Ferrara 2023/24

 

 

Title and Credits: Plane Cremona transformations, 4 CFU

Teacher: Alberto Calabri

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 2023/2024: 24h lectures +  home assignments, February-March 2024.

 


 

Title and Credits: Geometry of principal frequencies

Teacher: Lorenzo Brasco

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 2023/24: 24h lectures +  home assignments; precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.

 


 

Title and Credits: (Modal) Symbolic Learning, 2CFU+2CFU (optional, for some research work)

Teacher: Guido Sciavicco

Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.

Dates : January 2024, 4 lectures, 8 hours

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Title and Credits: An introduction to uncertainty quantification for PDEs 4CFU

Teacher:  Giulia Bertaglia, Elisa Iacomini

Syllabus: This co urse provides an overview of numerical methods of uncertainty quantification for evolutionary partial differential equations (PDEs). Both intrusive and non-intrusive methods will be presented and discussed, including Monte Carlo, Stochastic Galerkin, Stochastic Collocation, and Multi-fidelity approaches. Particular attention will be devoted to applications related to flow propagation and transport phenomena described by hyperbolic PDEs. In-depth studies will also be suggested for students through the reading of specific research articles.
Dates: June 2024, 10h lectures + reading course + home assignments
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Title and Credits: Symmetry of solutions to elliptic PDEs (4 CFU, 20h)
Teacher: Giulio Tralli
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.
Dates: April-May 2'024, 20h lectures

 


 

Title and Credits: Embodiment, Discourse and Technology in Mathematics Education Research (4CFU + 2CFU optional, for some research work)

Coordinator: Federica Ferretti

Syllabus: The aim of the course will be to acquire specific knowledge - and their use in research practices - of some of the most internationally popular theoretical frameworks in mathematics education. There will be plenary sessions, led by Ferdinando Arzarello (University of Turin), Paul Drijvers (Utrecht University), Anna Sfard (University of Haifa) and Osama Swidan (Ben-Gurion University of Negev), a round table discussion led by Ornella Robutti (University of Turin) and workshops - held by Federica Ferretti and other Italian researchers - in which doctoral students will be able to engage with the theoretical lenses discussed.

Dates: 18-19-20 April, 2024 :

 

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Title: internet Seminar on Harmonic Analysis (ISEM27), edition 27

Dates: october 2023- february 2024, local coordinator Prof. Michele Miranda

At the end of the course  (project phase (3CFU)), in-depth topics are proposed. Small groups of students from different universities are formed with the aim to study these topics and prepare a communication for the Final Workshop to be held in Luminy (Marseille, France).

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