The Reshetikhin–Turaev topological quantum field theories with gauge group ${SU}_2$ associate to any oriented surface $\Sigma$ a sequence of vector spaces $V_r\left(\Sigma \right)$ and to any simple closed curve $\gamma$ in $\Sigma$ a sequence of Hermitian operators $T_r^{\gamma }$ on the spaces $V_r\left(\Sigma \right)$. These operators are called curve operators and play a very important role in TQFT.

We show that the matrix elements of the operators $T_r^{\gamma }$ have an asymptotic expansion in orders of $1∕r$, and give a formula to compute the first two terms from trace functions, generalizing results of Marché and Paul for the punctured torus and the $4$–holed sphere to general surfaces.

We show that any smooth, closed, oriented, connected $4$–manifold can be trisected into three copies of ${♮}^k\left(S^1\times B^3\right)$, intersecting pairwise in $3$–dimensional handlebodies, with triple intersection a closed $2$–dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of $3$–manifolds. A trisection of a $4$–manifold $X$ arises from a Morse $2$–function $G:X\to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a $3$–manifold $Y$ arises from a Morse function $g:Y\to B^1$ and the obvious bisection of $B^1$.

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $E_{\infty }$–ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the nonperiodic and nonconnective $Tmf$. We find that $Pic\left(TMF\right)$ is cyclic of order $576$, generated by the suspension $\Sigma TMF$ (a result originally due to Hopkins), while $Pic\left(Tmf\right)=\mathbb{Z}\oplus \mathbb{Z}∕24$. In particular, we show that there exists an invertible $Tmf$–module which is not equivalent to a suspension of $Tmf$.

We extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$, and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.

We construct “barcodes” for the chain complexes over Novikov rings that arise in Novikov’s Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. Moreover, we prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and we use this to prove a $C^0$–robustness result for the fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather different from the standard methods of persistent homology, is based on a nonarchimedean singular value decomposition for the boundary operator of the chain complex.

We prove a conjectured decomposition of deformed ${\mathfrak{s}\mathfrak{l}}_N$ link homology, as well as an extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To this end, we use foam technology to give a completely combinatorial construction of Wu’s deformed colored ${\mathfrak{s}\mathfrak{l}}_N$ link homologies. By studying the underlying deformed higher representation-theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.

We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold $\left(M,\xi \right)$ admits a hypertight contact form ${\lambda }_0$ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on $\left(M,\xi \right)$ has positive topological entropy. Using this result, we provide numerous new examples of contact $3$–manifolds on which every Reeb flow has positive topological entropy.

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit $1$–parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a $1$–parameter family of pairwise nonconjugate spherical CR uniformizations of the figure eight knot complement.

We show that a decorated knot concordance $\mathcal{C}$ from $K$ to $K^{\prime }$ induces a homomorphism $F_{\mathcal{C}}$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}\left(S^3\right)\cong {\mathbb{Z}}_2$ that agrees with $F_{\mathcal{C}}$ on the $E^1$ page and is the identity on the $E^{\infty }$ page. It follows that $F_{\mathcal{C}}$ is nonvanishing on ${\widehat{HFK}}_0\left(K,\tau \left(K\right)\right)$. We also obtain an invariant of slice disks in homology 4–balls bounding $S^3$.

If $\mathcal{C}$ is invertible, then $F_{\mathcal{C}}$ is injective, hence

$$dim{\widehat{HFK}}_j\left(K,i\right)\le dim{\widehat{HFK}}_j\left(K^{\prime },i\right)$$

for every $i,j\in \mathbb{Z}$. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K^{\prime }$, then $g\left(K\right)\le g\left(K^{\prime }\right)$, where $g$ denotes the Seifert genus. Furthermore, if $g\left(K\right)=g\left(K^{\prime }\right)$ and $K^{\prime }$ is fibred, then so is $K$.