Speakers

(Joint work with Marta Pavelka)

When t is an integer that ranges from 1 to d, we introduce ''t-constructible'' and ''t-LC'' d-manifolds, interpolating between the known families of constructible and LC d-manifolds (case t = 1), and the family of all triangulated d-manifolds (in case t = d). All t-LC manifolds are also (t+1)-LC. 

Benedetti-Ziegler proved that LC d-manifolds are exponentially many: specifically, less than 2^{N d²}. We expand this result by showing that 2-LC d-manifolds are still exponentially many: specifically, less than 2^{N (d³)/2}. This is probably best possible, because if there are more than exponentially many 3-spheres, their suspensions yield more than exponentially many 3-LC manifolds.

Dunwoody manifolds are an intensively studied class of closed connected orientable  3-manifolds, introduced  by Dunwoody in 1994 using Heegaard diagrams. Notably, by results of Grasselli-Mulazzani 2001 and Cattabriga-Mulazzani 2004,  this class of manifolds coincides with that of  strongly-cyclic branched covering of genus one 3-manifolds having as branching set (1,1)-knots.  In the talk, after recalling the main achievements in the study of Dunwoody manifolds, I’ll present some new results, obtained in collaboration with Paolo Cavicchioli, on the equivalence moves for links, represented via plat closure, lying in Dunwoody manifolds.

Art and mathematics have developed in parallel throughout history. There are many examples where mathematics has played an essential role in art. In this talk I will discuss, as an example of this relationship, the topological approach to concepts used in the analysis of the crystalline symmetries of decorations on Islamic monuments.

Concerning art in mathematics, there is an aesthetic incentive in the study and research in mathematics that allows us to affirm that mathematics is an art form. I will present some examples to justify the above statement.

The undeniable success of Big Data analysis, by Machine Learning, promises and threatens a revolution of the scientific method itself. The substitution of rational models with effective but inscrutable oracles is a danger we cannot overlook. A possible countermeasure is the development of the so-called Explainable Artificial Intelligence, where mathematics, and topology in particular, may play a fundamental role.

Nel 1942 l'artista olandese M.C. Escher, studiando le tassellazioni, si imbatté in una curiosa proprietà di un certo tipo di esagoni, costruiti su un triangolo dato, che giustapposti riempiono il piano senza lasciare spazi vuoti. Escher enunciò la sua scoperta, che riguardava la costruzione di un punto notevole del triangolo, sotto forma di teorema, ma non riuscì a dimostrarlo rigorosamente. Il teorema di Escher è un caso particolare di un teorema dimostrato da Carl Friedrich Jacobi nel 1825. Verranno quindi esplorate le implicazioni del teorema, introducendo la nozione di punti complementari di Jacobi e alcune loro proprietà, come la collinearità con il circocentro e l'appartenenza a una speciale iperbole associata al triangolo dato.

After Donaldson's work in the 80s several techniques have been invented to produce exotic smooth structures on a given 4-manifold. In the simply connected case, it is well known that all smooth structures are related by an operation called "cork twist". This consists in embedding in a target 4-manifold certain contractible 4-manifolds with boundary endowed with an involution of their boundary, and then producing a new manifold by cutting them out and gluing them back using the involution. Unfortunately it is difficult to obtain general results about corks because we lack a classification of them. In order to solve this problem I will introduce a class of 4-manifolds, called protocorks, which are easily classified as they are in bijection with certain bipartite graphs. The idea is that to study a cork we can study a supporting protocork. By doing so we can cook up an invariant of corks defined as the U-torsion order of a special element in the cork's boundary Floer homology with Z/2Z coefficients. I will explain this and give some examples of applications.

Random tensor models have been introduced in the '90s in the context of the random geometry approach to quantum gravity. They indeed admit expansions over Feynman graphs that are dual to D-dimensional colored triangulations, and can therefore be seen as defining distributions on these triangulations. I will review the motivations for this approach, some of the questions it poses on the topological and geometrical properties of colored triangulations, and the current status of this domain.  

A distanza di nove secoli, e senza che sia giunta alcuna documentazione testuale o grafica, è possibile riproporre la facies del battistero pisano come pensato dal primo autore? Immaginare il modello iniziale con la perfetta euritmia dei volumi, la luce che vibra tra le logge perimetrali, il canto monodico che si diffonde tra le arcate intorno all’acqua della purificazione?

I tempi lunghi del cantiere, le varianti in corso d’opera, la diversa temperie culturale, la presenza di altri maestri rimandano un manufatto che, solo in parte, risponde all’idea originaria. La forte ingerenza del volume sullo skyline della piazza ne è testimonianza visiva.

