COURSES

Syllabus: The course aims to provide a historical description of some of the most relevant mathematical topics that were developed between the end of the 18th and the 19th centuries. The course is mainly based on the critical reading of original sources. The first part (15 hours) will be devoted to the history of the theory of functions and the development of analysis between the 18th and 19th centuries. Students will read original sources on the evolution of the concept of function, Cauchy’s foundations of calculus, the convergence of Fourier series, Riemann’s real and complex analysis, the arithmetization of analysis, and, if time permits, the introduction of distributions in the development of functional analysis.

The second part of the course (5 hours) will present the mathematical development of the science of waters by reconstructing the main milestones in the history of fluids mechanics in the period between the second half of the 18th and the first half of the 19th centuries. In this part of the course students will be guided through the critical reading of original sources from the history of fluid mechanics, in particular some extracts from the works by L. Euler, J. L. Lagrange and C.L. Navier.

Dates/Schedule: TBA Interested students are invited to contact the lecturers to arrange the schedule.

Venue: TBA

Syllabus: Analisi e approfondimento dei più consolidati metodi qualitativi, quantitativi e mixed methods della ricerca in educazione matematica. Scelta e utilizzo di approcci metodologici coerentemente agli obiettivi di ricerca e agli scopi degli studi. Combinazione e integrazione di diversi approcci metodologici.

Dates/Schedule: TBA Interested students are invited to contact the lecturers to arrange the schedule.

Venue: University of Ferrara.

Syllabus: Lie algebras are algebraic structures with wide-ranging applications in geometry and mathematical physics. The course will focus on semisimple Lie algebras and their real forms, addressing their structure and classification. We will introduce the fundamental concept of root systems and study finite reflection groups associated with them, i.e. the Weyl groups. These combinatorial tools will be used to classify semisimple Lie algebras and their real forms, classical and exceptional cases.

References: The primary reference for this course will be J.E. Humphreys’ book Introduction to Lie Algebras and Representation Theory. Additionally, Anthony W. Knapp’s book Lie Groups Beyond an Introduction will serve as a helpful supplementary resource.

Dates/Schedule: Precise dates will be determined in consultation with interested PhD students, who are encouraged to contact the teacher in advance.

Venue: University of Parma

Syllabus: The course focuses on the theory of Yangians and quantum loop algebras associated to symmetrizable Kac-Moody algebras (and more generally to quivers with potentials). In particular, we will discuss the construction of abelian R-matrices, proving a monodromy theorem for the abelian qKZ equations of the Yangians.

Dates/Schedule: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue: TBA

Syllabus: The aim of the course is to introduce the main techniques used in complex and algebraic geometry to study the geometry of compact complex surfaces. We will review the main results of the theory: the Riemann—Roch theorem with its implications, the effects of a blow up on the topology and the cohomology of a surface and Castelnuovo’s contraction theorem, the concept of minimal surface and the problem of classification (Castelnuovo’s rationality criterion and Enriques—Kodaira classification).

Dates/Schedule: TBA. Interested students are invited to contact the lecturer to arrange the schedule.

Venue: University of Parma

Syllabus: The first part of the course will provide an introduction to complex and almost complex geometry.There will be a concise overview of the theory of holomorphic functions of several complex variables, omitting analytical details, leading to basic examples of complex and almost complex manifolds. We will also discuss the Dolbeault, Bott-Chern and Aeppli cohomologies.

The second part of the course will focus on the Hermitian geometry, starting with the Kaehler manifold and describing their cohomological properties. Then, the existence of special Hermitian metrics on compact complex manifolds without Kaehler structure will be investigated. Explicit examples and computational techniques of cohomological invariants will be carefully described.

Notes for the course will be provided.

1. COMPLEX GEOMETRY.

1.1 Introduction to the theory of holomorphic functions of several complex variables. 1.2 Complex structures. Complex projective spaces. Complex tori. 1.3 Almost complex structures. Newlander-Nirenberg theorem. 1.4 (p,q)-forms on complex manifolds. del-bar operator. Dolbeault complex. Aeppli and Bott-Chern cohomology.

2. HERMITIAN MANIFOLDS AND SPECIAL STRUCTURES.

2.1 Hermitian and Kaehler metrics. Kaehler metrics in local coordinates. Examples of Kaehler manifolds. 2.2 Cohomological properties of compact Kaehler manifolds. The del-del-bar Lemma. 2.3 Chern and Bismut connections. 2.4 Strong Kaehler with torsion, Astheno Kaehler, balanced metrics. 2.5 p-Kaehler structures. Families of p-Kaehler structures. Cohomological properties of p-Kaehler manifolds.

