ELEMENTS OF ADVANCED GEOMETRY
Academic Year 2025/2026 - Teacher: Giuseppe ZAPPALA'Expected Learning Outcomes
1. Knowledge and understanding: The student will be able to understand and assimilate the basic concepts of Differential Geometry and Algebraic Geometry.
2. Ability to apply knowledge and understanding: The student will be able to acquire an appropriate level of autonomy in the basic theoretical knowledge of these disciplines.
3. Independent judgment: the ability to reflect and connect seemingly distant concepts.
4. Communication skills: the ability to communicate acquired concepts using appropriate scientific language.
5. Learning skills: the ability to deepen and develop acquired knowledge.
Course Structure
Information for students with disabilities and/or SLD To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and/or dispensatory measures, based on the didactic objectives and specific needs.
Required Prerequisites
Attendance of Lessons
Detailed Course Content
1) Categories. Morphisms. Monomorphisms and epimorphisms. Functors. Natural transformations. Examples. Equivalence of categories. Diagrams. Limits and colimits. Products and coproducts. Final and initial object. Complete and cocomplete categories.
2) Presheaves. Sheaves. Sheaf associated to a presheaf. Stalk of a presheaf at a point. Sheaf of continuous functions on a topological space. Morphisms of presheaves. Etalè space. Direct and inverse image of sheaves.
3) Ringed spaces. Structure sheaf. Locally ringed spaces. Sheaf of algebras of differentiable functions on a topological space. Morphisms of locally ringed spaces.
4) Topological varieties. Differentiable varieties. Diffeomorphisms. Category of differentiable varieties. Examples. Linearization of differentiable varieties. Derivations. Tangent space to a differentiable variety. Vector fields. Tangent bundle.
5) Spectrum of a commutative ring with unity. Zariski topology. Structure sheaf on a spectrum. Affine schemes. Category of commutative rings with unity. Schemes. Examples. Properties of schemes.
Teaching's Contribution to the Goals of the 2030 Agenda for Sustainable Development
Goal 4: QUALITY EDUCATION
Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all
Goal 5: GENDER EQUALITY
Achieve gender equality and the empowerment (increased strength, self-esteem, and awareness) of all women and girls
Goal 8: DECENT WORK AND ECONOMIC GROWTH
Promote sustained, inclusive, and sustainable economic growth, full and productive employment, and decent work for all
Textbook Information
[W] Wedhorn, Torsten. Manifolds, sheaves, and cohomology. Springer Spektrum, 2016.
[H] Hartshorne, Robin. Algebraic geometry. Vol. 52. Springer Science & Business Media, 2013.
[B] Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry, Revised. Vol. 120. Gulf Professional Publishing, 2003.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Category theory | [W] |
| 2 | Sheaves | [H] |
| 3 | Ringed spaces | [H] |
| 4 | Differentiable manifolds | [B], [W] |
| 5 | Schemes | [H] |
Learning Assessment
Learning Assessment Procedures
Grading Criteria:
- Failed: The student has not mastered the basic concepts.
- 18-23: The student demonstrates minimal mastery of the basic concepts, his/her presentation and connection skills are modest.
- 24-27: The student demonstrates good mastery of the course content, his/her presentation and connection skills are good.
- 28-30 with honors: The student has mastered all the course content and is able to present it thoroughly and connect it critically.
Examples of frequently asked questions and / or exercises
What is a functor?
How do you construct the sheaf associated to a presheaf?
How do you define the tangent space?
Show that the set of prime ideals of a commutative ring with unity forms a topological space (spectral topology).
Give examples of schemes.