Mathematics

Activity type

Formazione teorica avanzata

SSD

MAT/02

Year

1st year

Semester

1st semester

CFU

4

Hours distribution

32 classroom hours

Lecturers

DOMENICO FIORENZA
DOMENICO FIORENZA

Activity type

Formazione teorica avanzata

SSD

MAT/03

Year

1st year

Semester

1st semester

CFU

5

Hours distribution

40 classroom hours

Lecturers

DOMENICO FIORENZA
DOMENICO FIORENZA

General objectives of the course: The course has two objectives, one for each module;
1) To provide the algebraic-topological basis to understand the techniques of topological data analysis.
2) Deepen some aspects of classical projective geometry and theory of algebraic curves, usually neglected in the three-year degree courses.

Module: Module I - Foundations of Geometry
At the end of the course students will be familiar with the notion of persistent homology and its use in topological data analysis, and with the main prmality tests and factorization algorithms for large integers and their use in cryptography.


Module: Module II - Foundations of Algebra
At the end of the course students will be familiar with the notion of persistent homology and its use in topological data analysis, and with the main prmality tests and factorization algorithms for large integers and their use in cryptography.

Module: Module I - Foundations of Geometry
The students will have to be familiar with the notions from the courses Algebra I and Geometry I from Bachelor's Degree in Mathematics. In particular students will need to be familiar with all of the algebraic structures introduced in these courses (groups, rings,fields). These notions are necessary. Other useful notions are those coming from the Algebra II and Geometry II courses


Module: Module II - Foundations of Algebra
The students will have to be familiar with the notions from the courses Algebra I and Geometry I from Bachelor's Degree in Mathematics. In particular students will need to be familiar with all of the algebraic structures introduced in these courses (groups, rings,fields). These notions are necessary. Other useful notions are those coming from the Algebra II and Geometry II courses

Module: Module I - Foundations of Geometry
Finitely generated abelian groups
Homological algebra
Simplicial complexes
Simplicial homology
Persistent homology and topological data analysis
Primality tests
Factorization algorithms for large integers
Basic notions of cryptography


Module: Module II - Foundations of Algebra
Finitely generated abelian groups
Homological algebra
Simplicial complexes
Simplicial homology
Persistent homology and topological data analysis
Primality tests
Factorization algorithms for large integers
Basic notions of cryptography

Module: Module I - Foundations of Geometry
For homological algebra, simplicial homology and p[ersistent homology: notes for the Istituzioni di Algebra e Geometria course by professor Marco Manetti. For primality tests, factorization algorithms and cryptography: Neal Koblitz: A Course in Number Theory and Cryptography


Module: Module II - Foundations of Algebra
For homological algebra, simplicial homology and p[ersistent homology: notes for the Istituzioni di Algebra e Geometria course by professor Marco Manetti. For primality tests, factorization algorithms and cryptography: Neal Koblitz: A Course in Number Theory and Cryptography

Module: Module I - Foundations of Geometry
Marco Manetti: Dispense per il corso di Istituzioni di Algebra e Geometria
Neal Koblitz: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics, 114); Springer


Module: Module II - Foundations of Algebra
Marco Manetti: Dispense per il corso di Istituzioni di Algebra e Geometria
Neal Koblitz: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics, 114); Springer

Module: Module I - Foundations of Geometry
The course will consist in classroom lectures (if possible) or streamed video lectures (if necessary)


Module: Module II - Foundations of Algebra
The course will consist in classroom lectures (if possible) or streamed video lectures (if necessary)

Module: Module I - Foundations of Geometry
Attending lectures is not compulsory, but it is encouraged


Module: Module II - Foundations of Algebra
Attending lectures is not compulsory, but it is encouraged

Module: Module I - Foundations of Geometry
The exam will consist in a written examination followed by an oral examination. The written exam will consist in programming code or in a short essay. The oral examination will aim at testing the student's proficiency in all of the topics presented in the lectures (at the same level of detail).


Module: Module II - Foundations of Algebra
The exam will consist in a written examination followed by an oral examination. The written exam will consist in programming code or in a short essay. The oral examination will aim at testing the student's proficiency in all of the topics presented in the lectures (at the same level of detail).

Module: Module I - Foundations of Geometry
What are a chain complex and its homology?
How can we compute the homology of a sphere? and of a torus?
What is persistent homology? How can we use it to analyze a data set?
What is an Euler's pseudoprime?
How does the rho factorization method work?
How does the elliptic curves primality test work?


Module: Module II - Foundations of Algebra
What are a chain complex and its homology?
How can we compute the homology of a sphere? and of a torus?
What is persistent homology? How can we use it to analyze a data set?
What is an Euler's pseudoprime?
How does the rho factorization method work?
How does the elliptic curves primality test work?

Module: Module I - Foundations of Geometry

• Il corso si svolgerà da ottobre (o fine settembre) a gennaio
  • Books: Marco Manetti: Dispense per il corso di Istituzioni di Algebra e GeometriaNeal Koblitz: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics, 114); Springer

• Il corso si svolgerà da ottobre (o fine settembre) a gennaio
  • Books: Marco Manetti: Dispense per il corso di Istituzioni di Algebra e GeometriaNeal Koblitz: A Course in Number Theory and Cryptography (Graduate Texts in Mathematics, 114); Springer