Algebra

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
Michele D'Adderio
michele.dadderio@unipi.it
Alessandro Iraci
alessandro.iraci@unipi.it
Vassilis Dionyssis Moustakas
vasileios.moustakas@dm.unipi.it
Leonardo Patimo
leonardo.patimo@unipi.it
Collaborators
Houcine Ben Dali (Université de Lorraine)
Houcine.Ben-Dali@irif.fr
Mark Dukes (University College Dublin)
Giovanni Interdonato (SNS, Pisa)
Alexander Lazar (Université libre de Bruxelles)
Yvan Le Borgne (Université de Bordeaux)
Anton Mellit (Universität Wien)
Roberto Riccardi (SNS, Pisa)
Marino Romero (Universität Wien)
Anna Vanden Wyngaerd (Université libre de Bruxelles)
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 algebro-geometric 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 (L-functions).

Members
Andrea Bandini
andrea.bandini@unipi.it
Ilaria Del Corso
ilaria.delcorso@unipi.it
Davide Lombardo
davide.lombardo@unipi.it

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 higher-dimensional 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 Mumford-Tate and Sato-Tate 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 low-genus curves. We also work on related problems, including Kummer theory (the study of division points of non-torsion points on commutative algebraic groups) and the determination of rational points on curves.

Members
Collaborators
Victoria Cantoral Farfán (Georg-August-Universität Göttingen)
victoria.cantoralfarfan@mathematik.uni-goettingen.de
Andrea Gallese (SNS, Pisa)
Samuel Le Fourn (Université Grenoble Alpes)
Samuel.Le-Fourn@univ-grenoble-alpes.fr
Matteo Verzobio (IST Austria)
matteo.verzobio@gmail.com
John Voight (Dartmouth College)
David Zywina (Cornell University)
(Hopf) Galois module theory and skew braces

Hopf-Galois 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 Hopf-Galois structures on a $G$-Galois field extension are in one-to-one 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 Hopf-Galois structures and on the study of the properties of the skew braces as well as their equivalent from the Hopf-Galois 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
Ilaria Del Corso
ilaria.delcorso@unipi.it
Davide Lombardo
davide.lombardo@unipi.it
Collaborators
Elena Campedel (Università degli Studi di Milano-Bicocca)
Andrea Caranti (Università degli Studi di Trento)
Fabio Ferri (University of Exeter)
Lorenzo Stefanello (Università di Pisa)
lorenzo.stefanello@phd.unipi.it
Senne Trappeniers (Vrije Universiteit Brussel)
Paul Truman (Keele University)
p.j.truman@keele.ac.uk
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
Andrea Bandini
andrea.bandini@unipi.it
Collaborators
Francesc Bars (Universitat Autònoma de Barcelona)
Ignazio Longhi (Università di Torino)
ignazio.longhi@unito.it
Maria Valentino (Università della Calabria)
Stefano Vigni (Università di Genova)
Lie theory
Members
Giovanni Gaiffi
giovanni.gaiffi@unipi.it
Davide Lombardo
davide.lombardo@unipi.it
Andrea Maffei
andrea.maffei@unipi.it
Leonardo Patimo
leonardo.patimo@unipi.it
Francesco Sala
francesco.sala@unipi.it

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 Kac-Moody algebras associated with one-dimensional manifolds (‘real curves’) can be thought of as continuum colimits of quantized enveloping algebras and Kac-Moody 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
Francesco Sala
francesco.sala@unipi.it
Collaborators
Andrea Appel (Università di Parma)
Olivier Schiffmann (CNRS, Université de Paris-Saclay)
olivier.schiffmann@universite-paris-saclay.fr
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 quasi-simple 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 Feigin-Frenkel 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 Feigin-Frenkel 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 monodromy-free 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
Davide Lombardo
davide.lombardo@unipi.it
Andrea Maffei
andrea.maffei@unipi.it
Collaborators and Ph.D. students at other institutions
Luca Casarin (Sapienza Università di Roma)
Alberto De Sole (Sapienza Università di Roma)
Giorgia Fortuna (Reply)
Valerio Melani (Università degli Studi di Firenze)
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
Andrea Maffei
andrea.maffei@unipi.it
Collaborators
Rocco Chirivì (Università del Salento)
Jacopo Gandini (Università di Bologna)
Pierluigi Moseneder (Politecnico di Milano)
Paolo Papi (Sapienza Università di Roma)