Saranno l’analisi delle strutture e il rilevo delle partiture a svelare il processo, insieme artistico e geometrico, con cui è stato elaborato. Sarà il percorso matematico con la misura e il numero, parametri fondanti del costruire, a rivelare la trama compositiva e i messaggi scritturali che l’opera avrebbe dovuto trasmettere allora e per i secoli a seguire.

E proprio dall'identificazione della forma del battistero emerge chiara quell’ordinatio ad unum dei tre monumenti della piazza che solo un unico autore può aver programmato in sintonia con il progetto Divino della Creazione.

I fondi storici e speciali della Biblioteca Panizzi di Reggio Emilia custodiscono importanti testimonianze della cultura scientifica e tecnica del tardo medioevo, del Rinascimento e della prima modernità. Vi troviamo infatti manoscritti di Leonardo Fibonacci, Piero della Francesca e Francesco di Giorgio Martini, opere a stampa di Luca Pacioli e Leonardo da Vinci, il primo volgarizzamento di Vitruvio e i libri di architettura di Sebastiano Serlio. Alcune di queste opere sono state oggetto degli studi di Luigi Grasselli e nella relazione si forniranno in carrellata gli elementi essenziali relativi al loro aspetto e contenuto e alla loro storia.

Only recently a workable formula for the volume of the general hyperbolic tetrahedron was obtained by Prof. Mednykh and collaborators. The problem was started by Gauss and Prof. Sforza (from Reggio Emilia) found a formula based on Schlafli formula. In this talk I will trace the history of the development of this interesting topic.

Extraterrestrial life, such as microorganisms, has been hypothesized to exist in the Solar System and throughout the universe. This hypothesis relies on the vast size and consistent physical laws of the observable universe. The first basic requirement for life is an environment with non-equilibrium thermodynamics, which means that the thermodynamic equilibrium must be broken by a source of energy. The traditional sources of energy in the cosmos are the stars, such as for life on Earth, which depends on the energy of the sun. Life on Earth requires water in a liquid state as a solvent in which biochemical reactions take place, and to allow the transport of nutrients and substances required for metabolism. Sufficient quantities of carbon and other elements, along with water, might enable the formation of living organisms on terrestrial planets with a chemical make-up and temperature range similar to that of Earth. Some bodies in the Solar System have the potential for an environment in which extraterrestrial life can exist, particularly those with possible subsurface oceans. Extrasolar planets that may be conducive to life are usually thought to be terrestrial planets within the habitable zones of their stars. As of today, over five thousand exoplanets have been discovered, ranging in size from terrestrial planets similar to Earth to gas giants larger than Jupiter. There is an average of one planet per star, and about 1 in 5 Sun-like stars have an Earth-sized planet in the habitable zone. Assuming 100 billion stars in the Milky Way, that would be one billion potentially habitable Earth-sized planets in the Milky Way. The Fermi paradox, the discrepancy between the lack of conclusive evidence of advanced extraterrestrial life compared to the apparently high a priori likelihood of its existence, still challenges our understanding of life elsewhere in the universe.

Heegaard Floer homology was defined by Ozsváth and Szabó in the early 2000s. It consists of a package of invariants of closed oriented 3-manifolds and it has found many important and profound applications in low dimensional topology. I will introduce the L-space conjecture, that boldly predicts strong connections among properties relating Heegaard Floer homology, foliations and the fundamental group of an irreducible rational homology 3-sphere. I will then state a result concerning this conjecture and manifolds that arise as surgeries on fibered hyperbolic two-bridge links. Time permitting we will see how to apply this result to deduce that all non-meridional surgeries on Whitehead doubles of a non-trivial knot support coorientable taut foliations.

(Joint work with Luke Mizzi)

Mechanical metamaterials are artificial structured materials that exhibit unique mechanical properties that are not commonly observed in natural materials, such as a negative Poisson's ratio. Mechanical metamaterials find use in a wide range of applications, ranging from biomedical to aeronautical engineering, thanks to their high energy absorption, tunable mechanical properties, and lightweight nature. The mechanical properties of these materials are intrinsically linked to their geometric configuration, which can normally be expressed in terms of periodic tilings or tessellations, and by exploiting the relationship between the topology, symmetry and deformation mechanisms, new and more efficient metamaterials can be designed and implemented in engineering applications.

We illustrate how to construct branched coverings over certain PL 4-dimensional manifolds. In particular we characterise the 4-manifolds that are realisable as branched coverings of the complex projective plane. This is a joint work with Riccardo Piergallini.