In the academic year 2020/21 the organizers were: Ludovico Battista (UniPi), Alice Merz (UniPi), Diego Santoro (SNS) and Giovanni Italiano (SNS).

14 July

Leonardo Ferrari (University of Pisa)

Manifold covers of right-angled polytopes were first introduced by Davis and Januszkiewicz in 1991 as a simple, combinatorial method to build manifolds by gluing copies of some polytope along its facets. Since then a number of techniques have been added to their initial work, allowing for a better understanding of the geometry of such manifolds, and many important, recent examples of 4- and 5-dimensional hyperbolic manifolds have arisen from this setting. In this seminar, we will introduce the notion of right-angled polytopes as orbifolds, present the basic construction of manifold covers and give an overview of the additional tools developed in recent years. We will conclude the seminar with a few examples and open problems of interest.

30 June

Giuseppe Bargagnati (University of Pisa)

The simplicial volume is an invariant of manifolds introduced by Mikhail Gromov in his pioneering article of 1982. Although it is a homotopy invariant, it turns out to be very strongly related with some other geometric invariants. We should all agree that a very natural question can be the following: what numbers can arise as simplicial volume of compact (or non-compact) d-manifolds? In this seminar, after defining the simplicial volume both in the compact and the non-compact case and saying why we should care about computing it, we will try to give a (partial) answer to this question, flying over some recent results that try to shed light on this age-old matter. At the end of the talk, we will focus a little bit our attention on the case of contractible open 3-manifolds.

23 June

Stefan Mihajlović (Central European University, Alfred Renyi Institute of Mathematics)

One of the most exciting questions in 4-manifold topology is understanding different smooth structures on the same underlying topological manifold, and 'corks' provide one insight into this fun question. I will define this notion, present an example or two, and then try to roughly sketch the proof of the theorem which says that we can get any smooth structure of a given smooth 4-manifold by taking out some cork, and gluing it back with a different gluing map. Also, I will try to fly through some more recent results and possible further questions.

19 May

Federica Bertolotti (Scuola Normale Superiore)

Cartan-Hadamard theorem states that the universal cover of a complete, simply connected, negatively curved manifold is diffeomorphic to \( \mathbb{R}^n \). Thanks to this it is possible to define the visual boundary, a kind of boundary "at infinity". We will use this structure to define geometrically finite and infinite ends; we will see how to apply this constructions to negatively curved 3-manifold.

12 May

Francesco Milizia (Scuola Normale Superiore)

The talk will be an introduction to bounded cohomology and cohomology with bounded values, which have been defined by Gromov in 1982 and by Gersten in 1991 respectively. Bounded cohomology is obtained by modifying the definition of singular cohomology of a topological space: instead of considering all singular cochains, only bounded ones are taken into account. In a similar spirit, cohomology with bounded values is the bounded version of cellular cohomology (or the cellular version of bounded cohomology, if you prefer). We will see that these two cohomologies exhibit very different, sometimes opposite, behaviors. Furthermore, we will see how cohomology with bounded values is related to isoperimetric inequalities.

5 May

Roberto Ladu (Imperial College London)

Heegaard-Floer homology is an invariant of 3-manifolds defined by P. Ozsváth and Z. Szabó in 2003. Over the last twenty years it rapidly became one of the main subjects of investigation in geometric topology. This is due to the fact that it is relatively easy to compute -in practice combinatorial, in contrast with other Floer theories- and to its groundbreaking applications to the study of knots and surgery. But there are also many other reasons why you should care about it. In this talk I will focus on the definition of this invariant which roughly boils down to counting pseudoholomorphic curves connecting submanifolds constructed with the help of an Heegaard splitting.

28 April

Enrico Savi (Università di Trento)

One of the main topics in real algebraic geometry is the study of real algebraic sets and, in particular, the study of topological spaces admitting an algebraic model. That is the reason why in this seminar I will present the proof of Tognoli’s theorem, namely: Every compact connected smooth manifold admits a nonsingular affine algebraic model. This result was published in the Annali della Scuola Normale Superiore di Pisa in 1973 as the solution of a conjecture by John Nash. The techniques applied to prove this theorem must connect the algebraic and the topological nature of the statement, thus I will give an overview on the general results which are involved in the proof by Alberto Tognoli. In addition, these techniques gave rise to a very rich field of research, which is still active these days. Hence, I will conclude by presenting other remarkable results recently developed.

