# Geometry

The scientific activities of the group are focused on different aspects of Algebraic and Arithmetic Geometry, Complex and Differential Geometry, Combinatorics, and Geometric Topology.

The group runs the following Seminars Series:

Usually, the group organizes workshops, conferences, and summer schools. A list of the upcoming ones is available on this page.

#### Research Topics

##### Members:
• Luca Bruni
• Marco Franciosi
• Matthias Leopold Nickel
• Rita Pardini
• Gregory James Pearlstein
• Mattia Pirani
• Francesco Sala
• Tamás Szamuely
• Mattia Talpo
##### Cohomology of varieties and arithmetic questions

Given an algebraic variety $X$ defined over a field $k$, one can associate to it various cohomology groups: coherent, étale, $p$-adic, or even motivic. These groups reflect the geometry, and in case $k$ is of arithmetic interest, the arithmetic of the variety $X$. Using cohomological methods we study, among other things:

• algebraic cycles
• algebraic fundamental groups
• local-global principles for rational points
##### Members:
• Mattia Pirani
• Tamás Szamuely
##### Collaborators:
• Philippe Gille (Lyon)
• David Harari (Orsay)
• Damian Rössler (Oxford)
##### Moduli stacks, cohomological Hall algebras, and quantum groups

Quantum groups encode the ‘hidden symmetries’ of quantum physics via integrability and the geometric approach to them has been successful in representation theory (e.g. the theory of Maulik-Okounkov Yangians), the theory of moduli stacks and spaces (e.g. the proof of Beauville and Voisin’s conjectures of Maulik-Negut), and theoretical physics (e.g. the proof of the Alday-Gaiotto-Tachikawa conjecture).
The group aims at investigating quantum groups via their geometric incarnations in terms of Hall algebras and their refined versions (cohomological, K-theoretical, categorified). At the moment, we are focusing on:

• 2-dimensional cohomological Hall algebras of curves and surfaces;
• continuum Lie algebras and continuum quantum groups, and their relations to Hall algebras of constructible sheaves on real curves
##### Members:
• Luca Bruni
• Francesco Sala
##### Logarithmic and tropical algebraic geometry

Logarithmic geometry is an enhanced version of algebraic geometry, where spaces are equipped with an additional structure sheaf, which encodes information of a combinatorial nature (e.g. toric varieties). This recent theory has been fruitfully applied to questions regarding special kinds of degenerations of varieties or other more complicated objects, and compactifications of moduli spaces, for example in the context of mirror symmetry. There are also very interesting interactions with the field of tropical (and non-Archimedean) geometry. Our activity focuses for example on

• the study of moduli spaces of parabolic bundles (some notion of coherent sheaf, adapted to log schemes)
• sheaf-counting on log smooth varieties
• interactions between (log) algebraic and tropical moduli spaces (e.g. for curves with level structures)
• Mattia Talpo
##### Collaborators:
• Sarah Scherotzke (Luxembourg)
• Nicolò Sibilla (SISSA)
• Bernd Siebert (Austin)
• Richard Thomas (Imperial)
• Martin Ulirsch (Frankfurt)
##### Moduli spaces of surfaces

The moduli space of surfaces of general type is well known to have an intricate structure. Its “geography” has been extensively studied. Furthermore, a modular compactification of it is the moduli
space of stable surfaces, i.e. semi-log-canonical surfaces with ample canonical divisor. The activity of the group is focused on the analysis of the compactified moduli space $\mathcal{M}(a,b)$ (where $a=K^2$ and $b$ is the holomorphic Euler characteristic), with particular attention to the case of surfaces with low numerical invariants. Such analysis is given by studying log-canonical pairs via a classical approach and analyzing the singularities, via Deformation Theory, and a detailed study of the canonical ring.
Related goals are to extend the knowledge of the moduli space by analyzing $\mathbb{Q}$-Gorenstein surfaces and to study the Hodge theoretic approach, by associating to a variety its cohomology and analyzing the induced variation of Hodge structures.

