Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

This information is intended exclusively for students already enrolled in this course.
If you are a new student interested in enrolling, you can find information about the course of study on the course page:

Laurea in Matematica applicata - Enrollment from 2025/2026

The educational activities of type D are the student's choice, those of type F are additional knowledge useful for entering the world of work (internships, transversal skills, project works, etc.). According to the Didactic Regulations of the Course, some activities can be chosen and entered independently in the booklet, others must be approved by a special committee to verify their consistency with the study plan.

The following activities can cover type D or F training activities.

1. Teachings delivered at the University of Verona.

Include the teachings listed below and/or in the Catalog of Teachings (which can also be filtered by language of delivery through Advanced Search).

Booklet entry mode: if the teaching is included among those listed below, the student can enter it independently during the period in which the curriculum is open; otherwise, the student must make a request to the Secretariat, sending to carriere.scienze@ateneo.univr.it the form in the period indicated.

2. CLA certificate or language equivalency.

In addition to those required by the curriculum, the following are recognized for those matriculated from A.Y. 2021/2022:

English language: 3 CFU are recognized for each level of proficiency above that required by the course of study (if not already recognized in the previous course of study).

Other languages and Italian for foreigners: 3 cfu are recognized for each proficiency level starting from A2 (if not already recognized in the previous study cycle).

These cfu will be recognized, up to a maximum of 6 cfu in total, of type F if the teaching plan allows it, or of type D. Additional elective credits for language knowledge may be recognized only if consistent with the student's educational project and if adequately motivated. Those enrolled until A.Y. 2020/2021 should consult the information found here.

Way of inclusion in the booklet: request the certificate or equivalence from CLA and send it to the Student Secretariat - Careers for the inclusion of the exam in the career, by email: carriere.scienze@ateneo.univr.it

3. Soft skills

Discover the training paths promoted by the University's TALC - Teaching and learning center, intended for students regularly enrolled in the academic year of course delivery https://talc.univr.it/it/competenze-trasversali Booklet entry mode: the teaching is not expected to be included in the curriculum. Only upon obtaining theOpen Badge will the booklet CFUs be automatically validated. The registration of CFUs in career is not instantaneous, but there will be some technical time to wait.

4. Contamination lab

The Contamination Lab Verona (CLab Verona) is an experiential pathway with modules dedicated to innovation and business culture that offers the opportunity to work in teams with students from all courses of study to solve challenges launched by companies and institutions. The pathway allows students to receive 6 CFUs in the D or F area. Find out about the challenges: https://www.univr.it/clabverona

PLEASE NOTE: To be eligible to take any teaching activity, including electives, you must be enrolled in the course year in which it is offered. Therefore, it is recommended that undergraduates in the December and April sessions DO NOT undertake extracurricular activities of the new academic year, in which they are not enrolled, since these degree sessions are valid concerning the previous academic year. Therefore, for activities carried out in an academic year in which they are not enrolled, no recognition of CFU can be given.

5. Internship/internship period

In addition to the CFU stipulated in the curriculum (check carefully what is indicated on the Academic Regulations) here you can find information on how to activate the internship. Check in the regulations which activities can be Type D and which can be Type F.

Please also note that for internships activated from October 1, 2024, it will be possible to recognize excess hours in terms of Type D credits, limited only to internship experiences carried out at host institutions outside the University.

PROCEDURE FOR RECOGNIZING WORK ACTIVITY AS INTERNSHIP CREDITS

As per the resolution of the Teaching College of Mathematics and Data Science No. 4 -24/25, the student who intends to have hours of work activity recognized as internship credits, before the start of the activity, is required to send to the e-mail address of the student secretariat and in copy knowledge to the student practices committee (paolo.daipra@univr.it, luca.dipersio@univr.it, barbara.gaudenzi@univr.it) explicit request.

The request should specify the type of activity, name of the company and place of work, and hours/credits for which recognition is being sought. The activity must have taken place during the years of enrollment in the course of study to be recognizable. Once the student practices committee has ascertained that the current work activity is consistent with the course objectives, the student will be promptly notified, with the secretary's office as a copy.

At the end of the stipulated work period, the student sends the following documentation to the student secretariat: a detailed final report that is forwarded to the committee for final approval (signed by the student and a company contact person); a statement from the legal representative of the company/entity and/or documentation demonstrating the type of professional activity and the hourly commitment to it.

The student secretary's office will forward the documentation received to the student practices committee and record the CFUs (taf F and any additional taf D credits) decided by the committee.