REFERENCES

• [1] D. Huybrechts, Complex Geometry: An Introduction, Springer Universitext, 2014.
• [2] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton Legacy Library, 1987.
• [3] J. Morrow, K. Kodaira, Complex manifolds. Reprint of the 1971 edition AMS Chelsea Publishing, Providence, RI, 2006. x+194
• Notes for the course.

See more information HERE (PDF file)

Dates/Schedule: TBA Interested students are invited to contact the teacher to arrange precise dates.

Venue: University of Parma and online

Syllabus: Rational and birational maps of the complex projective plane. Fundamental points and exceptional curves of a plane Cremona transformation. Quadratic transformations and De Jonquières maps. Properties like equations of conditions and Noether’s inequality. Factorization of transformations and proofs of Noether-Castelnuovo theorem, Cremona equivalence of plane curves. Properties of the varieties parametrizing plane Cremona maps of fixed degree. Lengths in the plane Cremona group.

Dates/Schedule: PhD students who are interested should contact the instructor as soon as possible to decide on the days, time, and possible venue where the course will be held.

Venue: TBA

Syllabus: The course will consider some selected topics in the theory of algebraic curves over finite fields. Useful previous knowledge: elementary theory of algebraic curves.

Dates/Schedule: TBA Interested students are invited to contact the teacher to arrange precise dates.

Venue: University of Modena and Reggio Emilia and online.

Syllabus: The course will consist of a series of lectures on various topics in Graph Theory, aimed on the one hand at consolidating some basic knowledge, and on the other at introducing some advanced topics that might be investigated by the PhD students in the audience.

Dates/Schedule: TBA Interested students are invited to contact the lecturers to arrange precise dates.

Venue: University of Modena and Reggio Emilia.

Syllabus:

• Introduction to Deep Learning, basic steps of the algorithm, and analogies with mathematical models of the human visual system.
• Geometry of data space and parameter space; Kullback-Leibler (KL) divergence and information geometry interpretation.
• Geometric Deep Learning: Deep Learning algorithms on graphs (Graph Neural Networks, GAT, message passing via the heat equation and the Laplace operator on graphs).
• Practical sessions (hands-on on Google Colab).

Dates/Schedule: Intensive course of approximately 16 hours, held both in person (at the FIM Campus, Via Campi in Modena) and online. Period: Between December 2026 and January 2027. Interested students are invited to contact the lecturers to arrange precise dates.

Venue: University of Modena and Reggio Emilia and online.

Syllabus: Foundations: Review of Algebraic Geometry on affine and projective varieties, including coordinate rings, Zariski topology, and normality.

 Affine Toric Varieties: Definition of toric variety. Description of affine toric varieties via lattices (of characters and subgroups) and dual cones.

 Variety associated with Fan: Classical construction of the toric variety from a fan and atudy of key properties (smoothness, completeness, normality, and Cohen-Macaulay property).

Algebraic Methods: New constructions (e.g., quotients of affine spaces, monomial maps), introduction of homogeneous coordinates and toric ideals.

Polytopes: Relationship between lattice polytopes and projective toric varieties. Overview of Dehn-Sommerville equations and Ehrhart polynomials.

Final Assessment: short thematic essay.

Dates/Schedule: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue: Department FIM -University of Modena and Reggio Emilia

Syllabus: We shall introduce free-discontinuity problems such as the Mumford-Shah functional; we shall recall the classical Formulation and introduce the variational approach proposed by De Giorgi based on the theory of SBV functions comparing the two approaches, first of all by showing that a classical competitor is also a weak competitor.

We prove the existence of minima together with some regularity properties in order to show that weak competitors are also classical, so to fully compare the variational approach with the classical approach. We also study a Gamma convergence regularization proposed by Ambrosio and Tortorelli.

Remark: These lectures rely on the theory of BV and SBV functions, as treated in the lectures of Filippo Cagnetti “BV FUNCTIONS” which is a prerequisite for this course. Students that attend the course could be requested to study and present a couple of arguments and a final colloquium is mandatory at the end of the course.

Dates/Schedule: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue: TBA

Syllabus: Elementary results on prime numbers. The Riemann zeta-function and its basic properties: analytic continuation, functional equation, Euler product and connection with prime numbers, the Riemann-von Mangoldt formula, the explicit formula, the Prime Number Theorem. Prime numbers in all and “almost all” short intervals.

Dates/Schedule: Second Semester.

Venue: University of Parma.

