**Arrangements, configuration spaces and braid groups**

Local cohomology of the complement of an arrangement. Cohomology ring of the complement of a toric arrangement. Homology of braid groups with coefficients in symplectic representations. Homology of regular orbit spaces for groups of complex reflections for classifying the fundamental groups of these spaces. Combinatorics of wonderful models and some of their blow-ups, in relation to certain actions of the symmetric group. Homological stability for complex braid groups. Combinatorial methods for Artin and Coxeter groups and description of new algebraic structures for the infinite series of Coxeter groups**People**

Filippo Callegaro

Mario Salvetti

Giovanni Gaiffi

Ivan Marin (Univ. Picardie)

Fred Cohen (Univ. Rochester)

Emanuele Delucchi (Univ. Fribourg) Michele D’Adderio (Univ. Libre de Bruxelles)

Lorenzo Guerra (PhD student, SNS)

Giovanni Paolini (PhD student, SNS)

Roberto Pagaria (PhD student, SNS)

Oscar Papini (PhD student, Dept. of Mathematics, Univ. of Pisa)**Recent grants**

PRIN 2015-2016 “Moduli Spaces and Lie theory”

National coordinator: Kieran O’Grady

Local coordinator: Mario Salvetti

FIRB 2012 project ”Perspectives in Lie Theory”” (grant n. RBFR12RA9W 003)

Local coordinator: Filippo Callegaro

Supporting institution MIUR

**Algebraic Geometry**

Algebraic varieties of maximal Albanese dimension: eventual behaviour of linear systems and numerical inequalities of Clifford-Severi type. The KSBA compactification of the moduli space of surfaces of general type with small K2. Singular algebraic curves: numerical connectedness and properties of the dualizing sheaf. Stratifications of configuration spaces of k points in complex affine and projective space and Grassmannians.

Algebraic stacks. Logarithmic geometry and moduli spaces of parabolic sheaves.

Derived algebraic geometry. Shifted Poisson and symplectic structures.

Hall algebras (classical, cohomological, categorified), their relation to the theory of Yangians and quantum affine algebras, and their relation to the topology of moduli spaces of stable sheaves on smooth surfaces and Nakajima quiver varieties.

**People** Rita Pardini Marco Franciosi

Sandro Manfredini

Valerio Melani

Francesco Sala

Tamas Szamueli

Mattia Talpo

Miguel Ángel Barja (UPC, Barcelona)

Mauro Porta (Université de Strasbourg, France)

Sönke Rollenske (Univ. Marburg, Germania)

Sarah Scherotzke (Univ. Luxembourg)

Olivier Schiffmann (CNRS and Université Paris-Saclay, France)

Nicolò Sibilla (SISSA, Trieste)

Lidia Stoppino (Univ. Insubria)

**Recent Grants**

PRIN 2017- 2017SSNZAW “Advances in Moduli Theory and Birational Classification”

National coordinator: Lucia Caporaso

Local coordinator: Rita Pardini

Supporting institution: MIUR

PRIN 2015 “Geometria delle Varieta’ Algebriche”

Web page: http://ricerca.mat.uniroma3.it/GVA/

National coordinator: Alessandro Verra

Local coordinator: Rita Pardini

Supporting institution: MIUR

**Real and Analytic Algebraic Geometry**

Algebraic properties of algebra of global analytic functions on a real analytic space (Nullstellensatz, …). HIlbert’s 17th problem for global analytical functions. Global semianaliticity of notable subgroups (points of inconsistency, points of regularity of fixed dimension, ….) of C-spaces. Algebraic properties of quasianalytic germ function algebra (division and preparation theorem of Weierstrass, Nullstellensatz, Noetherian properties, …). Local study of semi-analytical, local algebraic approximation with conservation of geometric properties and multiplicity of a point. Semi-analytical and sub-analytical sets. Real algebraic curves.**People**

Francesca Acquistapace

Fabrizio Broglia

Jose’ Fernando (Univ. Complutense Madrid)

Carlos Ueno (Kodály Zoltán Gimnázium, Pécs, Hungary)

Andreea Nicoara (Trinity College, Dublin)

Elisabetta Fortuna

Massimo Ferrarotti (Polytechnic Univ. of Turin)

Les Wilson (University of Hawaii, Manoa, Honolulu, USA)

Margherita Galbiati**Recent grants**

PRIN 2010-11 “Varieta’ reali e complesse: geometria, topologia e analisi armonica”

National coordinator: Fulvio Ricci – SNS Pisa

Local coordinator: Riccardo Benedetti

Supporting institution: MIUR

GRAYAS 2015-2018 “Geometria Real y aplicaciones”, MTM2014-55565-P

Scientific coordinators: Jose Javier Etayo, Jose F. Fernando

Supporting institution: MINECO

STRANO 2018-2021 “Estructuras algebraicas, analiticas y ominimales”, MTM2017-82105-P