##### Members:
• Marco Franciosi
• Matthias Leopold Nickel
• Rita Pardini
• Gregory James Pearlstein
##### Collaborators:
• Sönke Rollenske (Univ. Marburg, Germany)
• Stephen Coughlan (Univ. Bayreuth, Germany)
• Julie Rana (Lawrence University, USA)
• Barbara Fantechi (SISSA, Italy)
##### Combinatorics of evolutionary structures

I am interested in combinatorial methods and structures of use in the study of the evolutionary relationships among or within groups of organisms. A particular focus is on coalescent models of evolution, in which gene trees, representing the evolutionary history of individual genes sampled from a set of species, evolve along the branches of species trees, reflecting the history of species divergences. In order to understand how features of the species tree can influence the distribution of the possible gene trees, the number and probability of the combinatorially different configurations that gene trees can assume within a given species tree are investigated. When individual gene copies are selected within a single species, the gene tree is modeled as a random coalescent tree that evolves independently of the branching pattern of the species tree, and the goal is to describe the distributive properties of its combinatorial parameters.

##### Member:
• Filippo Disanto
##### Collaborators:
• Michael Fuchs (Taipei)
• Noah Rosenberg (Stanford)
• Thomas Wiehe (Koeln)
##### Members:
• Giuseppe Bargagnati
• Riccardo Benedetti
• Federica Bertolotti
• Filippo Bianchi
• Filippo Gianluca Callegaro
• Pietro Capovilla
• Jacopo Guoyi Chen
• Carlo Collari
• Roberto Frigerio
• Giovanni Gaiffi
• Viola Giovannini
• Giovanni Italiano
• Paolo Lisca
• Domenico Marasco
• Bruno Martelli
• Alice Merz
• Matteo Migliorini
• Francesco Milizia
• Andrea Parma
• Carlo Petronio
• Mario Salvetti
• Diego Santoro
• Viola Siconolfi
• Andrea Tamburelli
##### Classical and higher rank Teichmüller theory

This area of research studies the geometric and dynamical properties of representations of the fundamental group of a surface $S$ (of negative Euler characteristic) into a Lie group $G$. For example, when $G=\mathbb{P}\mathrm{SL}(2, \mathbb{R})$, conjugacy classes of discrete and faithful representations are in bijection with the Teichmüller space of $S$, the space of marked hyperbolic (or complex) structures on $S$. More in general, and especially for Lie groups of rank $2$ (i.e., $G=\mathrm{SL}(3,\mathbb{R}), \mathrm{Sp}(4, \mathbb{R}, \mathrm{SO}(2,2), G_{2}$), researchers have identified special connected components of the character variety $\mathrm{Hom}(\pi_{1}(S), G)/G$ that parametrize geometric structures on $S$, or fiber bundles over $S$, and share a lot of similarities with the classical Teichmüller space.

The main goal of this research is to understand to which extent the classical Teichmüller theory generalizes to the higher rank. Some aspects include:

• the study of diverging sequences of representations and the definition of the analogue of Thurston’s boundary for higher rank Teichmüller spaces;
• the analysis of equivariant harmonic maps from the universal cover of $S$ into the symmetric space $G/K$ and the real Euclidean building modeled on $\mathfrak{g}$;
• the definition of natural (pseudo)-Riemannian metrics on these higher Teichmüller components and the study of their global geometry.
##### Member:
• Andrea Tamburelli
##### Collaborators:
• John Loftin (Newark)
• Charles Ouyang (Amherst)
• Michael Wolf (Georgia Tech)
##### Hyperbolic geometry

The uniformisation of surfaces of Koebe and Poincaré and the geometrisation of 3-manifolds of Thurston and Perelman have shown that every manifold of dimensions 2 and 3 admits a geometric structure (after cutting along some canonical spheres and tori in dimension 3). The prominent role among these geometric structures is played by hyperbolic geometry, that is by far the prevalent
structure. It is also the richest and most studied structure in dimensions 2 and 3.