Syllabus: The course is intended to provide an introduction to the theory of functions of bounded variation, which are of fundamental importance in the Calculus of Variations. The topics discussed will also set the necessary background for the course “BV and SBV FUNCTIONS WITH APPLICATIONS TO THE MUMFORD-SHAH FUNCTIONAL”, taught by Prof. Michele Miranda.

Main Topics:

• -The space BV: definition and examples.
• – BV functions in one variable.
• – Sets of finite perimeter.
• – Embedding theorems and isoperimetric inequalities.
• – Fine properties of BV functions.
• – SBV functions: definition and examples.

Assessment method: a presentation of a topic related to the course, lasting about 1 hour.

Dates/Schedules: TBA Interested students are invited to contact the lecturer to arrange the schedule

Venue: TBA

Syllabus: The course aims at analyzing the basic elements of the theory of nonlinear elliptic equations having signed measures as data. In particular, the following topics will be studied:

• – Existence of solutions for linear and nonlinear equations with measure data. Comparison between SOLAs, entropy and renormalized solutions.
• – Consequence of density conditions on the measure. Linear and nonlinear potentials.
• – Characterization of dual energy spaces in terms of Wolff potentials.
• – Bounds of solutions in terms of Wolff potentials.
• – Bounds of the gradient of solutions in terms of Riesz potentials.

Dates/Schedule: to be fixed, starting February 2027. Notes: interested participants are required to contact the teacher in advance, so to organize the first meetings.

Venue: Microsoft teams meetings if participants from Modena or Ferrara will participate, building of Mathematics of the University of Parma otherwise (in the case, the precise room will be fixed after the schedule for the second semester classes will be finalized).

Syllabus: The first eigenvalue of the Laplacian on an open set, and more generally of a second order elliptic operator, is an important object both from an applied and theoretical point of view. In Mathematical Physics, it usually plays the role of the ground state energy of a physical system. Despite its importance, for general sets it is not easy to explicitly compute it: thus, we aim at finding estimates in terms of simple geometric quantities of the sets, which are the sharpest possible. The most celebrated instance of this kind of problems is the so-called Faber-Krahn inequality. This course offers an overview of the methods and results on sharp geometric estimates for the first eigenvalue of the Laplacian and more generally of sharp Poincaré-Sobolev embedding constants (sometimes called “generalized principal frequencies”). In particular, we will present: supersolutions methods, symmetrization techniques, convex duality methods, the method of interior parallels, conformal transplantation techniques.

Dates/Schedule: Reading course

Venue:

Syllabus: This doctoral course introduces participants to the regularity theory for elliptic and parabolic PDEs with (strongly) singular or degenerate structure. The equations under consideration satisfy standard p-growth and p-ellipticity conditions, typical of p-Laplacian type operators, sometimes only away from the origin. This gives rise to a rich variety of analytical difficulties and phenomena. Wefirst discuss the elliptic setting, focusing on higher differentiability results for weak solutions, in both the subquadratic (1 < p < 2) and superquadratic (p ≥ 2) regimes. We then explore analogous results in the parabolic framework, addressing the higher spatial and temporal differentiability of the weak solutions under suitable assumptions on the data. Throughout the course, emphasis will be placed on the main ideas, techniques and tools underlying the proofs. The lectures aim to provide participants with both a conceptual understanding and a solid technical foundation for further research in nonlinear PDEs.

• – Review of some classical regularity results for the stationary and evolutionary p-Laplacian.
• – Introduction to Besov spaces.
• – Higher differentiability results for weak solutions to strongly singular/degenerate elliptic PDEs of p-Laplacian type away from the origin.
• – Higher differentiability results for weak solutions to strongly degenerate parabolic PDEs of p-Laplacian type away from the origin.

Dates/Schedule: Tentative Period: February-March 2027. Interested students are invited to contact the lecturer to arrange the schedule.

