Insegnamento a.a. 2025-2026

Classes: 1 (I sem.)

Instructors:
Class 1: JACOPO GIUSEPPE DE TULLIO

To ensure a productive and engaging learning experience, students are expected to have a basic understanding of fundamental calculus in one variable (including limits, derivatives, and integrals), as well as introductory concepts in linear algebra, such as vector spaces, linear transformations, and matrix operations.

This course provides a solid review of fundamental mathematical concepts that underpin modern applications in economics, social and data science. It supports the development of a shared knowledge base, enabling students to engage more effectively with advanced analytical methods. By bridging gaps in prior learning, the course contributes to the education program by fostering quantitative reasoning and enhancing readiness for graduate-level coursework.

Linear algebra:

• Euclidean spaces: geometric and algebraic approaches. Vectors in R^n. Operations with vectors. Matrices. Linear Space: linear dependance and independance. Dimension and bases of the linear space. Examples. Straight lines and planes in R^3. Linear systems: structure of solutions. Linear functions between euclidean spaces. Representation theorem. Eigenvalues and eigenvectors of a linear transformation. Spectral theorem for symmetric matrices.

Quadratic forms:

• Definitions and applications. Examples.

Curves in the plane and in the space:

• Straight lines in space. Parametric representation of a trajectory. Speed and tangent vector.

Functions in several variables:

• Level lines and contour map. Partial derivatives, gradient. Differential. Higher order derivatives. Derivative of a composite function. Hessian matrix. Implicit functions. Implicit function theorem. Jacobian matrix.

Optimization problems:

• Unconstrained optimization. The first order sufficient conditions. Fermat's theorem. Taylor polynomial of order two. Concavity and convexity. Second order sufficient conditions. Local-global theorem. Constrained oprimization. Lagrange multipliers technique. Meaning of multipliers.

At the end of the course student will be able to...

At the end of the course student will be able to...

— Carry out a formal mathematical proof.

— Recognize the abstract mathematical structures that underlie modern economic theories.

—  Master operations on functions and vectors.

At the end of the course student will be able to...

At the end of the course student will be able to...

— Apply to economics and to the social sciences the basics of mathematics.

— Work out both the quantitative and the qualitative perspectives.

• Lectures
• Practical Exercises

The course is delivered entirely online and is structured to support flexible, self-paced learning while ensuring a comprehensive understanding of the material. It combines several integrated methods:

 

• Lectures: Core concepts are introduced through a series of pre-recorded video lectures. These are organized thematically and designed to be concise, clear, and accessible, allowing students to revisit topics as needed.

• Practical Exercises: To reinforce theoretical knowledge, the course includes video tutorials that walk through selected exercises in detail, highlighting problem-solving techniques and common pitfalls.

• Self-Evaluation Tools: Each module offers a set of self-assessment exercises and practice tests that allow students to check their understanding and monitor their progress independently. These include both automated quizzes with instant feedback and downloadable problem sets for deeper practice.

 

This blended and modular approach enables learners to actively engage with the material, test their skills in real time, and build confidence before entering graduate-level courses in economics and data science.

	Continuous assessment	Partial exams	General exam
Active class participation (virtual, attendance)	x

This preparatory course does not include a final exam.

There is no formal assessment or grading process. Instead, student learning is supported through a system of continuous self-evaluation. Learners are encouraged to actively engage with the course content—video lectures, tutorials, and exercises—and to monitor their progress independently through self-assessment quizzes and practice tests. This approach promotes individual responsibility and allows students to identify areas for improvement at their own pace, in preparation for future graduate-level coursework.

All materials required for the course are provided online and are fully integrated into the course platform. These include:

 

• Video Lessons covering core theoretical concepts

• Video Tutorials focused on solving practical exercises

• Self-Assessment Quizzes available at the end of each module to support independent learning and progress tracking

• Open Practice Exercises designed to reinforce understanding and application of key topics

 

All resources will be made accessible at the beginning of the course and organized by topic within the digital learning environment.