The deformation spaces of hyperbolic 2- and 3-manifolds are the focus of a vast literature concerning Teichmueller spaces and hyperbolic fillings of open manifolds. Moreover, the topology of hyperbolic 3-manifolds is a central topic in low-dimensional topology. But hyperbolic manifolds are abundant in any dimension, and a major goal is to understand their topology as well as their deformation spaces. To this aim, the members of the research group rely on many techniques, from the decomposition of manifolds into hyperbolic polytopes to the study of the topology of fibrations over the circle, to the investigation of the variety of representations of discrete groups into the Lie group of the isometries of hyperbolic space.

##### Members:
• Jacopo Guoyi Chen
• Roberto Frigerio
• Viola Giovannini
• Giovanni Italiano
• Bruno Martelli
• Matteo Migliorini
• Carlo Petronio
• Diego Santoro
##### Collaborators:

Leone Slavich (Università di Pavia)

##### Hyperplane arrangements

The group investigates the combinatorial and topological properties of hyperplane arrangements. From such a point of view, we study the theory of Coxeter groups (seen as reflection groups), Artin groups (seen as fundamental groups of the complements of reflection arrangements), and the computation of cohomology groups of the complements of hyperplane arrangements, both in the linear and the toric cases.
To obtain an explicit characterization of the cohomology ring of the complement of a toric arrangement, the group is studying wonderful compactifications of the complements. This approach allows the definition of a certain differential graded algebra, which ‘governs’ the cohomology ring.
At the moment, the group is interested in the following topics:

• the $K(\pi, 1)$ conjecture for all possible Coxeter groups and the corresponding hyperplane arrangements (for example, the affine simplicial arrangements, which are a natural generalization of the affine reflection arrangements);
• the study of the so-called dual Coxeter groups, which depend on an element of the group and an interval formed by its divisors;
• the study of combinatorial properties (such as the shellability property) for intervals as above;
• the construction of an explicit basis of the integral ring cohomology of complements of toric arrangements and the explicit characterization of the corresponding differential graded algebra in specific examples.
##### Members:
• Filippo Gianluca Callegaro
• Giovanni Gaiffi
• Mario Salvetti
• Viola Siconolfi
##### Low-dimensional topology

This wide research area encompasses several subjects of interest for the research group.
In dimension 2, for example, we investigate the Hurwitz problem concerning the existence of branched covering between surfaces realizing a fixed combinatorial datum. To this aim, one may exploit Grothendieck’s dessins d’enfant, as well as the geometry of spherical, flat, and hyperbolic 2-orbifolds.

In dimension 3 some topics of interest for the group are the Heegaard-Floer homology of rational homology 3-spheres (with particular attention towards possible applications to the L-space conjecture) and the theory of knots and links in the sphere and in general 3-manifolds. A particular interest is devoted to Legendrian links, and to the study of Khovanov homology. 3-manifold topology is also involved in the study of apparent contours of surfaces in 3-space and in general 3-manifolds.

The topology of 4-manifolds is a very active research field, and the group is also interested in this area. Among the topics covered by the group, there are the study of handlebody decompositions of 4-manifolds, Heegaard-Floer homology, and  3-dimensional knot theory from a 4-dimensional viewpoint.

Members:

• Filippo Bianchi
• Carlo Collari
• Paolo Lisca
• Bruno Martelli
• Alice Merz
• Andrea Parma
• Carlo Petronio
• Diego Santoro
##### Simplicial volume and bounded cohomology

The simplicial volume is a homotopy invariant of manifolds defined by Gromov in 1982. Despite its purely topological definition, it is deeply related to the geometric structures that a manifold can carry.

Thanks to Thurston’s (now proved) Geometrization Conjecture, the simplicial volume of closed 3-manifolds is well understood. Much less is known in higher dimensions, or for open manifolds, and the group is interested in further investigating these research fields (with particular care devoted to aspherical manifolds).

A powerful tool for the computation of the simplicial volume is the so-called bounded cohomology (of groups and of spaces), which is itself a very active research field. Computing the bounded cohomology of groups is very challenging (for example, the problem of whether it vanishes or not for free groups in degrees bigger than 3 is still open), and the research group aims at achieving some progress in this direction, as well as at studying the relationship between bounded cohomology and other related areas like representation theory, group actions on the circle, ergodic theory of groups.