People

Faculty
Name Surname Email 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
Leonardo Patimo leonardo.patimo@unipi.it
Enrico Sbarra enrico.sbarra@unipi.it
Affiliate Members
Name Surname Email Personal Card
Roberto Dvornicich roberto.dvornicich@gmail.com
Former Members

there is no data

Postdoctoral Fellows
Name Surname Email Personal Card
Vassilis Dionyssis Moustakas vasileios.moustakas@dm.unipi.it
Ph.D. Students at the University of Pisa
Name Surname Email Personal Card
Lorenzo Furio lorenzo.furio@phd.unipi.it
Ph.D. Students at other institutions
Name Surname Affiliation
Luca Casarin Sapienza Università di Roma
Andrea Gallese SNS, Pisa
Giovanni Interdonato SNS, Pisa
Roberto Riccardi SNS, Pisa

Ph.D. Theses supervised by members of the group

awarded by the University of Pisa
Year Name Surname Title of the Thesis Supervisor(s)
2024 Lorenzo Stefanello Hopf-Galois structures, skew braces, and their connection Ilaria Del Corso and Paul Truman
2022 Michele Carmassi The B-orbits on a Hermitian symmetric variety: the characteristic 2 case and the Bruhat G-order 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 Computational Aspects of Line and Toric Arrangements 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 Jean-François Jaulent
2009 Marco Strambi Effective estimates for coverings of curves over number fields Yuri F. Bilu and Roberto Dvornicich
2008 Laura Paladino Local-global divisibility problems for elliptic curves Roberto Dvornicich
2007 Gabriele Ranieri Rang de l'image du groupe des unites; conjecture de Bremner Francesco Amoroso and Roberto Dvornicich
1997 Giampiero Chiaselotti Alcune presentazioni dei gruppi lineari speciali su campi finiti Roberto Dvornicich
awarded by another institution

there is no data

Grants

Current
Past

Visitors

Prospective

there is no data

Current

there is no data

Grouped by year
2024
First Name Last Name Affiliation From To
Matthew Bisatt University of Bristol Mar 10, 2024 Mar 16, 2024
Nigel Byott University of Exeter Apr 09, 2024 Apr 11, 2024
Francesco Campagna Université Clermont Auvergne Feb 26, 2024 Feb 28, 2024
Kestutis Cesnavicius Université de Paris-Saclay Jun 12, 2024 Jun 15, 2024
Giulia Codenotti Johann Wolfgang Goethe-Universität Feb 26, 2024 Apr 05, 2024
Nirvana Coppola Université de Strasbourg Apr 08, 2024 Apr 10, 2024
Nirvana Coppola Université de Strasbourg May 27, 2024 May 28, 2024
Alexander Lazar Université libre de Bruxelles May 13, 2024 May 24, 2024
Samuel Le Fourn Université Grenoble Alpes Mar 01, 2024 Mar 07, 2024
Ignazio Longhi Università di Torino May 20, 2024 May 25, 2024
Florian Schreier-Aigner Universität Wien Apr 10, 2024 Apr 12, 2024
Paul Truman Keele University Apr 09, 2024 Apr 11, 2024
Anna Vanden Wyngaerd Université libre de Bruxelles May 20, 2024 May 24, 2024
2023
First Name Last Name Affiliation From To
Houcine Ben Dali Université de Lorraine Feb 15, 2023 May 31, 2023
Petter Brändén KTH Royal Institute of Technology Dec 12, 2023 Dec 15, 2023
Petter Brändén KTH Royal Institute of Technology Dec 12, 2023 Dec 15, 2023
Patrizio Frosini Università di Bologna May 09, 2023 May 09, 2023
Daniel Gil-Munoz 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
Shin Hattori Tokyo City University Dec 10, 2023 Dec 15, 2023
Daniel Kaplan University of Hasselt Nov 22, 2023 Nov 25, 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
George H. Seelinger University of Michigan Oct 10, 2023 Oct 12, 2023
Dumitru Stamate University of Bucharest May 16, 2023 May 23, 2023
Maria Valentino Università della Calabria Dec 10, 2023 Dec 15, 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 Georg-August-Universität 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
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