Venue: University of Modena and Reggio Emilia

Syllabus: This course is devoted to the analysis and geometry of homeomorphisms in Sobolev spaces, with a focus on the subtle interplay between weak differentiability, topology, and orientation. Once we move from smooth diffeomorphisms to Sobolev homeomorphisms (for instance f \in W^{1,p}_ loc), several classical geometric facts become delicate: the Jacobian determinant may only be defined almost everywhere, the change-of-variables principle may require additional structural hypotheses, and even the very meaning of “preserving orientation” must be formulated carefully in weak settings. A central question is whether a Sobolev homeomorphism f : Ω ⊂ R^n → R^n can have a Jacobian determinant J_f that changes sign on a set of positive measure. This problem is deeply connected to the analytic formulation of orientation, to the stability of the topological degree under weak limits, and to the possibility (or impossibility) of approximating Sobolev homeomorphisms by smooth maps. Understanding when the sign of J_f is essentially constant is not merely a technical curiosity: it underpins common assumptions in geometric function theory and appears naturally in nonlinear elasticity, where orientation preservation is often encoded by constraints such as J_f ≥ 0 almost everywhere. The course develops the toolkit needed to address these issues. We will discuss the distinction between the pointwise Jacobian detDf (defined at points of approximate differentiability) and the distributional determinant DetDf, why they may fail to coincide under minimal assumptions, and which additional integrability or structural conditions restore the expected identities. A second major thread concerns the Lusin property (N) and related “size-control” properties for images of null sets. For Sobolev mappings, the implication |E| = 0 =⇒ |f(E)| = 0 is not automatic, yet it is intimately tied to change-of-variables and area-type formulas. We will see how condition (N) interacts with the Sobolev exponent, with the notion of finite distortion, and with borderline Orlicz–Sobolev frameworks that arise when one pushes approximation and regularity results to (or below) classical thresholds. A third pillar is approximation: when and how a Sobolev homeomorphism can be approximated by diffeomorphisms (or smooth homeomorphisms) in the same Sobolev topology. We will highlight both positive approximation theorems and obstructions, including the special role of the planar case, the appearance of bi-Sobolev assumptions (simultaneous control of f and f−1), and the role of Orlicz growth conditions in sharp approximation statements.

See more information HERE (PDF file)

Dates/Schedule: TBA Interested students are invited to contact the instructor to agree on the schedule.

Venue: TBA

Syllabus:The course covers the following topics:

• – review and complements of MeasureTheory;
• – covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems;
• – rectifiable sets and rectifiability criteria;
• – the theory of sets of finite perimeter;
• – applications to geometric variational problems;
• – the isoperimetric problem;
• – the partial regularity theory for quasi-minimiser of the perimeter.

Hand-written notes of the whole course are available in Italian on the Elly platform.

Dates/Schedule: Reading course

Venue:

Syllabus: Pseudodifferential calculus is a very useful tool in the study of partial differential equations in unbounded domains or manifolds, in particular when we deal with existence and uniqueness of a solution to the Cauchy problem for a certain pde with variable coefficients depending on time and space variables. In this course we will introduce first the symbolic calculus, and then the concept of pseudodifferential operator and the rules of pseudodifferential calculus (product formulas, formal adjoint, continuous action on Sobolev spaces,…). We shall conclude with applications to the study of pdes Pu=f in suitable Sobolev spaces.

Dates/Schedule: Reading course.

Venue:

Syllabus: In this course we present the theory of semigroups of bounded operators in Banach spaces, paying particular attention to analytic semigroups. Applications are given to the analysis of partial differential equations of parabolic type.

Dates/Schedule: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue: University of Parma

Syllabus: The course deals with some important methods for the study of boundary value problems to ordinary and partial differential equations. The Leray-Schauder topological degree will be briefly introduced, and its applications discussed in the study of periodic solutions and solutions satisfying Cauchy multi-point conditions in parabolic equations. The upper and lower solutions technique for ordinary differential equations will be then proposed and its application given to the study of traveling wave solutions of reaction-diffusion equations with degenerate diffusivities.

Dates/Schedule: TBA Students who are interested are invited to contact the instructor to agree on the schedule.

Venue: TBA

Syllabus: In this course we will address the question of well-posedness (existence and uniqueness of the solution, with continuous dependence on the data) of the Cauchy problem for the so-called class of p-evolution equations, a wide class of PDEs containing model equations of physical interest like Schrõdinger equation, Korteweg-de-Vries equation, Euler-Bernoulli vibrating beam equation. We are interested in equations with (t,x)-dependent coefficients: these pdes can be studied using the rules of pseudodifferential calculus, some microlocal analysis techniques and some functional analysis. We will focus on the main techniques used to study well-posedness for this class of pdes in various settings, and study the main results in literature on this topic.

Prerequisites: Knowledge of pseudodifferential calculus and the ability to apply its rules.

Dates/Schedule: The course schedule is as follows:

• Thursday, November 12, 19, and 26, 9:00–11:30 AM, Room C3, Palazzo Manfredini, via Muratori 9, Ferrara
• Wednesday, December 2, 9:00–11:30 AM, Room C2, Palazzo Manfredini
• Thursday, December 10, 10:30 AM–1:00 PM, Room C3, Palazzo Manfredini
• Thursday, December 17, 9:00–10:30 AM, Room C3, Palazzo Manfredini
• Monday, December 21 and Tuesday, December 22, 9:00–11:30 AM, Room 1, via Machiavelli 30

Venue: University of Ferrara, details listed next to course dates.