Classes: 2 (I sem.) - 3 (I sem.)

Instructors:
Class 2: JACOPO GIUSEPPE DE TULLIO, Class 3: JACOPO GIUSEPPE DE TULLIO

To ensure a productive and engaging learning experience, students are expected to have a basic understanding of fundamental calculus in one variable (including limits, derivatives, and integrals), as well as introductory concepts in linear algebra, such as vector spaces, linear transformations, and matrix operations.

This course provides a solid review of fundamental mathematical concepts that underpin modern applications in economics, social and data science. It supports the development of a shared knowledge base, enabling students to engage more effectively with advanced analytical methods. By bridging gaps in prior learning, the course contributes to the education program by fostering quantitative reasoning and enhancing readiness for graduate-level coursework.

Linear algebra:

• Euclidean spaces: geometric and algebraic approaches. Vectors in R^n. Operations with vectors. Matrices. Linear Space: linear dependance and independance. Dimension and bases of the linear space. Examples. Straight lines and planes in R^3. Linear systems: structure of solutions. Linear functions between euclidean spaces. Representation theorem. Eigenvalues and eigenvectors of a linear transformation. Spectral theorem for symmetric matrices.

Quadratic forms:

• Definitions and applications. Examples.

Functions in several variables:

• Level lines and contour map. Partial derivatives, gradient. Differential. Higher order derivatives. Derivative of a composite function. Hessian matrix. Jacobian matrix.

Optimization problems:

• Unconstrained optimization. The first order sufficient conditions. Fermat's theorem. Taylor polynomial of order two. Concavity and convexity. Second order sufficient conditions. Local-global theorem. Constrained oprimization. Lagrange multipliers technique. Meaning of multipliers.

At the end of the course student will be able to...

At the end of the course student will be able to...

— Carry out a formal mathematical proof.

— Recognize the abstract mathematical structures that underlie modern economic theories.

—  Master operations on functions and vectors.

At the end of the course student will be able to...

At the end of the course student will be able to...

— Apply to economics and to the social sciences the basics of mathematics.

— Work out both the quantitative and the qualitative perspectives.

• Lectures
• Practical Exercises

The course is delivered entirely online and is structured to support flexible, self-paced learning while ensuring a comprehensive understanding of the material. It combines several integrated methods:

 

• Lectures: Core concepts are introduced through a series of pre-recorded video lectures. These are organized thematically and designed to be concise, clear, and accessible, allowing students to revisit topics as needed.

• Practical Exercises: To reinforce theoretical knowledge, the course includes video tutorials that walk through selected exercises in detail, highlighting problem-solving techniques and common pitfalls.

• Self-Evaluation Tools: Each module offers a set of self-assessment exercises and practice tests that allow students to check their understanding and monitor their progress independently. These include both automated quizzes with instant feedback and downloadable problem sets for deeper practice.

 

This blended and modular approach enables learners to actively engage with the material, test their skills in real time, and build confidence before entering graduate-level courses in economics and data science.

	Continuous assessment	Partial exams	General exam
Active class participation (virtual, attendance)	x

This preparatory course does not include a final exam.

There is no formal assessment or grading process. Instead, student learning is supported through a system of continuous self-evaluation. Learners are encouraged to actively engage with the course content—video lectures, tutorials, and exercises—and to monitor their progress independently through self-assessment quizzes and practice tests. This approach promotes individual responsibility and allows students to identify areas for improvement at their own pace, in preparation for future graduate-level coursework.

All materials required for the course are provided online and are fully integrated into the course platform. These include:

 

• Video Lessons covering core theoretical concepts

• Video Tutorials focused on solving practical exercises

• Self-Assessment Quizzes available at the end of each module to support independent learning and progress tracking

• Open Practice Exercises designed to reinforce understanding and application of key topics

 

All resources will be made accessible at the beginning of the course and organized by topic within the digital learning environment.