We say that a 4-manifold X has an exotic smooth structure if X possesses more than one smooth structure, i.e., there exists a 4-manifold X′ that is homeomorphic but not diffeomorphic to X. Although examples of exotic smooth structures on closed 4-manifolds have been known since the 1980s, the definite case remained elusive for decades.

The first examples of exotic smooth structures on 4-manifolds with definite intersection form were produced by Levine–Lidman–Piccirillo in 2023 their construction yielded manifolds with fundamental group Z/2. Subsequently, Baykur–Stipsicz–Szabó and Harris–Naylor–Park developed related techniques to obtain examples with fundamental groups Z/(4k+2) and Z/2×Z/2, respectively.

The aim of this talk is to review the main ideas underlying these constructions and to explain how exotic phenomena can be produced via involutions on covering spaces. Finally, we will discuss how a related approach can be used to obtain examples with new fundamental groups, starting from appropriate exotic building blocks.

TBA

The rational blowdown operation in 4-manifold topology replaces a neighborhood of a configuration of spheres by a rational homology ball. Such configurations typically arise from resolutions of surface singularities that admit rational homology disk smoothings. Conjecturally, all such singularities must be weighted homogeneous and belong to certain specific families: Stipsicz–Szabó–Wahl constructed QHD smoothings for these families and used Donaldson’s theorem to obtain very restrictive necessary conditions on the resolution graphs for singularities with this property.

In particular, these results, as well as subsequent work of Bhupal–Stipsicz, show that for certain resolution graphs, the canonical contact structure on the link of the singularity cannot admit a QHD symplectic filling. By contrast, we exhibit Stein rational homology disk fillings for the contact links of an infinite family of rational singularities that are not weighted homogeneous, producing a new symplectic rational blowdown. Inspiration for our construction comes from de Jong–van Straten’s description of Milnor fibers of sandwiched singularities; we use a symplectic analog of de Jong–van Straten theory.

During this talk we give background and motivation for the study of symplectic rational homology disk fillings of contact three manifolds, the connections with isolated surface singularities, resolutions, plumbings and applications for constructing smooth exotic 4-manifolds.

As more and more powerful computational tools emerge, mathematicians must meet the moment and begin to decipher and prove what the new data is telling us. 

In this talk, we will focus on Szabó’s observation of a correlation for hyperbolic knots between:

• the rank of knot Floer homology, a state-of-the-art knot invariant defined by Ozsváth-Szabó and independently by Rasmussen using methods from symplectic geometry, and 

• the hyperbolic volume of the knot complement, a knot invariant which is well-defined by the Mostow rigidity theorem.

We will begin by introducing some relevant knot invariants and discussing connections between them which exist a priori, often by construction. We will then turn our attention to some unexpected connections that emerge when making use of computational and experimental tools. Finally, we will discuss what conjectures can be extracted from our results and how data-driven results can help us prove formal mathematical statements.

Recently, Diamantis-Kauffman-Lambropoulou published their seminal article on the theory of bonded knots and links (https://www.mdpi.com/2227-7390/13/20/3260), which are classical structures presenting bonding sites, i.e. arcs with endpoints lying on the knot(s) and non intersecting it.

Bonded knotted structures were first introduced in 2017 to offer a suitable model for protein chains, but quickly became an object of mathematical interest of its own.

We will begin from some bases of knot theory (definition, Reidemeister moves, Alexander and Markov theorems), then to dig deeper in the relation that knot theory has with the braid groups, with particular focus on the group of pure braids. After an accurate algebraic analysis of the two groups, we will continue with the theory of bonded knots and braids, stressing on their interplay, to conclude with the role classical braids play in bonded knots.

The power of dynamics in the study of simplicial volume was already understood by Gromov in his work. Along the years, many people developed different approaches to simplicial volume intertwining geometry, algebra and dynamics. In this talk, we will make a general overview about this philosophy, trying to understand its real advantages and the side effects that can arise.

Consider a surface bundle over a surface, which is a smooth, oriented fibre bundle whose fibre and base space are both closed surfaces of genus 2 or higher. It is called atoroidal if the fundamental group of the total space contains no subgroups that are isomorphic to Z^2. In 2024, Autumn E. Kent and Christopher J. Leininger constructed the first example of an atoroidal surface bundle over a surface. In this talk, we will motivate the importance of this result and outline its proof.

Explicit examples of aspherical 4-manifolds are notoriously difficult to construct, which explains the abundance of conjectures and open problems on them. A classical conjecture, attributed to Hopf and Thurston, predicts that the Euler characteristic of any closed aspherical 4-manifold is non-negative. While examples of closed oriented aspherical 4-manifolds with even Euler characteristic are easy to produce, explicit examples with odd Euler characteristic are much harder to find, as highlighted in recent work by Edmonds. The goal of this talk is to present a construction of closed oriented aspherical 4-manifolds with any odd Euler characteristic greater than 4. The underlying idea is to obtain them by gluing together some building blocks along their boundaries.