All courses have their Moodle page where I put all information for students.
Dipartimento: Dipartimento di Matematica
Docenti: Alessandra Bernardi, Stefano Canino
This course introduces some of the main algorithmic methods of algebra, combining mathematical theory with concrete computations, experiments, and computer-assisted verification. The aim is not simply to learn how to use particular algorithms or software commands, but to understand which mathematical problems they solve, why they work, and how their outputs should be interpreted.
The course is divided into two closely connected parts. The first part, taught by Dr. S. Canino, focuses on algorithms for integer factorisation and polynomial factorisation. The second part, taught by Prof. A. Bernardi, develops methods from computational algebra and algebraic geometry, including Gröbner bases, elimination, resultants, and the solution of polynomial systems. Joint classroom activities will help connect the two parts and highlight the relationships between their methods and perspectives.
The course will combine theoretical lectures, worked examples, computer-based activities, and group work. Part of the course will be delivered in a blended-learning format. These activities are not separate from the lectures: they are an integral part of the course and are designed to consolidate the concepts introduced in class, explore more substantial examples, and gradually develop greater independence in approaching computational problems.
The blended-learning activities will be supported by a tutor (Dr. E. Sorbera), who will accompany students during the practical work, help them address technical and computational difficulties, and support the organisation of collaborative activities and the final project.
We will mainly use Macaulay2 to experiment with the mathematical concepts studied in the second part of the course and to construct reproducible computational procedures. No previous knowledge of the software is required. A dedicated Macaulay2 Survival Kit will introduce the essential tools needed for the subsequent activities.
Particular attention will be devoted to the transition:
from the mathematical formulation of a problem to its computational implementation;
from software output to its mathematical interpretation;
from individual examples to the recognition of more general structures;
from computational experimentation to the verification and certification of results.
In the final part of the course, students will work in groups on a small project. Each group will be asked to formulate a precise mathematical question, select appropriate methods and examples or data, develop a reproducible script, analyse the results critically, and present the work to the class.
This Moodle page will be the main reference point for the course. It will contain the course schedule, lecture materials, instructions for the blended-learning activities, scripts, assignments, deadlines, and organisational announcements. Detailed information about assessment and individual activities will be provided in the relevant sections.
Willem de Graaf, Computational Algebra. Course notes.
Mateusz Michałek, Bernd Sturmfels, Invitation to Nonlinear Algebra, Graduate Studies in Mathematics, vol. 211, American Mathematical Society, 2021.
David Cox, John Little, Donal O’Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997.
David Eisenbud, Daniel R. Grayson, Michael E. Stillman, Bernd Sturmfels, eds., Computations in Algebraic Geometry with Macaulay 2, Algorithms and Computation in Mathematics, vol. 8, Springer-Verlag, Berlin/Heidelberg, 2002.
Hal Schenck, Computational Algebraic Geometry, London Mathematical Society Student Texts, vol. 58, Cambridge University Press, 2003.
Official Macaulay2 documentation, in particular the Getting Started section, tutorials and reference manual used for the computational activities.
Documentation of the software tools used during the course, for instance Macaulay2, SageMath, Singular, Bertini or equivalent tools indicated by the instructor.
Dipartimenti:
Dipartimento di Ingegneria e Scienza dell'Informazione, Laurea in Ingegneria informatica, delle comunicazioni ed elettronica [0533G] (L)
Dipartimento di Ingegneria e Scienza dell'Informazione, Laurea in Informatica [0532G] (L)
Docenti: Alessandra Bernardi
Assistente: Andrea Zambotti
Tutors: Alessandro Casagradne, Lorenzo Vigan
Strengthen foundations in linear algebra and analytic geometry
Learn MATLAB basics for computation and visualization
Develop the ability to interpret numerical outputs correctly
MATLAB is integral to the course. Students must complete online modules on MATLAB Academy (https://matlabacademy.mathworks.com/): MATLAB Onramp, Introduction to Symbolic Math with MATLAB, Introduction to Linear Algebra with MATLAB.
