(1st lesson) Mapping class groups of 4-manifolds

• Mark Powell (University of Glasgow)

Room: Aula Dini I will introduce the topic of mapping class groups of 4-manifolds, which can be studied using algebraic topology, gauge theory, and geometric methods. I will survey what is known, give an outline of some of the methods of proof, and highlight some open questions. I will particularly focus on pseudo-isotopy theory.

Using strong contraction to obtain hyperbolicity

• Stefanie Zbinden (Universität Bonn)

Room: Aula Dini If a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct an action on a hyperbolic space where said element acts loxodromically. However, the action depends on the chosen element and other strongly contracting elements are not necessarily loxodromic. It raises the questions whether there exists a single action on a hyperbolic space where all strongly contracting elements act loxodromically. In this talk, we answer the above question positively by introducing the contraction space construction. We then show that the contraction space can be used to extend the following dichotomy known for CAT(0) groups to other groups such as injective groups. Either the group has linear divergence, in which case all asymptotic cones are cut-point free, or the group has a Morse geodesic, in which case all asymptotic cones have cut-points and the group is acylindrically hyperbolic.

(1st lesson) CAT(0) cube complexes and cubulations

• Davide Spriano (Christ Church College, Oxford University)

Room: Aula Dini CAT(0) cube complexes are a central object in geometric group theory, and constitute a large class of objects with a rich structure. Roughly, a CAT(0) cube complex is the metric space obtained by gluing Euclidean cubes in a "non-positively-curved" way. The minicourse has three goals. The first is to provide an overview of why and how cube complexes are used in geometric group theory. The second is to explain the definitions and prove some basic results about them. The last is to describe Sageev's construction - a procedure that allows to "extract" a cube complex from a simple data - and some recent generalisations of it.

On the Topology of Virtual Artin Groups

• Federica Gavazzi (Université Bourgogne Europe, Dijon)

Room: Aula Dini Virtual Artin groups were introduced a few years ago by Bellingeri, Paris, and Thiel, with the aim of generalizing the well-studied structure of virtual braid groups to the broader context of Artin groups. These fascinating objects possess a rich algebraic structure that encompasses both Coxeter groups and classical Artin groups. In this talk, we will explore the topology of virtual Artin groups, focusing in particular on the construction of cell complexes that serve as promising candidates for classifying spaces of certain remarkable subgroups. We will also highlight a connection between the topological properties of these spaces and a well-known problem in the theory of Artin groups: the K(π,1) conjecture.

(2nd lesson) CAT(0) cube complexes and cubulations

• Davide Spriano (Christ Church College, Oxford University)

Room: Aula Dini CAT(0) cube complexes are a central object in geometric group theory, and constitute a large class of objects with a rich structure. Roughly, a CAT(0) cube complex is the metric space obtained by gluing Euclidean cubes in a "non-positively-curved" way. The minicourse has three goals. The first is to provide an overview of why and how cube complexes are used in geometric group theory. The second is to explain the definitions and prove some basic results about them. The last is to describe Sageev's construction - a procedure that allows to "extract" a cube complex from a simple data - and some recent generalisations of it.

An invitation to Local-to-Global rigidity

• Amandine Escalier (Université Claude Bernard Lyon 1)

Room: Aula Dini This talk is intended as an introduction to the so called “Local-to-Global rigidity” of graphs and aims to present the links of this notion with both topology and geometry. More precisely, a graph G is called Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in G is covered by G. We’ll talk about the motivations, discuss numerous examples and borrow topological tools to settle the basis. We will also see the known cases where LG-rigidity is invariant under quasi-isometry and, if time permits, discuss some strategies to prove this invariance.

(2nd lesson) Mapping class groups of 4-manifolds

• Mark Powell (University of Glasgow)

Room: Aula Dini I will introduce the topic of mapping class groups of 4-manifolds, which can be studied using algebraic topology, gauge theory, and geometric methods. I will survey what is known, give an outline of some of the methods of proof, and highlight some open questions. I will particularly focus on pseudo-isotopy theory.

(3rd lesson) Mapping class groups of 4-manifolds

• Mark Powell (University of Glasgow)

Room: Aula Dini I will introduce the topic of mapping class groups of 4-manifolds, which can be studied using algebraic topology, gauge theory, and geometric methods. I will survey what is known, give an outline of some of the methods of proof, and highlight some open questions. I will particularly focus on pseudo-isotopy theory.