##### Members:
• Giuseppe Bargagnati
• Federica Bertolotti
• Pietro Capovilla
• Roberto Frigerio
• Domenico Marasco
• Bruno Martelli
• Francesco Milizia
##### Collaborators:
• Michelle Bucher (Université de Genève)
• Clara Loeh (Regensburg University)
• Marco Moraschini (Università di Bologna)
• Maria Beatrice Pozzetti (Heidelberg University)
• Roman Sauer (Karlsruhe University)
• Alessandro Sisto (Heriot-Watt University, Edinburgh)
##### Holomorphic dynamical systems, complex differential geometry, and geometric function theory

The group focuses on different aspects of complex and differential geometry from both an analytical and a geometric viewpoint.

##### Holomorphic dynamical systems

In the last forty years, the study of holomorphic dynamical systems has become one of the most important topics in complex analysis and complex geometry of one and several variables, at the forefront of contemporary mathematical research. In Pisa we are particularly interested in studying:

• the global dynamics of holomorphic self-maps of hyperbolic manifolds and domains, and more generally of non-expanding self-maps of Gromov hyperbolic metric spaces;
• the local dynamics around a non-hyperbolic fixed point;
• the dynamics of meromorphic connections on hyperbolic Riemann surfaces.
##### Members:
• Marco Abate
• Matteo Fiacchi
##### Collaborators:
• Fabrizio Bianchi (Laboratoire Paul Painlevé, Lille, CNRS, France)
• Jasmin Raissy (Université de Bordeaux, France)
• Karim Rakhimov (Laboratoire Paul Painlevé, Lille, CNRS, France; Tashkent University, Uzbekistan)
##### Geometric function theory

A characteristic feature of complex analysis is the use of geometrical tools to study analytic phenomena. A typical example consists in using the behaviour of the natural invariant (under biholomorphisms) metrics and distances defined on complex manifolds to study the boundary behaviour of holomorphic functions or the action of integral operators on spaces of holomorphic functions. In particular, we are using the Kobayashi metric and distance in pseudoconvex and convex domains to study the boundary behaviour of the derivatives of a holomorphic function at a specific point in the boundary and, more recently, to study the mapping properties of Toeplitz operators on weighted Bergmann spaces using characterisations of Carleson measures expressed in terms of Kobayashi balls.

##### Members:
• Marco Abate
• Matteo Fiacchi
##### Collaborators:
• Fabrizio Bianchi (Laboratoire Paul Painlevé, Lille, CNRS, France)
• Jasmin Raissy (Université de Bordeaux, France)
• Karim Rakhimov (Laboratoire Paul Painlevé, Lille, CNRS, France; Tashkent University, Uzbekistan)
##### Complex differential geometry

We studied complex Finsler manifolds, having in mind as a guiding example hyperbolic manifolds endowed with the Kobayashi metric, and in particular Kahler-Finsler manifolds with constant holomorphic curvature. Using techniques coming from both differential geometry and algebraic topology we studied the rich geometrical structure of analytic varieties that can be obtained as fixed point sets of a holomorphic self-map, proving a number of index theorems generalising to this setting classical Baum-Bott and Lehmann-Suwa theorems known for holomorphic foliations, and with applications to holomorphic dynamics.