Syllabus: The course will present the duality approach to the study of Markov processes. This will combine, in a joint effort, probabilistic and algebraic tools. In particular we will consider several interacting particle systems that are used in (non-equilibrium) statistical mechanics, we will discuss “integrable probability”, we will show how (stochastic) PDE arise by taking scaling limits.

Dates/Schedules: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue: Department FIM -University of Modena and Reggio Emilia

Syllabus: This course introduces the foundational concepts of modern financial mathematics, focusing on the core principles and mathematical tools used to model the asset dynamics and apply risk-neutral valuation to financial instruments, including derivatives. Students will explore the fundamental theorems of asset pricing and their implications for arbitrage-free pricing in financial markets. The course also covers standard change-of-measure techniques, along with key examples that demonstrate their practical applications. By the end of the course, participants will have a solid understanding of the key mathematical principles that underpin modern financial theory and practice.

Program. Part I. Change of measure and Girsanov Theorem, key examples of its application such as the definition of the market price of risk and the risk neutral portfolio strategy.

Part II. Market models, self-financing portfolio, risk-neutral measures and numeraries. Arbitrages and the first fundamental theorem of asset pricing. Stochastic representation of the assets, European derivatives and the second fundamental theorem of asset pricing. Risk neutral pricing and the Black-Scholes equation as a key example.

Part III. Change of numeraire technique and its application in derivative pricing. Change of numeraire process and derivation of its dynamics.

Part IV. Bonds, forward measure and forward process. Forward and Future contracts, their market standards and related valuation. Forward-future spread, convexity adjustment and its implication in derivative pricing. Key examples with lognormal bond dynamics.

Preliminaries. Multidimensional Wiener process and the stochastic integral. The Lévy theorem for multidimensional martingales and the representation of square-integrable martingales in terms of the Wiener integral. Itô processes and Itô’s lemma for multidimensional processes (finite dimensions). Geometric Brownian motion.

Dates/Schedules: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue: TBA

Syllabus: The course is intended to provide an introduction to classical kinetic Boltzmann approach to rarefied gas dynamics, and some recent advances including the generalization of kinetic models to reactive gas mixtures and to socio-economic problems.

Dates/Schedule: Tentative period: February–March 2027. Please contact the teachers in advance.

Venue: TBA

Syllabus: This course focuses on the mathematical and physical principles governing fluid flow through porous media, an essential topic in many fields such as hydrology, biology, and environmental science. In particular, the course will explore the concept of asymptotic homogenization, a method used to derive effective macroscopic models that describe the behavior of fluid flow through heterogeneous media on the microscopic scale.

Fluid flow in porous media is inherently complex due to the spatial heterogeneities present in the medium. Asymptotic homogenization provides a systematic approach for averaging the microscale properties of the medium to derive a simpler, yet accurate, macroscopic model. This course will lay down the mathematical foundations of fluid flow in porous media, introduce the theory of asymptotic homogenization, prove key results, and show how these results lead to effective models for real-world applications. Some prerequisites of fluid dynamics will be recovered at the beginning of the course.

Dates/Schedule: TBA Interested students are invited to contact the lecturers to arrange the schedule.

Venue: The in-person venue is the University of Parma.

Syllabus: The objective of this course is to introduce the students to the mathematical theory of the Navier-Stokes initial-boundary value problem and to some of the solved and unsolved fundamental questions. As is well-known, this theory presents, to date, a number of unresolved basic questions, mostly, for the physically relevant case of three-dimensional (3D) motions. Among them, certainly, the most famous is that of proving or disproving existence of 3D regular solutions for all times, for data of arbitrary “size”. It should be observed that the analogous question in the two-dimensional (2D) case received a positive answer almost half a century ago. In this course, after deriving the Navier-Stokes equations from the general continuum theory, we shall prove the main known results of existence, uniqueness and regularity of solutions to the corresponding initial-boundary value problem. Moreover, we shall present a number of significant open questions and explain why the current mathematical approaches fail to answer them. In some cases, we shall also point out possible strategies of resolution. More specifically, the course will cover the following topics. 