The JSJ decompositions of generalized Baumslag-Solitar groups

• Dario Ascari (University of Basque Country)

Room: Aula Dini The theory of JSJ decomposition plays a key role in the classification of hyperbolic groups, in analogy with the case of 3-manifolds. The same strategy can also be adapted to larger families of groups, but with some obstructions arising, the most natural example being Generalized Baumslag-Solitar groups. We discuss the flexibility of JSJ decomposition for groups within this family, and its implications on the study of the isomorphism problem and of (outer) automorphism groups.

(3rd lesson) CAT(0) cube complexes and cubulations

• Davide Spriano (Christ Church College, Oxford University)

Room: Aula Dini CAT(0) cube complexes are a central object in geometric group theory, and constitute a large class of objects with a rich structure. Roughly, a CAT(0) cube complex is the metric space obtained by gluing Euclidean cubes in a "non-positively-curved" way. The minicourse has three goals. The first is to provide an overview of why and how cube complexes are used in geometric group theory. The second is to explain the definitions and prove some basic results about them. The last is to describe Sageev's construction - a procedure that allows to "extract" a cube complex from a simple data - and some recent generalisations of it.

The sl(n) polynomials for strongly invertible links

• Carlo Collari (Università di Pisa)

Room: Aula Dini Strongly invertible links are collections of disjoint oriented circles in the tri-dimensional space together with an involution preserving the components and reversing the orientation. In this talk we will introduce a family of invariants of strongly invertible links which are analogues of the sl(n) polynomials for links; in particular, they are Laurent polynomials in the variable q and are parameterised by a positive natural number. After a brief discussion on the effectiveness of these invariants, we will give a characterisation of them in terms of skein relations. We will conclude with some applications and a comparison with other known invariants of strongly invertible links.

Equivariant signature and unknotting number of strongly invertible knots

• Sarah Zampa (Budapest University of Technology and Economics)

Room: Aula Dini In this talk, I will first recall the definition of the knot's signature in the sense of Gordon-Litherland, computed from a possibly non-orientable surface (via the Goeritz pairing together with some appropriate correction term). I will then follow with an introduction of strongly invertible knots, and define the corresponding concept of equivariant signature, as well as the equivariant unknotting number for such knots.

Monodromies and visual primeness

• Lukas Lewark (ETH Zürich, Switzerland)

Room: Aula Dini If a knot decomposes as a connected sum, is this decomposition visible in a diagram of the knot? We will see that the answer is "yes", for so-called alternative diagrams. This partially resolves a conjecture posed by Cromwell in 1991. The proof relies on a new criterion for the existence of fixed arcs of (partial) monodromies. No prior knowledge of monodromies or knot diagrams will be assumed. Joint work with Peter Feller and Miguel Orbegozo Rodriguez.

Minimal volume entropy of mapping tori of 3-manifolds

• Giuseppe Bargagnati (Università di Bologna)

Room: Aula Dini The volume entropy of a closed Riemannian manifold is a number which measures the exponential rate of growth of balls in the Riemannian universal cover of the manifold. Taking the infimum of the volume entropy over all Riemannian metrics (up to normalization), one gets a homotopy invariant of the manifold, the minimal volume entropy. This invariant behaves in a quite mysterious way: for example, we do not know if it is multiplicative under finite covers. However, in 1982 Gromov proved that it is an upper bound for the simplicial volume, another (better-behaved) homotopy invariant which, intuitively, measures the difficulty of representing the fundamental class of the manifold via singular simplices. Gromov himself raised the question whether the vanishing of the simplicial volume implies the vanishing of minimal volume entropy. We prove that for mapping tori over oriented closed connected 3-manifolds, which are known to have vanishing simplicial volume by a 2020 result of Bucher and Neofytidis, this is indeed the case. One of our main technique involves finding covers of small cardinality of the mapping tori composed by sets of polynomial growth. This is a joint work with Alberto Casali, Francesco Milizia and Marco Moraschini.

Gordian distance bounds from Khovanov homology

• Laura Marino (University of Hamburg)

Room: Aula Dini The Gordian distance u(K,K') between two knots K and K' is the minimal number of crossing changes needed to relate K and K'. The unknotting number u(K) of a knot K arises as the Gordian distance between K and the trivial knot. Rasmussen was the first to find a connection between Khovanov homology and u: his invariant s, extracted from Khovanov homology, yields a lower bound for the slice genus and, as a consequence, for u(K). In this talk, I will introduce a new invariant λ, extracted from a universal version of Khovanov homology. Although it is not connected to the slice genus, λ is a lower bound for u, and in fact the inequality |s(K)| ≤ 2λ(K) always holds. Moreover, λ displays relations to a generalization of u, the proper rational Gordian distance. This is joint work with L. Lewark and C. Zibrowius.