Academic year:
Semester 1  From 10/1/24 To 1/31/25
years Modules TAF Teacher
1° 2° 3° Algorithms D Roberto Segala (Coordinator)
1° 2° 3° Basis of general chemistry D Silvia Ruggieri
1° 2° 3° Genetics D Massimo Delledonne (Coordinator)
1° 2° 3° Introduction to quantum mechanics for quantum computing D Claudia Daffara (Coordinator)
1° 2° 3° Python programming language [English edition] D Carlo Combi (Coordinator)
1° 2° 3° APP REACT PLANNING D Graziano Pravadelli (Coordinator)
Semester 2 From 3/3/25 To 6/13/25
years Modules TAF Teacher
1° 2° 3° Algebraic Geometry F Rosanna Davison Laking (Coordinator)
1° 2° 3° Algorithms D Roberto Segala (Coordinator)
1° 2° 3° LaTeX Language D Enrico Gregorio (Coordinator)
1° 2° 3° Python programming language [Edizione in italiano] D Carlo Combi (Coordinator)
1° 2° 3° Organization Studies D Serena Cubico (Coordinator)
1° 2° 3° Programming Challanges D Romeo Rizzi (Coordinator)
1° 2° 3° Tools for development of applications of virtual reality and mixed D Andrea Giachetti (Coordinator)
1° 2° 3° Protection of intangible assets (SW and invention)between industrial law and copyright D Mila Dalla Preda (Coordinator)
List of courses with unassigned period
years Modules TAF Teacher
Subject requirements: mathematics D Franco Zivcovich (Coordinator)

Teaching code

4S00244

Credits

9

Language

Italian

Also offered in courses:

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Courses Single

Authorized

The teaching is organized as follows:

Teoria

Credits

6

Period

Semester 2

Esercitazioni

Credits

3

Period

Semester 2

Learning objectives

The aim of the course is to introduce the theory and some applications of dynamical systems, which describe the time evolution of quantitative variables. By the end of the course, the students will be able to investigate the stability and the character of an equilibrium, the qualitative analysis of a system of ordinary differential equations, the phase portrait of a (parametric) dynamical system in dimension 1 and 2, and to analyse finite-dimensional Hamiltonian systems. Moreover, the students will be able to study some basic applications of dynamical systems arising from population dynamics, mechanics and traffic flows. Finally, students will be also able to produce proofs using the typical tools of modern dynamical systems and will be able to read and report specific books and articles on dynamical systems and related applications.

Prerequisites and basic notions

The material covered in first-year and second-year, first-semester courses - in particular, Mathematical Analysis 1 and 2, Linear Algebra.

Program

Part I
1. Topics in the theory of ordinary differential equations
Qualitative analysis of ODE: existence and uniqueness of solutions; maximal and global solutions; Gronwall’s Lemma; continuous dependence on the initial data.
2. Vector fields and ordinary differential equations
Vector fields: phase space, integral curves, orbits, equilibria, phase portrait. 1-dimensional examples of phase portraits. Second-order systems of differential equations; phase-space analysis and equilibria.
3. Linear systems
Linearisation of a vector field about an equilibrium. Classification of two-dimensional linear systems (over the real numbers) that are diagonalisable over the complex numbers. (If time permits, we will briefly discuss the nilpotent case as well.) n-dimensional linear systems: invariant subspace decomposition; the stable, unstable and central subspaces. Comparing a vector field with its linearisation about a hyperbolic equilibrium.
4. Flow of a vector field
Flow of a vector field. Change of coordinates: conjugate vector fields; pull-back and push-forward of a vector field by a diffeomorphism. Non-autonomous differential equations: time-dependent change of coordinates; scaling of vector fields and time reparametrisations. The local rectification theorem.
5. First integrals
Invariant sets; first integrals; Lie derivative. Invariant foliations; reduction of the order. First integrals and attractive equilibria.
6. Stability theory
Stability 'à la Lyapunov' of an equilibrium; the method of Lyapunov functions; the spectral method. Applications and examples.
7. 1-dimensional Newton equation.
Phase portraits of the 1-dimensional Newton equation, in the conservative case. Linearisation. Reduction of the order. Systems with friction.
Part II
8. Bifurcations
Bifurcatios from equilibria, with 1-dimensional examples; applications.
9. Introduction to the 1-dimensional Calculus of Variations
The indirect method for one-dimensional integral functionals. Necessary conditions for the existence of minimisers: the Euler-Lagrange equations. Jacobi integral; conservation laws. Geodesics on a surface.
10. Hamiltonian systems
Hamiltonian vector fields. Legendre transform. Poisson brackets. Canonical transformations. Lie conditions, generating functions. The Hamilton-Jacobi equations. Integrability. Geometry of the phase space: Liouville's theorem and Poincaré's recurrence theorem.

Bibliography

Visualizza la bibliografia con Leganto, strumento che il Sistema Bibliotecario mette a disposizione per recuperare i testi in programma d'esame in modo semplice e innovativo.

Didactic methods

Lectures and exercises (often, combined).

Learning assessment procedures

The exam consists of a written part, with exercises and/or questions on the course content, and an oral part.

Students with disabilities or specific learning disorders (SLD), who intend to request the adaptation of the exam, must follow the instructions given HERE

Evaluation criteria

Knowledge of the content of the course. Ability to solve simple problems. Logical rigour. Clarity of presentation.

Criteria for the composition of the final grade

Please see the Italian version.

Exam language

Italiano