• Derivation of the Navier-Stokes equations and their physical relevance; 
• Formulation of the initial-boundary value problem in an arbitrary domain of R^n, n = 2,3. 
• Considerations on the peculiar structure of the Navier-Stokes equations. 
• A priori estimates: The “Energy Equality” and fundamental differences between the 2D and 3D cases. 
• Weak Solutions: Definition and basic properties. 
• Weak solutions: Existence and Leray-Hopf class. 
• Weak solutions: Uniqueness criteria. 
• Weak solutions: Regularity criteria. 
• Existence and uniqueness of global regular solutions in 2D. 
• Classical solutions in 3D: Local and global existence results. 
• Leray’s Structure Theorem”: Behavior of 3D solutions at possible times of irregularity and associated fractal dimension property.

See more information HERE (PDF file)

Dates/Schedule: The one-hour lectures will be held from 1 to 28 February, 2027. The precise schedule will be available in due time.

Venue: University of Ferrara

Syllabus: The purpose of this course is to show how the non-linear analysis of structural elements subject to external actions cannot be reduced to simply checking a box on a drop-down menu in a commercial software such as Abaqus. Far from being exhaustive, I intend to present the basic principles that make it possible to acquire, at least, the nomenclature commonly used in non-linear structural mechanics. I hope that, after attending the course, the student will be able to make a more conscious use of the great computing capacities available today which, unfortunately, are too often abused. The topics that will be covered are summarized below.

Preliminaries. Euclidean space, points, vectors, tensors. Scalar product of vectors. Tensor product. Trace operator. Inverse of a tensor. Orthogonal tensors. Rotations. Oriented area and cross product of vectors. Volume and triple product of vectors. Determinant of a tensor. Symmetric and skew tensors. Green Theorem and transformation of surface integrals in volume integrals.

Analysis of the deformation. Actual and deformed configuration. Deformations defined by mappings. Homogeneous deformation. Pure deformations. Rotations. Polar decomposition theorem. Plane deformations. Infinitesimal deformations. 

Stress and balance laws. Forces. Contact forces. Euler separation axiom. Principle of local action. Simple materials. Cauchy’s theory and the definition of stress. Conservation of mass. Piola transformation. Balance of linear momentum. Balance of angular momentum. Cauchy stress tensor. First and second Piola-Kirchhoff stress tensors. 

Work of deformation. Work done by the external forces. Rate of change of kinetic energy. Work expended to deform the body. Work conjugation of the Cauchy, first and second Piola-Kirchhoff stress tensors, with the corresponding measures of deformations. 

Constitutive equations. General forms of the constitutive equations. Principle of determinism. Material frame indifference. Restrictions on the form of the equations. Correspondence with the molecular theory. Constitutive symmetries. Specialization to infinitesimal deformations.

Credit Requirement – Seminar Lecture: Students earn course credit by successfully presenting a lecture on an instructor-assigned research paper. This paper will delve into advanced topics connected to the course subject matter.

See more information HERE (PDF file)

Dates/Schedule: The schedule may be decided by mutual agreement with the attending students.

Venue: Facoltà di Ingegneria, Sede Scientifica, Campus Universitario, Parco Area delle Scienze 181/A, 43123 Parma

Syllabus:

Course Overview: Deep generative models represent one of the most transformative advancements in artificial intelligence in recent years, due to their remarkable ability to generate images, text, and other forms of data that are often indistinguishable from real-world counterparts. Despite their growing popularity and widespread adoption, these models are frequently treated as black boxes, with limited understanding of the principles that govern their behavior. This course aims to provide a comprehensive introduction to modern deep generative models, making the subject accessible even to students who do not have prior experience in the field. Throughout the course, both theoretical foundations and practical implementation aspects will be covered, with the goal of enabling students to critically understand and effectively use these models in their own research.

Assessment: A final assessment is required in order to obtain the PhD credits. The modalities of the exam will be discussed at the beginning of the course, and may include a small project, a practical assignment, or a short repo.

Schedule and Tentative Topics:

Lesson 1: Introduction to generative modeling, probabilistic foundations, and latent variable models. Lab: introduction to the coding environment and basic generative modeling examples.

Lesson 2: Variational Autoencoders (VAEs): theory and training objectives. Lab: implementation of a VAE for image generation.

Lesson 3: Generative Adversarial Networks (GANs): architecture, training dynamics, and challenges. Lab: implementation and experimentation with GANs.

Lesson 4: Diffusion and Flow Matching models: principles, training procedures, and recent advances. Lab: implementation of a basic model and discussion of applications.

Dates/Schedule: The course will be held on the following dates:

November 12, 2026; November 19, 2026; November 26, 2026; December 3, 2026, from 10:00 a.m. to 1:00 p.m.

Venue: Edificio Matematica P2- Sala riunioni MO 18-02-015, FIM Department – University of Modena and Reggio Emilia.

