In this extended note we give a precise definition of fully extended topological field theories à la Lurie. Using complete $n$–fold Segal spaces as a model, we construct an $\left(\infty ,n\right)$–category of $n$–dimensional bordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of bordisms.

This paper introduces parametrized homology, a continuous-parameter generalization of levelset zigzag persistent homology that captures the behavior of the homology of the fibers of a real-valued function on a topological space. This information is encoded as a “barcode” of real intervals, each corresponding to a homological feature supported over that interval; or, equivalently, as a persistence diagram. Points in the persistence diagram are classified algebraically into four classes; geometrically, the classes identify the distinct ways in which homological features perish at the boundaries of their interval of persistence. We study the conditions under which spaces fibered over the real line have a well-defined parametrized homology; we establish the stability of these invariants and we show how the four classes of persistence diagram correspond to the four diagrams that appear in the theory of extended persistence.

We study the following quantitative phenomenon in symplectic topology: in many situations, if a Lagrangian cobordism is sufficiently small (in a sense we specify) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms, which, by Biran and Cornea’s Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev’s theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.

For any $r\ge 1$ and $n\in {\mathbb{Z}}_{\ge 0}^r\setminus \left\{0\right\}$ we construct a poset $W_n$ called a $2$–associahedron. The $2$–associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. We prove that the completion ${\hat{W}}_n$ is an abstract polytope of dimension $\left|n\right|+r-3$. There are forgetful maps $W_n\to K_r$, where $K_r$ is the $\left(r-2\right)$–dimensional associahedron, and the $2$–associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, we work out the $2$– and $3$–dimensional $2$–associahedra in detail.

We develop techniques to compute the cooperations algebra for the second truncated Brown–Peterson spectrum $BP\langle 2\rangle$. We prove that the cooperations algebra $BP{\langle 2\rangle }_{\ast }BP\langle 2\rangle$ decomposes as a direct sum of an ${\mathbb{F}}_2$–vector space concentrated in Adams filtration $0$ and an ${\mathbb{F}}_2\left[v_0,v_1,v_2\right]$–module which is concentrated in even degrees and is $v_2$–torsion-free. We also develop a recursive method which produces a basis for the $v_2$–torsion-free component.

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\mathrm{\Xi }$ of trees, which carries a tight relationship to the Moerdijk–Weiss category of rooted trees $\mathrm{\Omega }$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $\mathrm{\Xi }$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.

We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We also bound the dilatation of pseudo-Anosov pure surface braids away from zero and give a constant upper bound in the case of a sufficient number of punctures.

We show that an important classical fixed-point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows (see Astérisque 333, Soc. Math. France, Paris (2010)), topological Hochschild homology and Morita-invariance in bicategorical generality.

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.