• Marco Abate

#### Members

##### Staff
FilippoDisanto[Mathscinet]
SandroManfredini[Mathscinet]
Gregory JamesPearlstein[Mathscinet] [Orcid]
MarioSalvetti[Mathscinet] [Orcid]
##### Postdoctoral Fellows
CarloCollari[Mathscinet] [Orcid]
MatteoFiacchi[Mathscinet]
ViolaSiconolfi[Mathscinet] [Orcid]
##### Ph.D. Students
GiuseppeBargagnati[Mathscinet]
FilippoBianchi
ViolaGiovannini
DomenicoMarasco
AliceMerz
AndreaParma
MattiaPirani
##### Ph.D. Students at Scuola Normale Superiore co-supervised by members of the group
FedericaBertolotti
LucaBruni
PietroCapovilla
Jacopo GuoyiChen
GiovanniItaliano
MatteoMigliorini
FrancescoMilizia
DiegoSantoro
##### 2022
• Ludovico Battista (Università di Pisa), “Hyperbolic 4-manifolds, perfect circle-valued Morse functions and infinitesimal rigidity”, supervised by Bruno Martelli.
##### 2021
• Federico Conti (Università di Pisa), “Surfaces close to the Severi lines”, supervised by Rita Pardini.
• Leonardo Ferrari (Università di Pisa), “Hyperbolic manifolds and coloured polytopes”, supervised by Bruno Martelli.
• Chiara Spagnoli (Università di Pisa), “The eventual map for irregular varieties”, supervised by Rita Pardini.
##### 2020
• Giulio Belletti (Scuola Normale Superiore di Pisa), “Asymptotic behavior of quantum invariants”, supervised by Francesco Costantino and Bruno Martelli.
• Karim Rakhimov (Università di Pisa), “Dynamics of geodesics for meromorphic connections on Riemann surfaces”, supervised by Marco Abate.
##### 2018
• Kirill Kuzmin (Università di Pisa), “Quasi-Isometric Rigidity for Universal Covers of Manifolds with a Geometric Decomposition”, supervised by Roberto Frigerio.
• Marco Moraschini (Università di Pisa), “On Gromov’s theory of multicomplexes: the original approach to bounded cohomology and simplicial volume”. supervised by Roberto Frigerio.
##### 2017
• Stefano Riolo (Università di Pisa), “Cone-manifolds and hyperbolic surgeries”, supervised by Bruno Martelli.
##### 2016
• Fabrizio Bianchi (Università di Pisa and Université de Toulouse III Paul Sabatier), “Motion of Julia sets and dynamical stability in several complex variables”, supervised by Marco Abate.
• Alessio Carrega (Università di Pisa), “Shadows and Quantum Invariants”, supervised by Bruno Martelli.
• Federico Franceschini (Università di Pisa), “Simplicial volume and relative bounded cohomology”, supervised by Roberto Frigerio.
##### 2014
• Leone Slavich (Università di Firenze), “Hyperbolic 4-manifolds and 24-cells”, supervised by Bruno Martelli.
##### 2013
• Valentina Disarlo (Scuola Normale Superiore di Pisa), “Combinatorial methods in Teichmüller theory”, supervised by Athanase Papadopoulos and Carlo Petronio.
• Mattia Pedrini (SISSA – Trieste), “Moduli spaces of framed sheaves on stacky ALE spaces, deformed partition functions and the AGT conjecture”, supervised by Ugo Bruzzo and Francesco Sala.
##### 2012
• Cristina Pagliantini (Università di Pisa), “Relative (continuous) bounded cohomology and simplicial volume of hyperbolic manifolds with geodesic boundary”, supervised by Roberto Frigerio.
• Michele Tocchet (Sapienza Università di Roma), “Generalized Mom-structures and volume estimates for hyperbolic 3-manifolds with geodesic boundary and toric cusps”, supervised by Katya Pervova and Carlo Petronio.
##### 2011
• Isaia Nisoli (Università di Pisa), “A general approach to Lehmann-Suwa-Khanedani index theorems: partial holomorphic connections and extensions of foliations”, supervised by Marco Abate.
• Fionntan Roukema (Università di Pisa), “Dehn Surgery on the Minimally Twisted Five-Chain Link”, supervised by Bruno Martelli and Carlo Petronio.
• Matteo Ruggiero (Scuola Normale Superiore di Pisa), “The valuative tree, rigid germs and Kato varieties”, supervised by Marco Abate.
• Vito Sasso (Università di Roma Tor Vergata), “Complexity of unitrivalent graph-pairs and knots in 3-manifolds”, supervised by Katya Pervova and Carlo Petronio.
##### 2010
• Tiziano Casavecchia (Università di Pisa), “Rigidity of holomorphic generators of one-parameter semigroups and a non-autonomous Denjoy-Wolff theorem”, supervised by Marco Abate.
• Maria Antonietta Pascali (Sapienza Università di Roma), “Branched covers between surfaces”, supervised by Carlo Petronio.
• Jasmin Raissy (Università di Pisa), “Geometrical methods in the normalization of germs of biholomorphisms”, supervised by Marco Abate.
##### 2007
• Daniele Alessandrini (Scuola Normale Superiore di Pisa), supervised by Riccardo Benedetti.
##### 2005
• Francesco Bonsante (Scuola Normale Superiore di Pisa), “Deforming the Minkowskian cone of a closed hyperbolic manifold”, supervised by Riccardo Benedetti.
• Roberto Frigerio (Scuola Normale Superiore di Pisa), “Deforming triangulations of hyperbolic 3-manifolds with geodesic boundary”, supervised by Carlo Petronio.
##### 2004
• Gennaro Amendola (Università di Pisa), “Minimal spines and skeleta of non-orientable 3-manifolds and bricks”, supervised by Carlo Petronio.
• Francesco Costantino (Scuola Normale Superiore di Pisa), “Shadows and branched shadows of 3 and 4-manifolds”, supervised by Riccardo Benedetti.
• Stefano Francaviglia (Scuola Normale Superiore di Pisa), “Hyperbolicity equations for cusped 3-manifolds and volume-rigidity of representations”, supervised by Carlo Petronio.
##### 2002
• Bruno Martelli (Università di Firenze), “Complexity of three-manifolds”, supervised by Carlo Petronio.
##### 1999
• Silvia Benvenuti (Università di Pisa), supervised by Riccardo Benedetti.
##### 1995
• Carlo Petronio (Scuola Normale Superiore di Pisa), “Standard spines and 3-manifolds”, supervised by Riccardo Benedetti.
##### 1992
• Domenico Luminati (Università di Pisa), “Immersions of surfaces and apparent contours”, supervised by Riccardo Benedetti.
##### External Collaborators
FrancescaAcquistapace[Mathscinet]
FabrizioBroglia[Mathscinet]
ElisabettaFortuna
MargheritaGalbiati[Mathscinet]