Syllabus: The course will give an introduction to the theory of the Optimal Transport (OT) problem, a branch of Mathematics studying the optimal way of moving resources. Over the last few decades, OT has transcended its origins to become a fundamental tool at the intersection of analysis, geometry, and probability. The course will also give an overview of the numerical methods developed in recent years.

The topics covered will be:

• – Monge and Kantorovich formulations of the OT problem.
• – Wasserstein distance and its properties.
• – PDE-based formulations of OT: the dynamical Benamou-Brenier formulation and the Monge-Kantorovich equations.
• – Numerical method for discrete, semi-discrete and continuous OT problems.

Exam: Oral examination or numerical project.

Prior knowledge Recommended:

• – elements of analysis (L^p spaces, dual spaces, Lebesgue measure, Dirac measures, etc.);
• – notions of classic PDEs e.g. Poisson and transport equations.

Suggested:

• – basic notions of optimizations: primal and dual problems;
• – discretizations method e.g.FEM, FV (for the numerics only).

Dates/Schedule: TBA Interested students are invited to contact the lecturer to arrange the schedule.

Venue:TBA

Syllabus: This PhD course introduces the basics of Neural Networks and Physics-Informed Neural Networks (PINNs) for solving forward and inverse problems governed by evolutionary differential equations. After covering neural-network fundamentals and the construction of PINNs, students explore augmented-data strategies, asymptotic-preserving PINNs (APNNs) tailored for multiscale systems, and PINNs for optimal control problems. Laboratory sessions in Python provide hands-on experience with diverse mathematical models and applications. In-depth studies will also be suggested for students through the reading of specific research articles.

Dates/Schedule: The course will be held from 2:30 PM to 4:30 PM, according to the schedule outlined below

• Lecture I  – 01/02/27: Basics on NNs and PINNs
  Lecture II  – 03/02/27: Python Lab on NNs VS PINNs (application to SIR model)
  Lecture III  – 05/02/27: Augmented data NNs
  Lecture IV  – 08/02/27: Python Lab on augmented data NNs (application to SIR model)
• Lecture V  – 10/02/27: PINNs for optimal control problems
  Lecture VI  – 12/02/27: Python Lab on OCP-PINNs (application to viscous Burgers’ equation)
  Lecture VII  – 15/02/27: APNNs for multiscale systems
  Lecture VIII  – 17/02/27: Python Lab on APNNs VS vanilla PINNs (application to Goldstein-Taylor model).

Venue: Room A2, Palazzo Manfredini at the Department of Mathematics and Computer Science of Ferrara

Syllabus: Advances in medical imaging strongly rely on mathematical modeling and analysis, which provide the theoretical foundation for image formation, reconstruction, and post-processing, through inverse problem frameworks. Mathematical tools, such as optimization and regularization, are essential for designing robust and efficient algorithms that allow to reconstruct images, enhance their quality and extract meaningful information with high accuracy, ultimately improving diagnostic capabilities and clinical outcomes. Recently, artificial intelligence has further expanded this landscape, with convolutional networks and diffusion models emerging as a powerful tools for imaging. However, their adoption requires explainability and integration into mathematical frameworks.

Topics:

• (1) introduction to computational imaging tasks and reformulation into inverse problem;
• (2) tailoring of optimization algorithms for medical applications;
• (3) deep learning for medical imaging and different possible approaches from end-to-end networks to GenAI (CNNs and diffusion models);
• (4) focus on tomographic reconstruction with inspection of real medical images and their analysis;
• (5) focus on semantic segmentation on optical non-invasive retinas scans;
• (6) focus on image denoising and contrast enhancement on magnetic resonance images with CNNs.

Dates/Schedule: TBA Interested students are invited to contact the lecturers to arrange the schedule.

Venue: TBA

Syllabus: The problem of recovering a quantity of interest from indirect measurements, mathematically described as an inverse problem, is ubiquitous in science and engineering. This course explores the theory and numerical methods for solving inverse problems, focusing on ill-posedness, regularization, and emerging strategies in deep learning. The course includes a mathematical introduction to the foundations of inverse problems, a dash of classical regularisation theory, modern regularisation techniques including data-driven models with applications to large scale problems in imaging.

Dates/Schedule: TBA Interested students are invited to contact the lecturers to arrange the schedule.

Venue: TBA

Syllabus:

Module 1 — Mathematical Methods for 6D Pose Estimation with Autoencoders (Applications to Industrial Objects)

Mathematical formulation of the 6D pose problem: rotation and translation spaces, geometric representations, metrics, and ambiguities. Autoencoders and variants: architecture, latent-space geometry, regularization, and representation learning. Techniques for learning robust latent embeddings: geometric constraints, manifold learning, and dimensionality reduction. Pose estimation through latent-space regression and dedicated optimization strategies. Performance analysis in industrial environments: stability, sensitivity to noise, robustness to occlusions and illumination changes.

