TO BE REVISED

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, ramification; 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.

### Members

lista dei membri

### Postdoctoral Fellows

lista dei postdocs

### Ph.D. Students

lista degli studenti di dottorato

### External Collaborators

lista dei collaboratori esterni

### Grants

lista dei grants in corso

**Number Theory in Pisa: webpage**

**Lines of research in progress:**

**Lie theory: nilpotent orbit, B-orbits in symmetric varieties, Bruhat order, Kazhdan-Lusztig polynomials, vertex algebras and opers with two singularities, decomposition of tensor products.**

Michele Carmassi

Rocco Chirivì (Univ. Lecce)

Giovanni Gaiffi

Jacopo Gandini (Univ. Bologna)

Davide Lombardo

Andrea Maffei

Valerio Melani

**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. Bologna)

Emanuela De Negri (Univ. Genova)

Alessandro De Stefani (Univ. Genova)

Giulio Caviglia (Purdue Univ.)

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

**Study of the ramification jumps 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

Davide Lombardo

Fabio Ferri (Univ. Exeter)

**Study of Hopf-Galois structures of field extensions: classification problems, existence/non-existence. Algebraic structures associated with the Yang-Baxter equation: the language of skew braces.**

Ilaria Del Corso

Andrea Caranti (Univ. Trento)

Elena Campedel (PhD student)

Lorenzo Stefanello (PhD student)

**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

Filippo Callegaro

Michele D’Adderio (Univ. Libre de Bruxelles)

Corrado De Concini (Sapienza Università di Roma)

Emanuele Delucchi (Univ. Fribourg)

Andrea Maffei

Oscar Papini (dottorando)

Mario Salvetti

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

Davide Lombardo

Elisa Lorenzo García (Univ. Rennes)

Jeroen Sijsling (Univ. Ulm)

John Voight (Dartmouth College)

**Research projects**

**PRIN 2017**

National coordinator: O’GRADY Kieran Gregory

Local Principal Investigator: SALVETTI Mario

Title: Moduli and Lie theory

**PRIN 2017**

National coordinator: CORVAJA Pietro

Local Principal Investigator: DVORNICICH Roberto

Title: Geometric, algebraic and analytic methods in arithmetic