Baby Geometri
anno accademico 2022/2023

We will review a classical problem in group theory, the so-called Word Problem, from a geometric perspective by relating it to an even older geometric problem, the Isoperimetric Problem. The main actor being Dehn functions. We will talk about an interesting class of groups and review what is known about their Dehn functions and present some recent progress. If time allows it we will discuss the ideas behind it which will serve as an excuse to talk about an interesting open problem.

Since the 1980s, the homology cobordism group has been a central object in the development of low-dimensional topology. In this talk, we will discuss its historical roots, present our current knowledge about its algebraic structure, and state open problems about the behaviors of homology spheres.

Introduced around 2000, Khovanov homology marked the start of a new field of study in knot theory. It was born as a 'categorification' (or generalisation) of the Jones polynomial, a classical link invariant, but it proved to be more than just that: despite its combinatorial nature, it contains a surprising amount of topological information about knots and links (for example, it provides lower bounds for the slice genus and the unknotting number). In this talk, we will give a description of Khovanov homology as well as some applications in low dimensional topology, such as a way to detect exotically knotted surfaces in the 4-ball.

La teoria di Teichmüller di rango superiore è un'area di ricerca relativamente nuova e in rapido sviluppo che studia proprietà geometriche e dinamiche di rappresentazioni di gruppi di superfici in gruppi di Lie di rango almeno 2, generalizzando così la teoria di Teichmüller classica che si occupa di rappresentazioni in PSL(2,R) (o PSL(2,C)). In questo seminario daremo vari esempi di spazi di Teichmüller di rango superiore e spiegheremo come alcuni risultati classici si estendono, con opportune modifiche, a questo contesto.

Il mapping class group è un oggetto di grande importanza in topologia di bassa dimensione. In questa presentazione, introdurremo alcuni aspetti importanti della teoria, concentrandoci in particolare sui Dehn twists e parleremo di sottogruppi generati da un numero finito di Dehn twists. En passant, presenteremo alcuni risultati noti e problemi ancora aperti e vedremo come tali gruppi appaiono naturalmente in svariati contesti.

Gromov and Lawson's surgery theorem states that positive scalar curvature (psc) is invariant under surgeries of codimension at least 3. Working with cobordism theory on a refined statement by Chernysh of the latter, Ebert and Frenck proved that the homotopy type of the space of psc metrics on a manifold M, if non empty, depends on a certain cobordism class of M. In this talk I will discuss a conjecture that claims that the homotopy is in fact only dependent on the normal 2-type of M. We will also see how this is related to the "isotopy implies concordance" conjecture.

La comprensione dei punti razionali di una varietà algebrica arbitraria è una ricerca tanto affascinante quanto complessa. Un esempio classico è fornito da "L'ultimo teorema di Fermat". Una possibile strategia in tale studio consiste nel cercare di costruire parametrizzazioni di tali punti razionali. La nozione di R-equivalenza è stata introdotta da Manin, nel 1972, con lo scopo di formalizzare questo tipo di questione. Il seminario sarà introduttivo. Prima di tutto verranno richiamate delle nozioni di base di geometria algebrica, dopodiché verrà data la definizione di R-equivalenza, il tutto accompagnato da esempi e motivazioni. Nella seconda parte verrà presentato un risultato, dovuto a Colliot-Thélène e Sansuc, sul calcolo delle classi di R-equivalenza di tori algebrici.

The Jones polynomial introduced in 1984 was the first so-called quantum invariant in knot theory. It associates to every knot a polynomial, which is not defined in a geometric but rather in a diagrammatic way. Fifteen years later, Khovanov homology was introduced as a so-called categorification of the Jones polynomial. In spite of its combinatoric origins, Khovanov homology has an ever-growing number of geometric applications in knot theory. In this mini-course, we will first work through the detailed construction of Khovanov homology, following a modern version of Bar-Natan’s tangle setup. Then, we will see some of the applications, such as Rasmussen’s proof of the Milnor conjecture (regarding the unknotting number of torus knots). Previous knowledge of knot theory and homological algebra is helpful, but will not be required.

