The scientific activity in this area has significant connections with geometry and computer science, and takes place mainly in the following fields: algebraic number theory, study of p-adic field extensions, branching; Galois module structure; study of the dynamics of polynomial maps; rational points of algebraic varieties; computational commutative algebra and algebraic geometry (with particular reference to the study and creation of algorithms for factorization analysis, determination of minimal systems of generators for projective algebraic varieties, correcting codes and cryptography, study of the topology of real algebraic varieties); Lie algebras and algebraic combinatorics; and representation theory (with particular reference to the study of geometric properties of algebraic groups, symmetrical and spherical varieties). Study of the cohomology of complex braid groups, ring cohomology of toric complementary arrangements, combinatorial and topology of wonderful models for arrangements of hyperplanes linked to groups of complex reflections.

**Lines of research in progress:**

**Lie theory, spherical varieties, symmetrical varieties, extensions of loop groups, with particular reference to studying the singularity of spherical and B orbits of an irreducible representation.**

Andrea Maffei

Paolo Bravi(Sapienza Università di Roma)

Rocco Chirivi (Univ. Lecce)

Giorgia Fortuna (Wolfram research)

Giovanni Gaiffi

Jacopo Gandini (SNS)

**Commutative algebra and its interactions with computational algebra and combinatorics. Classification of Hilbert functions, Eisenbud-Green-Harris's conjectures and Evans' Lex-Plus-Power. Jet-schemes of determinantal varieties and modules with straightening laws.**

Enrico Sbarra

Francesco Strazzanti (Univ. Sevilla)

Emanuela De Negri (Univ. Genova)

Giulio Caviglia (Purdue Univ.)

**Commutative algebra and real algebraic geometry, with special attention to construction and computational aspects. Correcting codes.**

Patrizia Gianni

Oscar Papini (PhD student)

**Study of the branching steps in Abelian extensions. Classification of wild extensions of p-adic fields (branching steps; wild extensions). Study of the dynamics of polynomial maps; rational points of variety of cyclotomic extensions (keywords: polynomial maps, Hilbert irreducibility theorem, cyclotomic extension).**

Roberto Dvornicich

Ilaria Del Corso

Maria Rosaria Pati (PhD student)

Laura Capuano (Univ. Oxford)

**Construction of (real and complex) models of subspace arrangements and toric arrangements, configuration spaces. Study of the cohomology rings of the the above mentioned objects and of the associated group actions (in the cases when Coxeter groups or complex reflection groups are involved). Combinatorics of nestohedra and permutonestohedra, nested set complexes.**

Giovanni Gaiffi

Andrea Maffei

Filippo Callegaro

Michele D’Adderio (Univ. Libre de Bruxelles)

Corrado De Concini (Sapienza Università di Roma)

Emanuele Delucchi (Univ. Fribourg)

Eva Feichtner (University of Bremen, ALTA Institute)

Alessandro Iraci (PhD student)

Mario Salvetti

**Arithmetic of abelian varieties. Elliptic curves and abelian varieties of higher dimension, representations of Galois groups, ****Mumford-Tate and Sato-Tate conjectures. Local-global principles for isogenies and torsion points; reduction of infinite order points. Low genus curves: theoretical aspects and computations of Jacobians, rational points.**

Davide Lombardo

Elisa Lorenzo García (Univ. Rennes)

Jeroen Sijsling (Univ. Ulm)

Antonella Perucca (Univ. Regensburg)

**Number Theory in Pisa: webpage**

**Research projects**

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

Local Principal Investigator: Filippo Callegaro

Funding institution: MIUR

Project begins-end: 21/03/2013-21/03/2016

**PRIN 2015 "Moduli spaces and Lie Theory"** **(grant n. 2015ZWST2C PE1)**

Local Principal Investigator: Mario Salvetti

Funding institution: MIUR

**PRIN 2015 "Geometry of Algebraic Varieties****" (grant n. 2015EYPTSB PE1)**

Local Principal Investigator: Rita Pardini

Funding institution: MIUR

**MIUR-DAAD 2016 Joint Mobility Program “The borderline between combinatorics and commutative algebra” (grant n. 57267452)**