#### Grants

• ##### Grant-in-Aid for Scientific Research (C): Yangians and Cohomological Hall algebras of curves (KAKENHI JSPS)

Principal Investigator: Francesco Sala

• ##### Advanced in Moduli Theory and Birational Classification (Prin 2017)

Principal Investigator: Lucia Caporaso (Università di Roma 3) | Coordinator of the Research Unit: Rita Pardini

Members of the Research Unit: Marco Franciosi, Enrico Sbarra, Tamás Szamuely, Mattia Talpo

• ##### Moduli and Lie Theory (Prin 2017)

Principal Investigator: Kieran G. O'grady | Coordinator of the Research Unit: Mario Salvetti

• ##### Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics (Prin 2017)

Principal Investigator: Filippo Bracci (Università di Roma Tor Vergata) | Coordinator of the Research Unit: Paolo Lisca

Members of the Research Unit: Marco Abate, Giuseppe Bargagnati, Filippo Bianchi, Carlo Collari, Roberto Frigerio, Paolo Lisca, Domenico Marasco, Bruno Martelli, Alice Merz, Andrea Parma, Carlo Petronio, Andrea Tamburelli

• ##### Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni (Progetti di Ateneo)

Coordinator of the Research Unit: Giacomo Tommei

Members of the Research Unit: Marco Abate, Riccardo Barsotti, Giulio Baù, Alessandro Berarducci, Gianluigi Del Magno, Mauro Di Nasso, Giovanni Federico Gronchi, Stefano Marò, Giacomo Lari, Daniele Serra, David Antonio Riccardo Mustaro

• ##### Spazi di moduli, geometria aritmetica e aspetti storici (Progetti di Ateneo)

Coordinator of the Research Unit: Mattia Talpo

Members of the Research Unit: Andrea Bandini, Alberto Cogliati, Valerio Melani, Francesco Sala, Tamás Szamuely

• ##### Sistemi dinamici in analisi, geometria, logica e meccanica celeste (Progetti di Ateneo)

Coordinator of the Research Unit: Marco Abate

Members of the Research Unit: Marco Abate, Claudio Bonanno, Mauro Di Nasso, Stefano Galatolo, Alessandro Berarducci, Giovanni Federico Gronchi, Carlo Carminati, Giacomo Tommei