21 April

Stavroula Makri (University of Caen)

The classic braid groups \( B_n \) were introduced by Artin in 1925. The notion of surface braid groups, which generalises the notion of the braids was first introduced by Zariski. Then, during the 1960’s the surface braid groups were rediscovered by Fox who proposed an equivalent topological definition in terms of the fundamental group of configuration spaces. In this talk we will begin with the basic definitions and results about the surface braid groups. We will continue with describing the problem of a possible splitting of the Fadell–Neuwirth short exact sequence of surface braid groups. Moreover, we will present the braid groups of the real projective plane \( RP^2 \), and we will describe the problem of the possible splitting of the Fadell–Neuwirth short exact sequence in this case. Finally, we will present partial results concerning this splitting problem.

14 April

Juan Luis Duran Batalla (UA Barcelona)

We will consider the deformation space of a complete, non-compact, hyperbolic 3-manifold of finite volume and which admits a geometric triangulation. In the orientable case, by results of Thurston and Neumann-Zagier, the deformation space can be parametrized by the generalized Dehn filling coefficients. The same strategy will be applied here to non-orientable 3-manifolds to parametrize deformations of the triangulation. If time allows it, we will also discuss the approach through the variety of representations, which, unlike the orientable case, gives rise to additional deformations which are unattainable through the triangulation.

07 April

Livio Ferretti (University of Bern)

In this talk we will introduce different topological and combinatorial methods for constructing knot invariants. Our main example will be the Levine-Tristram signature, a classical topological invariant for which a combinatorial interpretation has been recently conjectured by Kashaev. If time permits, we will present some partial results that provide evidence for the conjecture.

24 March

Luca Accornero (Utrecht University)

A codimension q-foliation is a partition of a manifold into submanifolds that locally looks like the fibers of a submersion onto the Euclidean space of dimension q. Foliations come with cohomological invariants - the most famous one being the Godbillon-Vey invariant/Thurston's "helical wobble" for codimension 1 foliations. We are going to present a modern version of Haefliger's construction of a characteristic map for foliations - organizing these invariants into a universal object. We will talk about groupoids, classifying spaces for groupoid principal actions, jet spaces and their canonical distribution, and the "Chern-Weil" construction of characteristic classes for foliations.

17 March

Dario Ascari (Oxford University)

We will talk about free groups, their bases, and their group of automorphisms. We will introduce Whitehead transformations and Whitehead's algorithm. We will introduce some spaces which are of fundamental importance in the study of Aut(Fn) and of Out(Fn), such as the Outer Space and the Free Factor Complex.

10 March

Benjamin Ruppik (Max Planck Institute)

In this talk I would like to give an exposition of Zeeman’s 1963 result that a nontrivial twist spin of a classical knot \( \kappa \) in \( S^3 \) is fibred by the punctured branched cover over \( \kappa \). We will also learn some fun facts about the fundamental group of the complement of a knotted surface, and if there is enough time will trisect these knotted surface groups.

03 March

Matteo Migliorini (Scuola Normale Superiore)

La norma di Thurston è una funzione che associa a ogni elemento di \(H_2(M)\), dove \(M\) è una 3-varietà, la minima complessità di una superficie embedded che lo rappresenta (data dalla parte negativa della caratteristica di Eulero). Nel seminario vedremo che oltre a essere una norma, ha una palla unitaria che è un poliedro, e le sue facce sono collegate con le fibrazioni della varietà. Definiremo inoltre il concetto di foliazione, che è una versione "locale" di fibrazione, e vedremo la relazione esistente tra classe di Eulero di una foliazione e norma di Thurston. Quest'ultima aveva portato Thurston a formulare una congettura, che è stata confutata solo nel 2016 (quarant'anni dopo) da Yazdi e Gabai.