Module 2 — Crop Classification Using Multiband Satellite Imagery (Classical Machine Learning)

Principles of remote sensing: spectral bands, spatial–spectral structure, and multi-source data characteristics. Mathematical preprocessing: dimensionality reduction, spectral transformations, vegetation indices, and feature extraction. Classical ML models: SVM, Random Forest. Industrial applications: crop monitoring, field mapping, and decision-support systems.

Module 3 — Object Detection and Semantic Segmentation in Thermal Imaging (Advanced Techniques for Harsh Operational Conditions)

Mathematical and statistical properties of thermal images: emissivity, noise characteristics, and non-Gaussian distributions. Detection and segmentation architectures: fully convolutional models, multiscale networks, transformer-based approaches. Domain-specific challenges: low contrast, limited texture, temporal variability, and sensor-dependent distortions. Enhancement strategies: physically plausible data augmentation, thermal normalization, and domain adaptation. Evaluation in real-world settings: imbalanced classes, sensitivity to thermal fluctuations, and robustness metrics for industrial applications.

Dates/Schedule: TBA Interested students are invited to contact the teacher to arrange precise dates.

Venue: FIM Department – University of Modena and Reggio Emilia.

Syllabus: The course is principally focused on Boundary Element Methods (BEMs).

Lectures involve:

• -Boundary integral formulation of elliptic, parabolic and hyperbolic problems
• – Integral operators with weakly singular, strongly singular and hyper-singular kernels
• – Approximation techniques: collocation and Galerkin BEMs
• – Quadrature formulas for weakly singular integrals, Cauchy principal value integrals and Hadamard finite part integrals
• – Convergence results
• – Numerical schemes for the generation of the linear system coming from Galerkin BEM discretization.

Dates/Schedule: PhD students who are interested should contact the instructor as soon as possible to decide on the days and time of the course.

Venue: TBA

Syllabus:

• – Introduction to differential model problems for option pricing in the Black-Scholes framework.
• – Analysis of peculiar troubles and advantages in application of standard numerical methods for partial differential problems: Finite Difference, Finite Element, Boundary Element, Binomial, Monte Carlo.

Dates/Schedule: PhD students who are interested should contact the instructor as soon as possible to decide on the days and time of the course.

Venue: University of Parma

Syllabus: This PhD course introduces the foundations of optimal control theory with a progressive focus on feedback design, numerical methods, and mean field models. Starting from the formulation of finite-dimensional optimal control problems, the course presents instantaneous control and greedy feedback strategies as computationally tractable approaches, motivating the transition to model predictive control. Classical theoretical tools such as Pontryagin’s Maximum Principle and the Hamilton–Jacobi–Bellman equation are then introduced to characterize optimal solutions and feedback laws. The second part of the course is devoted to mean field optimal control, with particular emphasis on Boltzmann-type descriptions arising as limits of large interacting particle systems. Applications from applied mathematics and physics, including plasma models, opinion dynamics, and fluid-related systems, illustrate the theoretical and numerical methods discussed. Numerical experiments and laboratory-style demonstrations support the theoretical content throughout the course.

Dates/Schedule: All lectures will be held from 2:30 to 4:30 p.m. on the following dates:

• Lecture I  – 02/03/27 
• Lecture II  – 03/03/27
• Lecture III  – 04/03/27 
• Lecture IV  – 09/03/27 
• Lecture V  – 10/03/27 
• Lecture VI  – 11/03/27 
• Lecture VII  – 16/03/27 
• Lecture VIII  – 17/03/27

Venue: Room A2, Palazzo Manfredini, University of Ferrara

Syllabus: The course introduces students to the topics of Decision Theory that are relevant for Artificial Intelligence. In particular, the course discusses decision-theoretic planning and learning through the following agenda: brief review of random variables and stochastic processes (if needed), discrete-time Markov chains, Markov decision processes, base algorithms for automated planning using Markov decision processes (e.g., value iteration and policy iteration), base algorithms for machine learning using Markov decision processes (e.g., Q-learning and SARSA), brief overview of additional topics (e.g., partially-observable Markov decision processes, game-theoretic planning).

Dates/Schedule: The course will take place between mid-February and the end of March 2027. Specific dates will be announced in due course. Interested students are invited to contact the lecturer to arrange the schedule.

Venue: University of Parma.