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Title and Credits: Galois Theory: from groups and forms to descent theory and extensions

Teacher: Prof. Daniel Bulacu, University of Bucarest

Syllabus:  The classical Galois theory dates back to 1830, but it took more than 100 years for it to be reformulated (by Artin) in the language of module theory. Artin's criterion that decides when a field extension K/k is Galois allows to extend the classical Galois theory to Hopf algebras. This was initiated by Chase and Sweedler in 1969 in the commutative case, the general case being considered by Kreimer and Takeuchi in 1981. Today Hopf-Galois extensions appear in various branches of mathematics and physics, being also known as dual algebraic versions of non-commutative fiber spaces (the notion of quantum fiber space can be introduced as a module associated with a Hopf-Galois extension).

The purpose of this course is to make the transition from the classical Galois theory to the Hopf-Galois theory and to present some directions of study for the latter. Briefly, the content of the course is as follows:

• Classical Galois theory.
• Extensions of fields, extensions of separable Galois fields without groups, strongly graded rings, cross products, affine group schemes.
• Relevant examples of Hopf algebras.
• Hopf-Galois extensions and examples.
• Descent theory.
• Hopf-Galois theories in various categories.

Dates 2022/2023: 4, 6, 11, and 13 July 2023, 16:00-18:00 (4 lectures, 8 hours).

Main lecturer
Special speaker
Professor Patrizio Bianchi  Chair Holder of the UNESCO Chair on Education, Growth and Equality
Professor Valentina Mini   Adjunct Professor of Statistics, Advanced Statistics and Statistics for Economics and Business
Location
Department of Mathematics and Computer Science
University of Ferrara, Palazzo Manfredini
Via Muratori 8 / 9 - 44121 Ferrara

Lecture 1. Global Vision and Globalization
Global economic, political, and cultural integration: benefits and drawbacks. Interdisciplinary approaches to contemporary challenges.
27 September 2024 | 09:00 – 12:00 | Classroom B1 | Professor Patrizio Bianchi
Lecture 2. Geopolitics: Focus on Africa and Europe
Geopolitical dynamics of Africa and Europe: historical contexts, contemporary challenges, and global ramifications.
30 September 2024 | 15:00 – 18:00 | Classroom C1 | Manfo Zangmo
Lecture 3. Introduction to Transdisciplinarity Theory
Transdisciplinarity: theoretical foundations, basic concepts, and principles.
02 October 2024 | 15:00 – 18:00 | Classroom C1 | Professor Valentina Mini
Lecture 4. Quantitative Methods for Transdisciplinary Approaches
Theoretical competencies in mathematics and statistics for interpreting scientific literature, bridging interdisciplinary gaps, and enhancing organizational capabilities.
04 October 2024 | 15:00 – 18:00 | Classroom C1 | Professor Valentina Mini
Lecture 5. Qualitative Methods for Transdisciplinary Approaches
Qualitative research methods for data collection, analysis, and interpretation in interdisciplinary research.
07 October 2024 | 15:00 – 18:00 | Classroom C1 | Professor Valentina Mini
Lecture 6. Long-Term Projection and Temporal Vision
Temporal perspective and scenario planning for understanding long-term implications in transdisciplinary studies.
09 October 2024 | 15:00 – 18:00 | Classroom C1 | Professor Patrizio Bianchi and Professor Valentina Mini
Lecture 7. Round Table Lab
Collaborative discussions with entrepreneurs to apply theoretical knowledge to realworld interdisciplinary issues.
11 October 2024 | 15:00 – 18:00 | Classroom C1 | Professor Patrizio Bianchi, Professor Valentina Mini and Guest Entrepreneurs