Hyperbolic spaces and groups were introduced by Gromov and have been studied extensively over the past few decades from a huge variety of points of view, and with applications to various fields such as the study of hyperbolic 3-manifolds. Many groups of interest in geometry and topology, most notably mapping class groups, are however not hyperbolic, and it is therefore natural to try to generalise the notion of the hyperbolic group to encompass as many of these examples as possible. The notion of hierarchical hyperbolicity, inspired by the deep work of Masur and Minsky, achieves just that. Roughly speaking, a hierarchically hyperbolic structure on a metric space or group is a coordinate system where the coordinates take values in hyperbolic spaces. I will give an introduction to hierarchically hyperbolic spaces, thereby describing the (coarse) geometry of mapping class groups, and explain what this kind of structure is good for and how to show that a given space or group is hyerarchical hyperbolic.

Goodwillie-Weiss embedding calculus can be used to study diffeomorphism grous, finite type invariants, and isotopy classes of surfaces in 4-manifolds. We aim to give a rough outline of the theory.

Data N una 3-varietà iperbolizzabile compatta con bordo, il volume rinormalizzato è una funzione a valori reali sullo spazio delle strutture iperboliche convesse cocompatte su N, le quali hanno volume iperbolico infinito. Poiché il Teichmüller del bordo di N parametrizza le strutture convesse cocompatte su N, possiamo pensare al volume rinormalizzato come una funzione definita sul Teichmüller del bordo.

Dopo un’introduzione alla geometria delle metriche convesse cocompatte su N, vedremo come si definisce il volume rinormalizzato ed alcune sue proprietà. Concluderemo enunciando alcuni risultati che descrivono come il volume rinormalizzato sia correlato al volume del convex core della varietà.

Quantum invariants of three-manifolds were discovered and defined in the early 1990s by Witten-Reshetikhin-Turaev. The physical approach to these invariants suggested they satisfy interesting links to the geometry of three-manifolds through perturbation theory.

On the other hand the mathematical description does not make these links at all clear. In exploring these perturbative properties numerically, Zagier noticed that these quantum invariants satisfy strange and new kinds of modular properties. This has led to the idea of quantum modular forms understood in subsequent work of Garoufalidis-Zagier. Then work of Garoufalidis-Gu-Mariño has used this idea of quantum modularity to give a method of computing Borel resummation of associated asymptotic series and their Stokes phenomenon. This story has been studied in the case of some simple hyperbolic knots and I will describe this, along with the extension to the case of simple closed hyperbolic three-manifolds.

Data un'estensione di campi K<F ed un elemento a in F, un oggetto naturale da studiare è l'ideale I(a) di K[x] dei polinomi p(x) tali che p(a)=0; tale ideale è non-banale esattamente quando a è algebrico su K, e in tal caso risulta essere un ideale principale, con una struttura quindi estremamente semplice.

Un'analoga domanda può essere posta nel contesto dei gruppi: dati due gruppi H<G ed un elemento g in G, siamo interessati a studiare l'insieme I(g) delle equazioni w(x) in H*<x> tali che w(g)=1. La risposta risulta essere parecchio più complicata rispetto al caso dei polinomi. Nel seminario, andiamo ad affrontare il problema nel caso particolare di un'estensione di gruppi liberi, e studiamo l'ideale I(g) con particolare attenzione al grado delle equazioni che contiene. Mostriamo come diversi risultati al riguardo possono essere ottenuti utilizzando i lingauggi liberi dal contesto, uno strumento che è stato largamente studiato in informatica.

A common theme in algebraic geometry is degeneration: we consider a family of smooth spaces which degenerates into a singular one. A complementary viewpoint is that of compactification: the total family can be seen as the union of the open locus formed by the smooth fibres and the singular fibre. Logarithmic geometry is a tool that unites these viewpoints. In this talk I will give an introduction to the subject, show how they extend algebraic geometry and why it is useful in studying degeneration and compactification problems.

Dato un grafo, come si può partizionare in due pezzi "grossi", tagliando il minor numero di archi possibile? La divisione ottimale è rivelata dalla costante di Cheeger, che è strettamente legata al secondo autovalore del Laplaciano del grafo.

Un circle packing è una collezione di cerchi su una superficie, di raggio eventualmente diverso, che possono essere tangenti tra di loro. Così come un grafo può essere un'approssimazione discreta di una varietà, una mappa tra circle packing su due superfici è un'approssimazione discreta di una mappa olomorfa tra le superfici.

