Information on courses

Teacher: Adriano Tomassini

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

Schedule of the course: (32 hours, 8 CFU)
The lessons will take place at the Mathematics building, Dipartimento di Scienze Matematiche, Fisiche e Informatiche - Università di Parma, starting from April 8, 2025, with the following weekly schedule.:
Tuesday- 14.30 - Sala Riunioni
Wednesday - 14.30, room C (unless Wednesday April, 9 2025, room F)
The course will end on June 4, 2025.  

Note: At the request of participants, it is possible to attend the lessons online.

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

Teacher : Dr. Enrico Ferri

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.

Schedule of the course (8 hours, 2 CFU)): April 7, 2025, classroom M2.5 Modena (14:00-18:00) - April 9, 2025, classroom L1.7 Modena (14:00:18:00) 

Assessment method:  A final exam is scheduled to take place in the FIM Department meeting room on Friday, April 11th, at 11:00 AM.
At the request of participants, it is possible to attend the lessons online.
For more information, please contact Professor Sergio Polidoro (sergio.polidoro@unimore.it)

Teacher:  Carlo Mercuri

Syllabus: This course is an introduction to the language, concepts, and methods for nonlinear PDEs in unbounded domains. We will review classical approaches mainly based on the direct method of the Calculus of Variations, to study some classes of nonlinear elliptic PDEs on R^N. We will mainly consider two `toy problems': the nonlinear Schrödinger equation, and the nonlinear Schrödinger-Poisson-Slater equation. Although these PDEs share some features, they require a separate analysis. Emphasis will be given to a suitable variational formulation of these equations, their functional setting, and relevant compactness properties. Known results and suggestions for new research projects in this area will be also discussed.
The first part of the course will be devoted to some basic concepts of the variational approach to nonlinear analysis, based on extending Calculus to infinite dimensional Banach spaces. Some elementary tools in critical point theory will be introduced, such as the Lagrange multipliers rule and the Mountain Pass Theorem. These can help identify nontrivial solutions to PDEs characterized by 'energy levels’.

Main references for the first part of the course:

A. Ambrosetti, A. Malchiodi “Nonlinear Analysis and Semilinear Elliptic Problems’’ (2007)
D. Costa “An invitation to Variational Methods in Differential Equations” (2007)
O. Kavian “Introduction à la théorie des points critiques et applications aux problèmes elliptiques” (1994)

Syllabus: The course will present classical techniques based on the maximum principle to infer symmetry properties for solutions to elliptic partial differential equations. We will mainly focus on the case of radial symmetry and on the characterizations of balls via overdetermined boundary value problems as well as via the prescription of constant mean curvature for their boundaries. In particular two approaches will be discussed: the moving planes technique and an integral method based on sharp inequalities.

The course will be taught from room B3 in Palazzo Manfredini at the University of Ferrara (except the last two lectures, that will take place in Aula 8), and all the lectures will be streamed online from the Google Meet link of the course (Classroom code: q3s2ehg).

• 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

• There will be NO lectures in the week 1-4 April 
• The lecture of Tuesday 25^th March will be in "aula F"
• Interested students should contact the teacher

Teachers: Marzia Bisi and Maria Groppi (University of Parma)
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.
Main topics:
- distribution function and Boltzmann equation for a single gas: collision operator, collision invariants, Maxwellian equilibrium distributions;
- kinetic theory for gas mixtures: extended Boltzmann equations and BGK models;
- kinetic models for reacting and/or polyatomic particles;
- hydrodynamic limits, Euler and Navier-Stokes equations;
- Boltzmann and Fokker-Planck equations for socio-economic phenomena, as wealth distribution or opinion formation.
Period and venue:
Lectures will be delivered online on Teams platform in February-March 2025 in the following dates:
- Tuesday, 25 February at 2.30 - 4.30 pm;
- Wednesday, 26 February at 11.00 am - 1.00 pm;
- Tuesday, 4 March at 3.30 - 5.30 pm;
- Thursday, 6 March at 10.30 am - 0.30 pm;
- Wednesday, 12 March at 11.00 am - 1.00 pm;
- Thursday, 13 March at 11.00 am - 1.00 pm;
- other lectures will be scheduled in the week 17-21 March upon agreements with participants.
Verification of the acquired skills: students will give a talk on a topic related to the arguments of the course.

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.
Knowledge of basic notions of Numerical Analysis and, in particular, of numerical
approximation of partial differential equations is required.
References will be provided directly during the course.
Exercises, given at the end of every lecture, will be discussed by the students in the following
lecture, in order to assess the comprehension of the subject.
Dates: Lectures will be delivered online on Teams platform in the following Tuesdays, from
9 am to 12 am:
11/03/2025; 18/03/2025; 25/03/2025; 01/04/2025; 08/04/2025; 15/04/2025; 29/04/2025; 06/05/2025

Docente: Paolo Baroni
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.
Calendar: March and April 2025, Tuesday, 14:30-16:30, 4 (March 4, 11, 18, 25)+4 (April 1, 8, 15, 22) two-hour lessons
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)
Verification of the acquired skills: an oral presentation of an in-depth analysis on a topic developed throughout the course, lasting approximately one hour.
Notes: interested participants are required to contact the teacher in advance, so to organize the first meetings.

Content: 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.

Period: 01/11/2024 - 31/03/2025
Calendar: 12 hours scheduled upon agreement with participants
CFU: 3
Assessment Method: production of a short paper/discussion on a significant follow-up topic.

(20 hours, 5 CFU)  Michela Eleuteri, Maria Giulia Lugaresi

Syllabus: The course aims to describe some methods of research in the history of mathematics, showing how the study of different themes and historical sources requires different approaches and tools of inquiry.
In the first part of the course (10 hours) we will present elementary methods of historical research that can be applied in the critical examination of printed mathematical texts of the past. We will provide examples of critical reading taken from some important Italian mathematical works of the 18th and 19th century devoted to the foundations of infinitesimal calculus.
In the second part of the course (10 hours) we will introduce some unpublished original sources in the history of mathematics in order to explain how to approach the critical reading, transcription and analysis of them. In particular, we will focus on some pure and applied mathematical works by Paolo Ruffini.

Interested students are invited to contact the teachers.

Date:
Friday, 28/02/2025, ore 10-13; 14-16. Modena. The early period of the Calculus of Variations.
Tuesday, 11/03/2025, ore 10-13; 14-16. Ferrara. Mathematics applied to the study of the fluid-dynamics during the 18th century.
Tuesday, 18/03/2025, ore 10-13; 14-16. Modena. The Calculus of Variations after Lagrange.
Tuesday, 25/03/2025, ore 10-13; 14-16. Ferrara. The manuscripts by Teodoro Bonati at the Ariostea Library of Ferrara.

Verification of the acquired skills: presentation of a written paper regarding one of the themes
developed during the course.