Baby Geometri
anno accademico 2024/2025

Bounded cohomology was introduced in the 1970s by Johnson and Trauber in the context of Banach algebras and has since become a widely studied group invariant. In this talk, we give an introduction to the bounded cohomology of groups, focusing particularly on its vanishing properties. We define the wreath product construction and see some finiteness properties of the resulting groups. Finally, we will present a proof of Monod's vanishing result for the bounded cohomology of (restricted) wreath products.

Delta complexity is a numerical invariant of closed manifolds which describes how complicated a manifold can be topologically. First, we define this invariant in general, and then we focus on a particular class of 3-manifolds called mapping tori. Finally, we will present a proof of a result of Lackenby and Purcell that provides an upper bound on the delta complexity of a mapping torus built from a 2-dimensional torus and an Anosov homeomorphism. To do this, we will introduce beautiful objects such as the Farey tessellation and continued fractions. 

Convex projective geometry was introduced by Hilbert in 1894 as a generalization of hyperbolic geometry. Over the past two decades the interest in this theory has increased due to its connections with the theory of Anosov representations. In this talk, we will first define convex projective manifolds and provide some basic examples. We will then focus on closed convex projective manifolds in dimensions 2 and 3. Finally, we will discuss a result by Benoist regarding the JSJ decomposition of a 3-dimensional closed irreducible convex projective manifold and examine the structure of its flats.

Uno degli aspetti che rendono la (co)omologia singolare interessante è il fatto che per spazi topologici particolarmente buoni, presenta simmetrie non banali: per esempio la dualità di Poincaré per varietà topologiche orientabili, e molte altre simmetrie sorprendenti (il cosiddetto Kahler package) per varietà complesse compatte con una buona metrica. Purtroppo, queste simmetrie coomologiche falliscono in presenza di singolarità. In questo talk, introdurremo l'omologia d'intersezione, che coincide con l'omologia singolare classica per varietà topologiche ma ha il vantaggio di produrre gruppi di omologia che rispettano ancora le simmetrie che trovavamo per spazi lisci.

Two surfaces in a 4-manifold are called exotic if they are topologically ambient isotopic but smoothly not. In this talk we will start by taking a quick general tour of the 4-dimensional knot theory, we will discuss some examples of exotic surfaces in closed 4-manifolds and investigate some of their properties with more focus on their 2-knot groups.

The K(π,1) conjecture for Artin groups goes back to the work of Arnol’d, Brieskorn, Pham, and Thom in the '60s. Recently, the dual approach has been instrumental in proving that conjecture in the affine and rank 3 cases. In this talk, we will introduce Coxeter groups and Artin groups, providing some basic examples and stating the K(π,1) conjecture. We will then introduce the standard and dual structures for Artin groups, highlighting the results in the spherical, affine and rank 3 cases. Finally, we will give a few results about the isomorphism between the standard and dual Artin groups.

The theory of groups acting on trees—attributed to Bass and Serre—provides a rich framework for the understanding of structural and homological properties of discrete groups. In this talk, I will introduce the profinite analogue of this theory, developed by Melnikov, Zalesskii and Ribes. I will outline some of the difficulties encountered when passing from the discrete to the profinite world, as well as some solutions. I will also introduce three notions of accessibility in the context of profinite/pro-p groups acting on profinite/pro-p trees and discuss a result of mine as well as two open questions regarding accessibility. Time permitting, I will outline some applications to other topics in profinite rigidity.

Consider the set of representations of Hom(π1(M),G) for M some manifold and G an interesting Lie group. How can their nature be understood? An approach for a classification of these was kickstarted by Goldman in the 80s leading to the emergence of Higher Teichmueller theory, nowadays an actively growing field. In the talk we shall outline the essence of this approach where the so-called Toledo invariant takes a central role. Then, realizing that the Toledo invariant is actually a cohomological invariant, is the point of view that allows to generalize aspects of Goldman's classification. If time permits, we shall briefly discuss how bounded cohomology comes into play and proves to be a powerful tool.

The Arf invariant is a modulo 2 integer associated to a non singular quadratic form over the field F_2. In 1965, Robertello applied this theory to Seifert forms, leading to an invariant of a certain class of oriented links. Colored links are a natural generalisation of oriented links, where the components are grouped into sublinks. Several invariants of oriented links, such as the Alexander module and the Levine-Tristram signature, admit natural multivariable extensions to colored links via so-called "generalised Seifert surfaces".  In this talk, I will define these notions, and briefly explain how to use generalised Seifert surfaces to extend the Arf-Robertello invariant to colored links. Unfortunately, these extensions turn out to be determined by the linking numbers.

Holonomy was introduced by Élie Cartan in 1926 and has since then provided a way to apply the machinery of Lie groups representation to study Riemannian manifolds. In this talk we will start by introducing holonomy in vector and principal bundles, providing some examples, to arrive at the main result obtained by Berger for the classification of the holonomy of Riemannian manifolds.

