The scientific activities of the group are focused on different aspects of number theory and representation theory, with ramifications and applications in Lie theory and algebraic combinatorics.
The group runs the Seminar on Combinatorics, Lie Theory, and Topology.
Research Topics
Algebraic combinatorics
Algebraic combinatorics studies mainly discrete structures that occur in algebraic problems. Our research focuses on combinatorics arising from representation theory. More specifically, in recent years we studied $q,t$combinatorics arising from Macdonald polynomials, plethystic operators on symmetric functions, and diagonal coinvariants.
Members
Collaborators
anna.vanden.wyngaerd@ulb.be
Arithmetic geometry and number theory
Number theory studies arithmetic problems by using tools from different areas of mathematics. Our research focuses on arithmetically relevant actions of Galois groups on algebrogeometric objects (such as number rings and their class groups, or abelian varieties and their Selmer groups). Via Iwasawa theory, algebraic invariants of such actions can sometimes be related to analytic objects (Lfunctions).
Members
Below is the list of the specific topics studied in this research area, each of them with a detailed description and the lists of members and collaborators.
Arithmetic of abelian varieties
Elliptic curves and higherdimensional abelian varieties are fundamental objects of study in number theory. Our investigations focus mainly on their Galois representations, especially over number fields, in the spirit of the MumfordTate and SatoTate conjectures. We consider in particular the possible existence of nontrivial endomorphisms on a given abelian variety (with special attention given to the case of Jacobians), with implications for the arithmetic of lowgenus curves. We also work on related problems, including Kummer theory (the study of division points of nontorsion points on commutative algebraic groups) and the determination of rational points on curves.
Members
Collaborators
victoria.cantoralfarfan@mathematik.unigoettingen.de
(Hopf) Galois module theory and skew braces
HopfGalois module theory is a generalisation of Galois module theory, where the action of a Galois group is replaced by that of a $K$Hopf algebra. This generalisation is fruitful both for enlarging the set of extensions under consideration and for better describing wild extensions. We work on questions in both these settings.
The HopfGalois structures on a $G$Galois field extension are in onetoone correspondence with suitable subgroups of the permutation group on $G$, and are closely connected with skew braces. The focus of our research is both on the classification of HopfGalois structures and on the study of the properties of the skew braces as well as their equivalent from the HopfGalois point of view.
We are also employing the language and the properties of the braces in the study of some problems of commutative algebra.
Members
Collaborators
Iwasawa theory and Drinfeld modular forms
Iwasawa theory studies modules (like class groups and Selmer groups) associated to global fields and abelian varieties in all characteristics, in an attempt to link algebraic invariants (like orders of class groups and ranks of elliptic curves) to analytic ones (i.e. special values of $L$functions).
The study of Drinfeld modular forms is currently focused on their structure in an attempt to define old forms and new forms, and on their slopes with respect to the action of the Hecke algebra. The main goal is to establish an analog of Hida theory for function fields.
Members
Collaborators
Lie theory
Members
Below is the list of the specific topics studied in this research area, each of them with a detailed description and the lists of members and collaborators.
Continuum Lie algebras and quantum groups
The theories of continuum quantum groups and continuum KacMoody algebras associated with onedimensional manifolds (‘real curves’) can be thought of as continuum colimits of quantized enveloping algebras and KacMoody Lie algebras, respectively. These algebraic structures have rather peculiar properties, e.g. they are governed by a continuum root system with infinitely many elements and no simple roots.
At the moment, we are investigating ‘continuum’ analogs of representations, integrable systems, and affinizations of quantum groups.
Members
Collaborators
Opers on the punctured disc
Frenkel and Gaitsgory proposed a version of the Langlands correspondence in the case of the field $\mathbb{C}((t))$. Given a quasisimple group $G$ such a proposal relates the geometry of the space of $G$ connections on the disk with a singularity in the origin and representations of the Langlands dual group of $G$. The tool that allows one to build a bridge between these objects is the FeiginFrenkel theorem which describes the center of the enveloping algebra of an affine Lie algebra at the critical level as the functions on a space of connections for the dual algebra, called the space of Opers, introduced by Beilinson and Drinfeld. In this theory, vertex algebras play a key role in both the proof of the FeiginFrenkel theorem and the development of the theory of Frenkel and Gaitsgory.
The local geometric correspondence is largely conjectural, but in the case of “spherical” representations, which corresponds to the case of monodromyfree connections, Frenkel and Gaitsgory gave complete proof.
We are studying a version of this correspondence in the case of families of connections on the disk with two singular points $0$ and $a$. In this case, we consider an analogue of the affine algebra, which we denote by $\mathfrak{g}(2)$, on the ring $\mathbb{C}[[a]]$. In the case $\mathfrak{g}=\mathfrak{sl}_2$ we were able to construct a family of elements that generalize Sugawara operators and generate the center of the enveloping algebra of $\mathfrak{g}(2)$ at the critical level. We have also provided a description of endomorphisms of a Weyl module similar to the one given by Frenkel and Gaitsgory in the usual case. We intend to continue the study started with this work by extending the results in various directions.
Members
Collaborators and Ph.D. students at other institutions
Symmetric and spherical varieties
If $G$ is a reductive group and $H$ is the subgroup of points fixed under an involution then $G/H$ is called a symmetric variety. They have been extensively studied in relation to problems coming from representation theory or algebraic geometry and in particular enumerative geometry. Symmetric varieties are a particular case of spherical varieties: normal $G$ variety with a finite number of $B$ orbits. The classification of spherical varieties has been obtained recently. We study different aspects of symmetric and spherical varieties: compactifications, description of the coordinate ring, applications to the study of normality of particular classes of $G$ variety important in representation theory.
An important theme is a study of the classification of $B$ orbits in these varieties and their Bruhat order.
Members
Collaborators
People
Faculty
Name  Surname  Personal Card  