Submit certificates at least 3 working days before the exam. Without these certificates, admission to the exam is not possible.
We'll use MATLAB Grader (accessible at grader.mathworks.com) for assessments:
What is MATLAB Grader? A browser-based platform for creating, delivering, and auto-grading MATLAB coding assignments. It integrates seamlessly with your LMS and provides instant feedback to students.
Usage and integration:
No installation needed: authoring and submission happen in the browser.
Supports integration with Moodle (and other LMS) via LTI standards, sending automated grades into your gradebook.
Requirements:
Instructor and students need a MathWorks Account and a valid MATLAB license (Campus-Wide License works fine)
Written Exam (MATLAB Grader-based)
A required MATLAB-based exam on linear algebra and analytic geometry
Graded out of 30, with a maximum of 28/30
Admission depends on submitted certificates for MATLAB Academy modules
Outcomes
Grade ≥ 15:
Accept as final grade, or
Opt for oral exam (especially advisable if grade is 15–18)
Oral Exam (optional):
Begins with a student-chosen topic
If unprepared → exam ends (must be retaken)
If presented → continues with further questions
Final grade based solely on performance in oral, regardless of MATLAB score
Mathematical correctness
Suitability of methods
Proper use and interpretation of MATLAB outputs
Clarity and rigor in exposition
Academic integrity
(English):
Strang, Introduction to Linear Algebra
Anton/Rorres, Elementary Linear Algebra: Applications Version
Leon/de Pillis, Linear Algebra with Applications.
Farin–Hansford, Practical Linear Algebra: A Geometry Toolbox.
Jim Hefferon, Linear Algebra
(Italian):
Algebra lineare e geometria analitica, A. Bernardi & A. Gimigliano, CittàStudi Edizioni, 2ª ed., 2018.
Manuale di algebra lineare, Cosimo Flavi, Città Studi 2026.
LINK TO BIBLIOGRAPHIC RESOURCES
English
Strang, G. (2023). Introduction to linear algebra (6. ed.). Wellesley-Cambridge press.
Anton, H., Rorres, C., & Kaul, A. (2025). Elementary linear algebra : applications version (12. ed., international adaptation, revised&updated edition). Wiley.
Leon, S., Leon, S. J., & De Pillis, L. (2021). Linear algebra with applications (10. ed., global ed.). Pearson.
Farin, G. E., & Hansford, D. (2024). Practical linear algebra : a geometry toolbox (4. ed.). CRC press.
Hefferon, J. (2020). Linear Algebra (4. ed.). [s.n.].
Italian
Bernardi, A., & Gimigliano, A. (2018). Algebra lineare e geometria analitica (2. ed.). CittàStudi.
Flavi, C. (2026). Manuale di algebra lineare. D Scuola.
Dipartimento di Matematica
Docente: Alessandra Bernardi
The 42-hour course combines:
regular lectures,
a 10-hour masterclass with Prof. J Cohen (Lyon),
four structured blended activities with the tutor Dr. E. Sorbera,
and a final group project.
The course begins with Tucker decomposition and HOSVD, continues with the masterclass, and then develops tensor rank, CP decomposition, algebraic-geometric methods, and Sylvester-type algorithms.
The blended activities involve guided individual or group work, short Moodle submissions, and a report-back session in class. This Moodle page is organised chronologically: each section corresponds to a lesson or activity and contains the relevant slides, readings, assignments, and deadlines.
The official syllabus contains the learning outcomes, prerequisites, course contents, teaching methods, assessment criteria, and study materials.
Suggestes readings
1) Tensors: Geometry and Applications J.M. Landsberg Graduate Studies in Mathematics, Vol 128, American Mathematical Society
2) Invitation to Nonlinear algebra, Mateusz Michalek, Bernd Sturmfels, Springer
3) Tensor Decompositions for Data Science, 2025, Grey Ballard and Tamara G. Kolda.
4) Nonnegative Matrix Factorization, Nicolas Gillis, SIAM.
5) Notes of the course