In this talk, we will explore the world of SU(2) representations applied to 3-manifolds. First, we will introduce the importance of SU(2)-representations as an invariant of non-triviality. Then, we will describe this theory's relations to the L-space conjecture, particularly to instanton Floer homology. Finally, we will apply some of these techniques to the case of graph manifolds with small order.

Simplicial volume is a homotopy invariant for oriented, closed, connected manifolds defined by Gromov. Even if its definition is purely topological, the simplicial volume of a manifold is deeply related to the geometric structures it can support.  In this talk, we will recall the definition of simplicial volume and its basic properties, as well as its connections to other invariants. Special attention will be given to Gromov's Vanishing Theorem, which links the simplicial volume of a manifold to its amenable category providing a powerful vanishing criterion. As an application, we will discuss the role of simplicial volume in the Chern conjecture for closed affine manifolds and we will present a vanishing result for the simplicial volume of certain complete affine manifolds. This is based on a joint work with Marco Moraschini.

How can we get to know a 4-manifold? We can't take it out for dinner or even embed it nicely in our universe. Comparatively, 2-manifolds are much easier to get our hands on. By examining how spheres and surfaces behave within a 4-manifold, we can uncover clues about its structure. In this talk, we explore the interplay between the algebraic and geometric properties of these surfaces to shed light on the nature of 4-manifolds. 

Given 3d-1 points in general position in the complex projective plane, a classical enumerative question is how many plane rational curves of degree d pass through them. While elementary to state, the answer to this problem was known only for a few values of d until, in the 1990s, Kontsevich found a general recursive formula using groundbreaking techniques.

In this talk, we will use this problem as a guideline to explore some key ideas in algebraic geometry: moduli spaces of stable curves and maps, intersection theory, and the need for virtual fundamental classes. We will ultimately arrive at the definition of Gromov-Witten invariants and use them to prove Kontsevich's result.

A connected n-manifold is called aspherical if all its higher homotopy groups vanish, except possibly for the fundamental group.

Aspherical manifolds in dimensions 1, 2, and 3 are well understood: their universal covers are always homeomorphic to R^n. This naturally leads to the question of whether the same holds in higher dimensions. In 1983, Michael W. Davis proved that for every n>3, there exists a closed aspherical n-manifold whose universal cover is not homeomorphic to the Euclidean space. In this talk, I will present a proof of Davis’s theorem.

TBA.

The core idea of geometric group theory is trying to understand the algebraic property of a group G via its actions by isometries on "nice" metric spaces. Here "nice" means that the more structure the space has, the more properties we can infer of the group. One such structure is the existence of a coarse median, which roughly is a way to assign a center to every triangle "by majority vote". 

In this talk we'll focus on building exotic coarse median structures on right-angled Artin groups. If time allows, I'll do a full retcon and explain how my supervisor Alessandro Sisto and I got to this result while we were looking for something completely different.

In all dimensions n ≥ 5, we prove the existence of closed orientable hyperbolic manifolds that do not admit any spinᶜ structure, and in fact, we show that there are infinitely many commensurability classes of such manifolds. These manifolds all have non-vanishing third Stiefel-Whitney class w₃ and are all arithmetic of simplest type. More generally, we show that for each k ≥ 1 and n ≥ 4k+1, there exist infinitely many commensurability classes of closed orientable hyperbolic n-manifolds M with w₄ₖ₋₁(M) ≠ 0.

Let R be a normed ring, e.g. the real numbers or the integers. Then the ℓ1-norm on the singular chain complex with R-coefficients given by the basis of singular simplices induces a semi-norm on singular homology. In this talk, we discuss why the semi-norm of products of classes in singular homology cannot be estimated from below by the product of the norms of the two classes. This shows that the norm of products over the integers behaves differently than over the real numbers. Ongoing work with Clara Löh.

The study of the rationality of the L^2-Betti numbers of a countable group has led to the development of a rich theory in L^2-homology with deep implications in structural properties of the groups. For decades almost nothing has been known about the general question of whether the strong Atiyah conjecture passes to free products of groups or not. In this talk, we will confirm that the strong Atiyah conjecture is stable under the graph of groups construction provided that the edge groups are finite. We will also discuss how universal localizations can be used to describe the division rings of fundamental groups of certain low dimensional manifolds and show applications of this theory. 

A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and examine their key properties.

Two smoothly embedded surfaces in a smooth 4-manifold are called exotic if they are topologically isotopic but not smoothly isotopic. The phenomenon of exotic surfaces in dimension 4 is an interesting topic in low-dimensional topology. In this talk, we start with an introduction to equivariant knot concordance theory in the context of exotic disks. Then we recall how involutive Heegaard Floer theory can work as a machinery for equivariant concordance. Then we demonstrate our recent progress on equivariant concordance of Whitehead doubles, which produces exotic disk pairs. This is joint work with Sungkyung Kang and JungHwan Park.