Andrea  Bandini  andrea.bandini@unipi.it  
Massimo  Caboara  massimo.caboara@unipi.it  
Michele  D'Adderio  michele.dadderio@unipi.it  
Ilaria  Del Corso  ilaria.delcorso@unipi.it  
Giovanni  Gaiffi  giovanni.gaiffi@unipi.it  
Alessandro  Iraci  alessandro.iraci@unipi.it  
Davide  Lombardo  davide.lombardo@unipi.it  
Andrea  Maffei  andrea.maffei@unipi.it  
Enrico  Sbarra  enrico.sbarra@unipi.it 
Affiliate Members
Name  Surname  Personal Card  

Roberto  Dvornicich  roberto.dvornicich@gmail.com  
Patrizia  Gianni  patrizia.gianni@gmail.com 
Former Members
there is no data
Postdoctoral Fellows
there is no data
Ph.D. Students at the University of Pisa
Name  Surname  Personal Card  

Lorenzo  Furio  lorenzo.furio@phd.unipi.it  
Lorenzo  Stefanello  lorenzo.stefanello@phd.unipi.it 
Ph.D. Students at other institutions
Name  Surname  Affiliation 

Luca  Casarin  Sapienza Università di Roma 
Ph.D. Theses supervised by members of the group
awarded by the University of Pisa
Year  Name  Surname  Title of the Thesis  Supervisor(s) 

2022  Michele  Carmassi  The Borbits on a Hermitian symmetric variety: the characteristic 2 case and the Bruhat Gorder  Andrea Maffei 
2021  Matteo  Verzobio  Primitive divisors of elliptic divisibility sequences  Roberto Dvornicich 
2020  Alessio  Moscariello  Lacunary polynomials and compositions  Roberto Dvornicich and Umberto Zannier 
2019  Alessandro  Iraci  Macdonald polynomials and the Delta conjecture  Michele D'Adderio and Giovanni Gaiffi 
2018  Oscar  Papini  Giovanni Gaiffi and Mario Salvetti  
2016  Francesco  Strazzanti  A family of quotients of the Rees algebra and rigidity properties of local cohomology modules  Enrico Sbarra 
2009  Luca  Caputo  On the structure of étale wild kernels of number fields  Roberto Dvornicich and JeanFrançois Jaulent 
2009  Marco  Strambi  Effective estimates for coverings of curves over number fields  Roberto Dvornicich and Yuri F. Bilu 
2008  Laura  Paladino  Localglobal divisibility problems for elliptic curves  Roberto Dvornicich 
2007  Gabriele  Ranieri  Rang de l'image du groupe des unites; conjecture de Bremner  Roberto Dvornicich and Francesco Amoroso 
1997  Giampiero  Chiaselotti  Alcune presentazioni dei gruppi lineari speciali su campi finiti  Roberto Dvornicich 
awarded by another institution
there is no data
Grants
Current

Semiabelian varieties, Galois representations and related Diophantine problems (Prin 2022)
Principal Investigator: Davide Lombardo
Project period: Sep 28, 2023 – Sep 27, 2025