Nel seminario vedremo come realizzare un circle packing sulla sfera per poter stimare la costante di Cheeger di un grafo planare. Vedremo poi come generalizzare questo risultato su superfici iperboliche di genere più grande.

The theory of Drinfeld modules, pioneered by Anderson and Thakur in the 90's, was conceived as a possible analogue to the theory of complex elliptic curves in finite characteristic, where the role of the ring of integers Z is assumed by the ring of regular functions of some curve X/F_q. We will introduce the analogues of the real and complex numbers, of the period lattices, and of the exponential map in this context.

Two novel objects - special functions and Pellarin L-functions - arise in this theory, which have no parallel in characteristic zero, and can be conceived as interpolations of Gauss sums and Dirichlet L-functions, respectively; we will present some results about their relation, and compare them to the classical functional equation for Dirichlet L-functions.

Equivariant topology studies topological spaces that satisfy a given notion of symmetry. 

We introduce the study of knots in the equivariant setting, focusing on strongly invertible knots. We provide all the basic definitions and tool and define both classic and novel invariants.

Finally, we use a new invariant to study the equivariant concordance order of 2-bridge knots. This is joint work with Alessio Di Prisa. 

In 1969 Levine defined a surjective homomorphism from the knot concordance group to the so called algebraic concordance group, which is a Witt group of Seifert forms.

Studying symmetric knots and in particular strongly invertible knots, a natural question is whether it is possible to define an appropriate equivariant version of algebraic concordance.

In this talk we briefly recall Levine's construction and we highlight some of the problems occuring when trying to define its equivariant analogous.

Finally, we define a notion of equivariant algebraic concordance for strongly invertible knots and we show some of the differences with the classical case.

The theory of braid foliations was developed to study knots and links, as well as surfaces in 3-manifolds. The original idea goes back to Bennequin and his works on exotic contact structures on $R^3$ (1989) and was then developed by Birman and Menasco in a series of papers during the 90’s. These techniques have proved to be very useful to solve many foundational problems in braid theory and contact topology, such as Jones conjecture, the Legendrian grid number conjecture, Bennequin’s inequality, transverse Markov theorem and many others. In this talk we will see a proof by means of braids foliations of Markov theorem, a fundamental result connecting braids and links. This proof is due to Birman and Menasco (2002). 

Hironaka’s desingularization results are very powerful tools that allow to desingularize algebraic sets defined over fields of characteristic zero. If S ⊂ Rn is a semialgebraic set (i.e. the set of solutions of finitely many systems of equations and inequalities), what does it mean ‘desingularize S’? In this talk we will see how to desingularize (a certain class of) semialgebraic sets and some applications of this result.

Il talk prevede una panoramica sulle proprietà di rigidità del Mapping Class Group di una superficie. Ci focalizzeremo in particolare su come molte proprietà algebriche di questo gruppo possano essere verificate studiando la sua azione sul "Curve graph", un grafo che cattura il comportamento delle curve sulla superficie. Parallelamente mostreremo come queste proprietà siano ereditate da un particolare quoziente del MCG e possano essere riconosciute tramite l'azione sul corrispondente quoziente del Curve graph. I risultati di questa seconda parte nascono da una collaborazione con Alessandro Sisto. 

In this seminar I will present some notions of algebraic geometry over subfields introduced and studied in depth in a recent work by Fernando and Ghiloni. I will give an idea of why the most interesting cases to study are those in which the ground field is a real closed field, such as ℝ, and the subfield is not real closed, such as ℚ. We will introduce several notions of real algebraic sets to be `described over ℚ' and explain some characterizations via Galois theory proved by Fernando-Ghiloni and Ghiloni and the speaker. Time permitting, I will give an insight into which class of algebraic sets `described over ℚ' is adequate for solving the ℚ-algebrization problem by means of algebraic approximation techniques. 

Circa venti anni fa, Fomin-Zelevinsky introdussero un metodo puramente algebrico per generare polinomi di Laurent con coefficienti dalle proprietà curiose. Darò una visione panoramica di un programma, in parte realizzato e in parte congetturale, il cui scopo è usare questi polinomi per risolvere un problema di topologia contemporanea: la classificazione dei tori Lagrangiani in una varietà simplettica compatta.