This course provides key skills for a BSc in Data Science and Artificial Intelligence (and, possibly, for other BSc) by teaching advanced modeling techniques and mathematical tools essential for analyzing complex systems. Students will learn to model natural phenomena, understand dynamical systems, and apply computational methods, bridging the gap between theoretical knowledge and practical applications. The acquired skills are crucial for careers in data analysis, predictive modeling, and AI. From the didactic operative viewpoint, starting from applicative examples, increasingly complex mathematical tools will be introduced Knowledge and Understanding: The student will gain a basic understanding of the building and alayzing mathematical models in natural sciences. Application of Knowledge: The student will be able to develop, test and simulate from scratch simple mathematical models, by using various analytical and computational tools, Communication Skills: The student will acquire the ability to present the results of model applications and provide an explanation of the motivations behind the chosen approach. Learning Skills: The student will navigate the literature related to the course themes and will be capable of comparing and improving the chosen modeling strategy. In particular, students will be able to extend the knowledge acquired to topics not directly covered in the lectures.

Basic Newtonian Physics, Calculus (knowing how to fidn the seolution of a linear differential euation) and Linear Algebra. Knowledge of Python.

**Art and Science of Complex Systems, with Applications in Biomedicine and Physics** **Premise:** This course is not pure mathematics but modelling — we start from real-world phenomena (biology, physics) to show how mathematical tools come to life and prove useful in practice. * **Population dynamics** (single species, multiple species, metapopulations, discrete models with noise) * **Physical systems and oscillators** (damping, Ginzburg–Landau, linear and parametric resonances) * **Dynamic epidemiology** including human behaviour * **Chaos and spatiotemporal structures** (Lorenz, Chua, Turing patterns, travelling waves) * **Cellular automata, Ising model, and biological neural networks** **Tools covered:** ODEs/PDEs, stability analysis, linearisation, multistability, bifurcations, spatiotemporal models, Fourier series and transforms.

Main Textbooks: N Britton “Essential Mathematical Biology” Springer (2003) C Gros Complex and Adaptive Dynamical Systems Springer (2024) Va Edizione Hiromi Seno - “A Primer on Population Dynamics Modeling” Springer (2022) Steven Strogatz, “Nonlinear dynamics and chaos, with Applications to Physics, Biology, Chemistry, and Engineerin” CRC Prss (2015); Daniel Kaplan and Leon Glass "Understanding nonlinear dynamics", Springer (1995); Further readings and sources of exercises: JD Murray “Mathematical Biology” Springer (2002) M Iannelli, Andrea Pugliese “An Introduction to Mathematical Population Dynamics” Springer (2014) David G Costa, Paul J Schulte - An Invitation to Mathematical Biology-Springer (2023) HDI Abarbanel et al “Introduction to nonlinear dynamics for physicists” World Scientific (1993) Rössler, Otto E., and Christophe Letellier. Chaos: The World of Nonperiodic Oscillations. Springer ( 2020).

**Premise:** This is not a mathematics course, but a course on modelling. Each topic is introduced through concrete phenomena, mainly in biology and physics. Beyond modelling itself, the other goal is to show how certain mathematical objects, once “placed” in reality, “come alive” and become useful. * **Single-species population dynamics**, with ecological and oncological applications: growth and control of populations and tumours * **One-dimensional damped physical systems:** the overdamped approximation and the Ginzburg–Landau model * **Multi-species and metapopulation dynamics.** Ecological and oncological applications. We will examine predator models (Lotka–Volterra and more realistic extensions), competition, and cooperation. * **Discrete-time population dynamics with stochastic perturbations** * **Dynamics and control of infectious diseases:** from classical models to those that include human behaviour and information flow (vaccine hesitancy, social distancing) * **Linear and nonlinear oscillators in physics** * **Lagrange and Hamilton equations** * **Linear and parametric resonance** * **Chaotic dynamical systems:** Lorenz, Chua, chaotic epidemiology * **Movement in biology:** Turing patterns * **Movement in biology:** travelling waves * **Deterministic and stochastic cellular automata,** with various applications to biological and ecological problems * **Cellular automata and the Ising model** * **Modelling biological neural networks** To model the phenomena listed above, we will gradually introduce the following mathematical tools alongside the models: * One-dimensional continuous and discrete dynamical systems * The concepts of equilibrium and global/local stability * A classic technique: linearisation of nonlinear dynamical systems * Multistability: the Ginzburg–Landau model * *n*-dimensional dynamical systems * Bifurcations * Simple spatiotemporal models * Chaotic dynamical systems: non-autonomous and autonomous * Fourier series and transform: what they are used for in practice and why they are even beloved

Lectures will contain both Frontal and “hands on” sessions.

Bring your Laptop

Written and oral exams. The written exam is composed by one specific problem related to the course content, requiring the application reasoning, modeling and math . The oral exam is composed by an oral presentation and by a session of questions. Namely: 1. Presentation: Students present a final project for the exam, typically by analyzing a scientific paper or section of a book The student must briefly (12 min) present the work and replicate its results.. 2. Viva Voce: questions to ascertain depth of knowledge and critical thinking

The course introduces students to modern techniques of analysis and modeling in the natural sciences, as well as the related mathematical and simulation techniques. Mathematical and computational modeling of natural phenomena is one of the pillars of sustainable development, and all the techniques learned in this course can be applied in this context.