Scientific coordinators: Jose Javier Etayo, Jose F. Fernando

Supporting institution: MINECO

**Low-dimensional Geometry and Topology**

Analytical families of “quantum-hyperbolic” invariants of 3-manifolds (with particular attention to families that dominate the geometric component of the range of characters of hyperbolic 3-manifold with a cusp); asymptotic behavior when the level approaches infinity (variants of “Volume conjecture”); study of the anomaly and functional identity of the symmetrization factor; the relationship with other “quantum” invariants of 3-manifolds ‘. Limited cohomology of discrete groups, with particular reference to simplicial volume applications, and on geometric group theory, with particular attention to metric spaces with curvature constraints (CAT(k) spaces). Hyperbolic ranges in dimension 3 and above; complexity of 3- and 4-manifolds, quantum invariants of 3-manifolds. Complexity of torus knots in lens spaces: extension of the techniques of Tillman et al. Existence of branched coverings between surfaces which induce covers between hyperbolic orbifolds with high-dimensional Teichmuller space. Apparent contours of surfaces on 3-manifold spines with respect to generic vector fields. Contact 3-manifolds, Stein fillings, Heegaard Floer homology, Heegaard Floer homology, Khovanov homology and its variations, Legendrian and transverse link invariants. Symmetric unions. Topological spin models. Classical concordance group. Casson-Gordon invariants. Rational homology 4-balls and their embeddings.**People**

Riccardo Benedetti

Roberto Frigerio

Paolo Lisca

Bruno Martelli

Carlo Petronio

Former students and collaborators:

Stephane Baseilhac (Universita’ di Montpellier, France)

Giulio Belletti (post-doc, Università di Heidelberg)

Michelle Bucher (Università di Ginevra)

Carlo Collari (post-doc, New York University Abu Dhabi)

Marco Golla (Nantes University)

Marco Moraschini (post-doc, Università di Regensburg)

Brendan Owens (Glasgow University)

Maria Beatrice Pozzetti (Università di Heidelberg)

Stefano Riolo (assegnista PRIN)

Alessandro Sisto (Heriot-Watt University)

Leone Slavich (Università di Pavia)

PhD students:

Ludovico Battista, Giuseppe Bargagnati, Leonardo Ferrari, Domenico Marasco, Alice Merz (University of Pisa)

Federica Bertolotti, Giovanni Italiano, Matteo Migliorini, Francesco Milizia, Diego Santoro (Scuola Normale Superiore)**Recent grants**

PRIN 2010-11 “Varieta’ reali e complesse: geometria, topologia e analisi armonica”

National coordinator: Fulvio Ricci – SNS Pisa

Local coordinator: Riccardo Benedetti

Supporting institution: MIUR

FIRB 2011 project “Geometria e topologia delle varietà in bassa dimensione (Geometry and topology of low-dimensional manifolds)”

National coordinator: Bruno Martelli

Local coordinators:

Francesco Bonsante (Pavia) Gabriele Mondello (Rome) Bruno Martelli (Pisa)

Supporting institution: MIUR

PRA 2016-2017 “￼Geometria e topologia delle varietà”.

Supporting institution: Università di Pisa

MIUR PRIN project 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics”

National coordinator: Filippo Bracci – Roma Tor Vergata

Local coordinator: Paolo Lisca

Web page of the project

*Holomorphic dynamical systems*

Parabolic holomorphic dynamic and, more generally, non-hyperbolic, from a topological, holomorphic and formal point of view; discrete and continuous dynamic in convex and pseudoconvex and, more generally, in taut varieties. Study of the geodesic flow of meromorphic connections on Riemann surfaces. Geometric theory of functions, with particular reference to the study of Carleson’s measures, Toeplitz’s operators and boundary behavior of analytical functions in pseudoconvex domains. Index theorems index for applications and holomorphic foliations.**People**

Marco Abate

Jasmin Raissy (Universite’ Paul Sabatier, Toulouse, France)

Fabrizio Bianchi (Imperial College, London)

Karim Rakhimov (Ph.D. student, Pisa)**Recent grants**

FIRB 2012 project ”Geometria Differenziale e Teoria Geometrica delle Funzioni” (grant n. RBFR12W1AQ 002).

Local coordinator: Jasmin Raissy (Universite’ Paul Sabatier, Toulouse, France)

Supporting institution: MIUR

PRA 2015-2016 “Connessioni fra dinamica olomorfa, teoria ergodica e logica matematica nei sistemi dinamici”

PRA 2017-2018 “Sistemi dinamici in analisi, geometria, logica e meccanica celeste”

Supporting institution: University of Pisa