LIT Workshop: Abstracts

Broadly speaking, volume conjectures relate the asymptotic growth of certain quantum invariants of topological objects (manifolds, links, graphs) to the volume of some hyperbolic structure on the object. I will give an overview of the topic and I will discuss some related results in hyperbolic geometry that are motivated by these sorts of questions, such as the Stoker conjecture or finding the maximum volume of hyperbolic polyhedra. 

Bounded cohomology is a powerful and by now well-established tool to study properties of groups and spaces. However, it is often difficult to compute and, beyond the case of amenable groups, many questions remain widely open.

In joint work with Francesco Fournier-Facio, Yash Lodha and Marco Moraschini, we present a new algebraic condition that implies the vanishing of the bounded cohomology of a given group. This condition is satisfied by many non-amenable groups of topological origin.

Roughly speaking, a weight system is a function from a space of (chord) diagrams to the complex numbers. Weight systems can be used to recover invariants for the relevant kind of knotted object (eg. knots, links, braids etc.) from the Kontsevich integral. The work of Sati and Schreiber highlighted the connection between (horizontal) chord diagrams and higher observables in quantum brane physics. This motivates the question: "which weight systems are quantum states?" Corfield, Sati and Schreiber showed that all gl(n) weight systems associated to the defining representation are indeed quantum states. In this talk I will present an extension of their result to more general weight systems.

The talk will be organised as follows; first, we start with a short presentation of weight systems and chord diagrams. Then, after translating the main question in mathematical terms, we will see the proof of Corfield-Sati-Schreiber's result. Finally, we will see how to extend their result to all fundamental weight systems.

Motivated by a recent example of Hayden, Kim, Miller, Park, and Sundberg of a pair of genus 1 Seifert surfaces for a knot K---oriented connected surfaces in the 3-sphere with boundary K---that can be shown to be non-isotopic in the 4-ball by a classical algebraic topology invariant, we essentially characterize in what cases this invariant can be used to distinguish Seifert surfaces of genus 1 in the 4-ball.

The key algebraic player will be Gauss's group $G_D$: primitive integral binary quadratic forms with fixed non-zero discriminant $D$ considered up to the natural $\mathrm{SL}_2(\mathbb{Z})$ action. We will see how this group naturally pops up when studying pairs of genus 1 Seifert surfaces of a knot K.

Based on joint work in progress with M. Akka, A. Miller, and A. Wieser. No knowledge of knot theory or number theory will be assumed. 

A manifold is said to bound geometrically if it is the totally geodesic boundary of another manifold. Such manifolds are quite rare as, already in dimension 2, the hyperbolic surfaces of fixed genus that bound geometrically form a countable subset of the moduli space. The only method for the construction of geometric bordism, moreover, relies on the existence of a kind of involution which most surfaces do not admit, but it is known by non-constructive methods that there are surfaces that bound geometrically without such involution. This talk will present a new construction for the case of surfaces that tessellate into regular polygons, share some partial results and comment on open problems.

A conjecture attributed to Singer stipulates that most L^2 Betti numbers of an aspherical manifold vanish. In dimension 4, this implies a conjecture of Gromov: the Euler characteristic of an aspherical 4-manifold bounds its signature. I will talk about a proof of Gromov’s conjecture for geometrically decomposable 4-manifolds. This is joint work with Luca F. Di Cerbo.

A question in knot theory motivated by the smooth 4-dimensional Poincaré conjecture is to classify which knots bound a smooth disc in X - int(B^4), where X is a given closed smooth 4-manifold. We study the case when X is any K3 surface, and prove that every knot that can be unknotted with less than 22 crossing changes bounds a smooth disc in X - int(B^4). Our proof is constructive and based on the existence of a plumbing tree of 22 spheres embedded in X. This is joint work with Stefan Mihajlović. 

We consider a notion of multisection for closed manifolds which generalizes Heegaard splittings of 3-manifolds and Gay-Kirby trisections of smooth 4-manifolds: a multisection of a closed manifold is a decomposition into 1-handlebodies, where any subcollection meets along a 1-handlebody, except the global intersection which is a closed surface. We will discuss existence and uniqueness of such decompositions and see how they can be represented by diagrams on closed surfaces. Joint work with Fathi Ben Aribi, Sylvain Courte and Marco Golla.

The theory of continuous bounded cohomology developed in the early 2000s by Burger-Monod has given fruitful applications in several contexts, for instance in Rigidity Theory. It has been later employed in the measurable context to study rigidity for cocycles by means of numerical invariants, following Burger-Iozzi’s approach. 

We will see how (continuous) bounded cohomology of (topological) groups has a natural measurable counterpart that can be defined for measured groupoids (e.g. standard Borel equivalence relation or probability measure preserving actions). This theory has been recently formalized in a joint work with A. Savini, and it turned out to be invariant under Measure Equivalence and Orbit Equivalence. This suggests applications in Measured Group Theory, but also in the study of purely topological invariants such as the simplicial volume.

We show that every non-trivial strongly quasipositive knot is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive knots. In contrast to our result, Baader, Dehornoy and Liechti showed that every concordance class contains at most finitely many positive knots. Moreover, Baker conjectured that smoothly concordant strongly quasipositive fibered knots are isotopic. Our construction uses a satellite operation with companion a slice knot with maximal Thurston-Bennequin number -1.

In the talk, we will define the relevant terms necessary to understand the theorem in the title, and explain the context of this result. If time permits, we will say a few words about how the construction extends to links.