Algebraic and geometric aspects of Lie theory (Prin 2022)
Principal Investigator: Alberto De Sole
Coordinator of the Research Unit: Mario Salvetti
Project period: Sep 28, 2023 – Sep 27, 2025

ALgebraic and TOPological combinatorics (ALTOP) (Prin 2022)
Principal Investigator: Luca Moci
Coordinator of the Research Unit: Michele D'Adderio
Project period: Sep 28, 2023 – Sep 27, 2025

Spazi di moduli, rappresentazioni e strutture combinatorie (Progetti di Ricerca di Ateneo (PRA) 20222023)
Principal Investigator: Davide Lombardo
Project period: Oct 10, 2022 – Dec 31, 2024

Quantum Structures and Enumerative Geometry (MIT  Italy Università di Pisa Seed Fund)
Principal Investigator: Francesco Sala
Project period: May 01, 2023 – Dec 31, 2024
Past

Geometric, algebraic and analytic methods in arithmetic (Prin 2017)
Principal Investigator: Pietro Corvaja
Coordinator of the Research Unit: Roberto Dvornicich
Project period: Aug 01, 2019 – Feb 19, 2023

Spazi di moduli, geometria aritmetica e aspetti storici (Progetti di Ricerca di Ateneo (PRA) 2020  2021)
Principal Investigator: Mattia Talpo
Project period: Jul 07, 2020 – Dec 31, 2022

Geometria e topologia delle varietà (Progetti di Ricerca di Ateneo (PRA) 2018  2020)
Principal Investigator: Bruno Martelli
Project period: Jul 09, 2018 – Jul 08, 2020
Visitors
Prospective
there is no data
Current
there is no data
Grouped by year
2023
First Name  Last Name  Affiliation  From  To 

Houcine  Ben Dali  Université de Lorraine  Feb 15, 2023  May 31, 2023 
Patrizio  Frosini  Università di Bologna  May 09, 2023  May 09, 2023 
Daniel  GilMunoz  Charles University  Univerzita Karlova  May 10, 2023  May 13, 2023 
Heidi  Goodson  Brooklyn College, City University of New York  Apr 24, 2023  Apr 28, 2023 
Alexander  Lazar  Université libre de Bruxelles  Mar 30, 2023  Apr 08, 2023 
Alexander  Lazar  Université libre de Bruxelles  Jun 19, 2023  Jun 21, 2023 
Yvan  Le Borgne  Université de Bordeaux  Apr 24, 2023  Apr 27, 2023 
Samuel  Le Fourn  Université Grenoble Alpes  May 19, 2023  May 26, 2023 
Philippe  Nadeau  Université Claude Bernard Lyon 1  Mar 20, 2023  Apr 28, 2023 
Kostas  Psaromiligkos  University of Chicago  Jan 16, 2023  Jan 21, 2023 
Dumitru  Stamate  University of Bucharest  May 16, 2023  May 23, 2023 
Anna Vanden  Wyngaerd  Institut de recherche en Informatique Fondamentale (IRIF)  Jun 19, 2023  Jun 21, 2023 
2022
First Name  Last Name  Affiliation  From  To 

Victoria  Cantoral Farfán  GeorgAugustUniversität Göttingen  Nov 28, 2022  Dec 02, 2022 
Mark Andrea  De Cataldo  Stony Brook University  May 29, 2022  Jun 16, 2022 
Fabio  Ferri  University of Exeter  Dec 11, 2022  Dec 15, 2022 
Rouven  Frassek  Università di Modena e Reggio Emilia  Nov 16, 2022  Nov 16, 2022 
Anton  Mellit  Universität Wien  Nov 23, 2022  Nov 25, 2022 
Roberto  Pagaria  Università di Bologna  Nov 14, 2022  Nov 17, 2022 
Flavio  Perissinotto  University of Luxembourg  Nov 21, 2022  Nov 25, 2022 
Miroslav  Rapcak  CERN  Nov 03, 2022  Nov 04, 2022 
Michael  Wemyss  University of Glasgow  Oct 19, 2022  Oct 21, 2022 
Lauren K.  Williams  Harvard University  Dec 05, 2022  Dec 05, 2022 
Anna Vanden  Wyngaerd  Institut de recherche en Informatique Fondamentale (IRIF)  Oct 30, 2022  Nov 04, 2022 
Anna Vanden  Wyngaerd  Institut de recherche en Informatique Fondamentale (IRIF)  Nov 20, 2022  Nov 25, 2022 