1) we calculate the Seiberg–Witten and Furuta–Ohta invariants for the mapping tori of all semifree actions of $\mathbb{Z}∕n$ on integral homology $3$–spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in $S^3$ being an $L$–space; and (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.

We extend the concept of singular Ricci flow by Kleiner and Lott from 3D compact manifolds to 3D complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3D complete Riemannian manifold with nonnegative Ricci curvature, there exists a smooth Ricci flow starting from it. This partially confirms a conjecture by Topping.

A recent theorem of Hyde proves that the factorization statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of $n$ distinct ordered points in ${ℝ}^3$. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde’s theorem as an instance of the Grothendieck–Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to any Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

Properly embedded simplices in a convex divisible domain $\mathrm{\Omega }\subset ℝ{\text{P}}^d$ behave somewhat like flats in Riemannian manifolds (Geom. Dedicata 33 (1990) 251–263), so we call them flats. We show that the set of codimension-$1$ flats has image which is a finite collection of disjoint virtual $\left(d-1\right)$–tori in the compact quotient manifold. If this collection of virtual tori is nonempty, then the components of its complement are cusped convex projective manifolds with type $d$ cusps.